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Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type A

2010-03-18
Jun Hu , Andrew Mathas
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Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

2007-09-25
Richard Dipper , Stephen Doty , Jun Hu
In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the … In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer algebra $B_{n}(-2m)$ to the endomorphism algebra of the tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_{n}(-2m)$, is derived as an easy consequence of S. Oehms's results [S. Oehms, J. Algebra (1) 244 (2001), 19–44].
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Verma modules over generalized Virasoro algebras Vir[G]

2002-11-04
Jun Hu , Xiandong Wang , Kaiming Zhao

Schur-Weyl duality for orthogonal groups

2008-11-05
Stephen Doty , Jun Hu
We prove Schur–Weyl duality between the Brauer algebra 𝔅n(m) and the orthogonal group Om(K) over an arbitrary infinite field K of odd characteristic. If m is even, then we show … We prove Schur–Weyl duality between the Brauer algebra 𝔅n(m) and the orthogonal group Om(K) over an arbitrary infinite field K of odd characteristic. If m is even, then we show that each connected component of the orthogonal monoid is a normal variety; this implies that the orthogonal Schur algebra associated to the identity component is a generalized Schur algebra. As an application of the main result, an explicit and characteristic-free description of the annihilator of n-tensor space V⊗ n in the Brauer algebra 𝔅n(m) is also given.
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Quiver Schur algebras for linear quivers

2015-04-13
Jun Hu , Andrew Mathas
We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras R n Λ of type A when e = 0 (the linear quiver) or e > n . … We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras R n Λ of type A when e = 0 (the linear quiver) or e > n . We prove that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When e = 0 , we show that the Khovanov–Lauda–Rouquier grading on the quiver Hecke algebras is compatible with the Koszul grading on the blocks of parabolic category O Λ given by Backelin, building on the work of Beilinson, Ginzburg and Soergel. As a consequence, e = 0 our cyclotomic quiver Schur algebras are Koszul over fields of characteristic zero. Finally, we give an Lascoux–Leclerc–Thibon-like algorithm for computing the graded decomposition numbers of the cyclotomic quiver Schur algebras in characteristic zero.
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On tensor spaces for Birman–Murakami–Wenzl algebras

2010-09-17
Jun Hu , Zhankui Xiao

Schur-Weyl reciprocity between quantum groups and Hecke algebras of type $G(r,1,n)$

2001-11-01
Jun Hu

Seminormal forms and cyclotomic quiver Hecke algebras of type A

2015-07-03
Jun Hu , Andrew Mathas
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Modular representations of Hecke algebras of type G(p,p,n)

2004-02-24
Jun Hu

On involutions in symmetric groups and a conjecture of Lusztig

2015-11-20
Jun Hu , Jing Zhang
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Graded Induction for Specht Modules

2011-04-22
Jun Hu , Andrew Mathas
Brundan, Kleshchev, and Wang have introduced a -grading on the Specht modules of the degenerate and nondegenerate cyclotomic Hecke algebras of type G(ℓ,1,n). In this paper, we show that the … Brundan, Kleshchev, and Wang have introduced a -grading on the Specht modules of the degenerate and nondegenerate cyclotomic Hecke algebras of type G(ℓ,1,n). In this paper, we show that the induced Specht modules have an explicit filtration by shifts of graded Specht modules. This proves a conjecture of Brundan, Kleshchev, and Wang. To prove these results, we first establish several strong results about the transition matrices between the seminormal, standard, and homogeneous bases of these algebras and similar results for the action of the Jucys–Murphy elements on these bases.
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A Morita equivalence theorem for Hecke algebra ℋ q (D n ) when n is even

2002-08-01
Jun Hu

Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A

2009-01-01
Jun Hu , Andrew Mathas
This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov--Lauda--Rouquier algebras of type $A$. This paper constructs an explicit homogeneous cellular basis for the cyclotomic Khovanov--Lauda--Rouquier algebras of type $A$.
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Crystal bases and simple modules for Hecke algebra of type Dn

2003-06-30
Jun Hu
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BMW algebra, quantized coordinate algebra and type 𝐶 Schur–Weyl duality

2011-01-10
Jun Hu
We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper B Subscript n Baseline left-parenthesis minus q Superscript 2 m … We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper B Subscript n Baseline left-parenthesis minus q Superscript 2 m plus 1 Baseline comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {B}_n(-q^{2m+1},q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the quantum algebra associated to the symplectic Lie algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German s German p Subscript 2 m"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {sp}_{2m}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In particular, we deduce that this Schur–Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang in the symplectic case. As a byproduct, we show that, as a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z left-bracket q comma q Superscript negative 1 Baseline right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>q</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}[q,q^{-1}]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript q Superscript double-struck upper Z Baseline left-parenthesis g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mi>q</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>g</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A_q^{\mathbb {Z}}(g)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev–Reshetikhin–Takhtajan construction.
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On double centralizer properties between quantum groups and Ariki–Koike algebras

2004-01-16
Jun Hu , Friederike Stoll

The number of simple modules for the Hecke algebras of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

2008-02-01
Jun Hu

On involutions in Weyl groups

2016-01-01
Jun Hu , Jing Zhang
Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set … Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set of twisted involutions relative to $\ast$ in $W$. In this paper we consider the case when $\ast=\text{id}$ and study the braid $I_\ast$-transformations between the reduced $I_\ast$-expressions of involutions. If $W$ is the Weyl group of type $B_n$ or $D_n$, we explicitly describe a finite set of basic braid $I_\ast$-transformations for all $n$ simultaneously, and show that any two reduced $I_\ast$-expressions for a given involution can be transformed into each other through a series of basic braid $I_\ast$-transformations. In both cases, these basic braid $I_\ast$-transformations consist of the usual basic braid transformations plus some natural "right end transformations" and plus exactly one extra transformation. The main result generalizes our previous work for the Weyl group of type $A_{n}$.

Branching rules for Hecke algebras of type <i>D<sub>n</sub></i>

2006-12-20
Jun Hu
Abstract In this paper we study the branching problems for the Hecke algebra ℋ︁( D n ) of type D n . We explicitly describe the decompositions into irreducible modules … Abstract In this paper we study the branching problems for the Hecke algebra ℋ︁( D n ) of type D n . We explicitly describe the decompositions into irreducible modules of the socle of the restriction of each irreducible ℋ︁( D n )‐representation to ℋ︁( D n –1 ) by using the corresponding results for type B Hecke algebras. In particular, we show that any such restrictions are always multiplicity free. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)
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Crystal bases and simple modules for Hecke algebras of type $G(p,p,n)$

2007-02-01
Jun Hu
We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type $G(p,p,n)$. This yields a classification of … We apply the crystal basis theory for Fock spaces over quantum affine algebras to the modular representations of the cyclotomic Hecke algebras of type $G(p,p,n)$. This yields a classification of simple modules over these cyclotomic Hecke algebras in the non-separated case, generalizing our previous work [J. Hu, J. Algebra 267 (2003), 7–20]. The separated case was completed in [J. Hu, J. Algebra 274 (2004), 446–490]. Furthermore, we use Naito and Sagaki’s work [S. Naito & D. Sagaki, J. Algebra 251, (2002) 461–474] on Lakshmibai–Seshadri paths fixed by diagram automorphisms to derive explicit formulas for the number of simple modules over these Hecke algebras. These formulas generalize earlier results of [M. Geck, Represent. Theory 4 (2000) 370-397] on the Hecke algebras of type $D_n$ (i.e., of type $G(2,2,n)$).
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Specht filtrations and tensor spaces for the Brauer algebra

2007-10-22
Jun Hu
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Seminormal forms and cyclotomic quiver Hecke algebras of type $A$

2013-01-01
Jun Hu , Andrew Mathas
This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying … This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit "integral" closed formula for the Gram determinants of the Specht modules in terms of the combinatorics which utilizes the KLR gradings. We then use seminormal forms to give a deformation of the KLR algebras of type $A$. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As an application we construct a new distinguished graded cellular basis of the cyclotomic KLR algebras of type $A$.

Dual partially harmonic tensors and Brauer-Schur-Weyl duality

2010-04-19
Jun Hu
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Quantum double of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi mathvariant="normal">U</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi mathvariant="fraktur">sl</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo>⩽</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false…

2007-08-29
Jun Hu , Yinhuo Zhang

Involutions in Weyl group of type F 4

2017-06-03
Jun Hu , Jing Zhang , Yabo Wu

β-character Algebra and a Commuting Pair in Hopf Algebras

2007-06-11
Jun Hu , Yinhuo Zhang

Morita equivalences of cyclotomic Hecke algebras of type G(r, p, n)

2009-01-01
Jun Hu , Andrew Mathas
We prove a Morita reduction theorem for the cyclotomic Hecke algebras ℋr, p, n(q, Q) of type G(r, p, n) with p > 1 and n ≧ 3. As a … We prove a Morita reduction theorem for the cyclotomic Hecke algebras ℋr, p, n(q, Q) of type G(r, p, n) with p > 1 and n ≧ 3. As a consequence, we show that computing the decomposition numbers of ℋr, p, n(Q) reduces to computing the p′-splittable decomposition numbers (see Definition 1.1) of the cyclotomic Hecke algebras ℋr′, p′, n′(Q′), where 1 ≦ r′ ≦ r, 1 ≦ n′ ≦ n, p′ | p and where the parameters Q′ are contained in a single (ɛ′, q)-orbit and ɛ′ is a primitive p′th root of unity.
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On a generalisation of the Dipper–James–Murphy conjecture

2010-01-05
Jun Hu
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Schur–Weyl reciprocity between quantum groups and Hecke algebras of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

2005-07-22
Jun Hu , Toshiaki Shoji

Decomposition numbers for Hecke algebras of type <i>G</i> (<i>r</i> , <i>p</i> , <i>n</i> ): the (Îľ, <i>q</i> )-separated case

2012-01-13
Jun Hu , Andrew Mathas
The paper studies the modular representation theory of the cyclotomic Hecke algebras of type G(r, p, n) with (ε, q)-separated parameters. We show that the decomposition numbers of these algebras … The paper studies the modular representation theory of the cyclotomic Hecke algebras of type G(r, p, n) with (ε, q)-separated parameters. We show that the decomposition numbers of these algebras are completely determined by the decomposition matrices of related cyclotomic Hecke algebras of type G(s, 1, m), where 1⩽s⩽r and 1⩽m⩽n. Furthermore, the proof gives an explicit algorithm for computing these decomposition numbers. Consequently, in principle, the decomposition matrices of these algebras are now known in characteristic zero. In proving these results, we develop a Specht module theory for these algebras, explicitly construct their simple modules and introduce and study analogues of the cyclotomic Schur algebras of type G(r, p, n) when the parameters are (ε, q)-separated. The main results of the paper rest upon two Morita equivalences: the first reduces the calculation of all decomposition numbers to the case of the l-splittable decomposition numbers and the second Morita equivalence allows us to compute these decomposition numbers using an analogue of the cyclotomic Schur algebras for the Hecke algebras of type G(r, p, n).
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Brauer's centralizer algebras, symplectic Schur algebras and Schur-Weyl duality

2005-03-24
Richard Dipper , Stephen Doty , Jun Hu
In this paper we study Schur-Weyl duality between the symplectic group and Brauer's centralizer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer's … In this paper we study Schur-Weyl duality between the symplectic group and Brauer's centralizer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer's centralizer algebra $B_n(-2m)$ to the endomorphism algebra of tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivitity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_n(-2m)$, is derived as an easy consequence of S. Oehms' results.

Graded induction for Specht modules

2010-01-01
Jun Hu , Andrew Mathas
Recently Brundan, Kleshchev and Wang introduced a $\Z$-grading on the Specht modules of the degenerate and non-degenerate cyclotomic Hecke algebras of type $G(\ell,1,n)$. In this paper we show that induced … Recently Brundan, Kleshchev and Wang introduced a $\Z$-grading on the Specht modules of the degenerate and non-degenerate cyclotomic Hecke algebras of type $G(\ell,1,n)$. In this paper we show that induced Specht modules have an explicit filtration by shifts of graded Specht modules. This proves a conjecture of Brundan, Kleshchev and Wang.
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SOME IRREDUCIBLE REPRESENTATIONS OF BRAUER'S CENTRALIZER ALGEBRAS

2004-09-01
Jun Hu , Yichuan Yang
HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Let $m, n\in\Bbb … HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. Let $m, n\in\Bbb N$, $V$ be a $2m$-dimensional complex vector space. The irreducible representations of the Brauer's centralizer algebra $B_n(-2m)$ appearing in $V^{\otimes{n}}$ are in 1–1 correspondence to the set of pairs $(\,f,\lamda)$, where $f\in\Z$ with $0\leq f\leq [n/2]$, $and$ $\lam\vdash n-2f$ satisfying $\lam_1\leq m$. In this paper, we first show that each of these representations has a basis consists of eigenvectors for the subalgebra of $B_n(-2m)$ generated by all the Jucys-Murphy operators, and we determine the corresponding eigenvalues. Then we identify these representations with the irreducible representations constructed from a cellular basis of $B_n(-2m)$. Finally, an explicit description of the action of each generator of $B_n(-2m)$ on such a basis is also given, which generalizes earlier work of [15] for Brauer's centralizer algebra $B_n(m)$.
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On the decomposition numbers of the Hecke algebra of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>D</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:math> when n is even

2008-12-03
Jun Hu
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Fayers’ conjecture and the socles of cyclotomic Weyl modules

2018-06-27
Jun Hu , Andrew Mathas
Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> … Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-restricted partitions. We prove an analogue of this result in the very general setting of “Schur pairs”. As an application we show that the socle of a Weyl module of a cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we prove a cyclotomic analogue of the Carter-Lusztig theorem.
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Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

2023-10-14
Jun Hu , Lei Shi
In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra [Formula: see text] associated to an arbitrary symmetrizable Cartan matrix [Formula: see … In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra [Formula: see text] associated to an arbitrary symmetrizable Cartan matrix [Formula: see text], where [Formula: see text] and [Formula: see text]. As applications, we obtain some necessary and sufficient conditions for the KLR idempotent [Formula: see text] (for any [Formula: see text]) to be nonzero in the cyclotomic quiver Hecke algebra [Formula: see text]. We prove several level reduction results which decompose [Formula: see text] into a sum of some products of [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] for each [Formula: see text]. Finally, we construct some explicit monomial bases for the subspaces [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is arbitrary and [Formula: see text] is a certain specific [Formula: see text]-tuple defined in (5.1).
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On a theorem of Lehrer and Zhang

2012-01-01
Jun Hu , Zhankui Xiao
Let K be an arbitrary field of characteristic not equal to 2. Let m, n\in\mathbb N and V be an m dimensional orthogonal space over K . There is a … Let K be an arbitrary field of characteristic not equal to 2. Let m, n\in\mathbb N and V be an m dimensional orthogonal space over K . There is a right action of the Brauer algebra {\mathfrak B}_n(m) on the n -tensor space V^{\otimes n} which centralizes the left action of the orthogonal group O(V) . Recently G.I. Lehrer and R.B. Zhang defined certain quasi-idempotents E_i in {\mathfrak B}_n(m) (see (1.1)) and proved that the annihilator of V^{\otimes n} in {\mathfrak B}_n(m) is always equal to the two-sided ideal generated by E_{[(m+1)/2]} if \mathrm{char} K=0 or \mathrm{char} K&gt;2(m+1) . In this paper we extend this theorem to arbitrary field K with \mathrm{char} K\neq 2 as conjectured by Lehrer and Zhang. As a byproduct, we discover a combinatorial identity which relates to the dimensions of Specht modules over the symmetric groups of different sizes and a new integral basis for the annihilator of V^{\otimes m+1} in {\mathfrak B}_{m+1}(m) .
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Induced Modules of Semisimple Hopf Algebras

2007-12-01
Jun Hu , Yinhuo Zhang
Let K be a field. Let H be a finite-dimensional K-Hopf algebra and D(H) be the Drinfel'd double of H. In this paper, we study Radford's induced module H β … Let K be a field. Let H be a finite-dimensional K-Hopf algebra and D(H) be the Drinfel'd double of H. In this paper, we study Radford's induced module H β , where β is a group-like element in H ∗ . Using the commuting pair established in [7], we obtain an analogue of the class equation for [Formula: see text] when H is semisimple and cosemisimple. In case H is a finite group algebra or a factorizable semisimple cosemisimple Hopf algebra, we give an explicit decomposition of each H β into a direct sum of simple D(H)-modules.

Quasi-parabolic subgroups of the Weyl group of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>D</mml:mi></mml:math>

2006-01-07
Jun Hu

A combinatorial approach to representations of quantum linear groups

1998-01-01
Jun Hu
In this paper we deal with Manin's quantization of GLn. Using the bideterminant bases, we prove that for a particular kind of elements , the canonical morphism from to Itelndωλ … In this paper we deal with Manin's quantization of GLn. Using the bideterminant bases, we prove that for a particular kind of elements , the canonical morphism from to Itelndωλ (see Section 1) is surjective, and, H0(λ) is isomorphic to Itelnd wλif ω=ω 0, the longest element in . Our approach is to construct a basis consisting of bidetermi-nants for the above iterated induced modules. We believe that such a basis is also interesting and useful. Moreover, most of the well-known homological properties of GLn-q, such as Grothendieck vanishing, Kempf vanishing, De-mazure character formula and Bott-Borel-Weil Theorem, are reobtained as consequences of the above surjectivity.

On the generalised Dipper–James–Murphy conjecture in quantum characteristic 2

2016-05-18
Jun Hu , Kai Zhou , Danni Wang

Branching rules for Hecke Algebras of Type $D_{n}$

2005-01-01
Jun Hu
In this paper we study the branching problems for Hecke algebra $\H(D_n)$ of type $D_n$. We explicitly describe the decompositions of the socle of the restriction of each irreducible $\H(D_n)$-representation … In this paper we study the branching problems for Hecke algebra $\H(D_n)$ of type $D_n$. We explicitly describe the decompositions of the socle of the restriction of each irreducible $\H(D_n)$-representation to $\H(D_{n-1})$ into irreducible modules by using the corresponding results for type $B$ Hecke algebras. In particular, we show that any such restrictions are always multiplicity free.

On simple modules of Hecke algebras of type Dn

2001-08-01
Jun Hu , Jianpan Wang

Extending the Dehn quandle to shears and foliations on the torus

2014-01-01
Reza Chamanara , Jun Hu , Joel Zablow
The Dehn quandle, $Q$, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, $\mathbb {T}^2$, we generalize … The Dehn quandle, $Q$, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, $\mathbb {T}^2$, we generalize this to show the existence of a quandle $\hat Q$ extending $Q$ and whose elements a
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On the structure of cyclotomic nilHecke algebras

2018-05-01
Jun Hu , Xinfeng Liang
In this paper we study the structure of the cyclotomic nilHecke algebras $\HH_{\ell,n}^{(0)}$, where $\ell,n\in\N$. We construct a monomial basis for $\HH_{\ell,n}^{(0)}$ which verifies a conjecture of Mathas. We show … In this paper we study the structure of the cyclotomic nilHecke algebras $\HH_{\ell,n}^{(0)}$, where $\ell,n\in\N$. We construct a monomial basis for $\HH_{\ell,n}^{(0)}$ which verifies a conjecture of Mathas. We show that the graded basic algebra of $\HH_{\ell,n}^{(0)}$ is commutative and hence isomorphic to the center $Z$ of $\HH_{\ell,n}^{(0)}$. We further prove that $\HH_{\ell,n}^{(0)}$ is isomorphic to the full matrix algebra over $Z$ and construct an explicit basis for the center $Z$. We also construct a complete set of pairwise orthogonal primitive idempotents of $\HH_{\ell,n}^{(0)}$. Finally, we present a new homogeneous symmetrizing form $\Tr$ on $\HH_{\ell,n}^{(0)}$ by explicitly specifying its values on a given homogeneous basis of $\HH_{\ell,n}^{(0)}$ and show that it coincides with Shan--Varagnolo--Vasserot's symmetrizing form $\Tr^{\text{SVV}}$ on $\HH_{\ell,n}^{(0)}$.
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The number of simple modules for the Hecke algebras of type G(r,p,n)

2006-01-24
Jun Hu

SYMMETRIZERS AND ANTISYMMETRIZERS FOR THE BMW-ALGEBRA

2013-02-01
Richard Dipper , Jun Hu , Friederike Stoll
Let n ∈ ℕ and B n (r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that B n (r, q) has … Let n ∈ ℕ and B n (r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that B n (r, q) has two distinct linear representations generated by two central elements of B n (r, q) called the symmetrizer and antisymmetrizer of B n (r, q). These generate for n ≥ 3 the only one-dimensional two sided ideals of B n (r, q) and generalize the corresponding notion for Hecke algebras of type A. The main result, Theorem 3.1, in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of B n (r, q).
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Modified affine Hecke algebras and quiver Hecke algebras of type A

2019-10-03
Jun Hu , Fang Li
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On involutions in symmetric groups and a conjecture of Lusztig

2015-01-01
Jun Hu , Jing Zhang
Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown … Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally endowed with a module structure of the Hecke algebra of $(W, S)$. Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter $u^2$) associated to $(W,S)$ generated by the element $X_{\emptyset}:=\sum_{w^\ast=w}u^{-\ell(w)}T_w$. In this paper we prove this conjecture in the case when $\ast=\text{id}$ and $W$ is the symmetric group on $n$ letters.
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On a generalization of Dipper--James--Murphy's Conjecture

2009-01-01
Jun Hu
Let $K$ be a field and $q\in K^{\times}$. Let $e$ be the multiplicative order of $q$; or 0 if $q$ is not a root of unity. Let $\bQ:=(q^{v_1},...,q^{v_r})$. Let ${K}_r(n)$ … Let $K$ be a field and $q\in K^{\times}$. Let $e$ be the multiplicative order of $q$; or 0 if $q$ is not a root of unity. Let $\bQ:=(q^{v_1},...,q^{v_r})$. Let ${K}_r(n)$ be the set of Kleshchev $r$-multipartitions with respect to $(e;\bQ)$. In this paper, we consider an extention of Dipper--James--Murphy's Conjecture to the Ariki--Koike algebra $H_{r,n}(q;\bQ)$ with $r&gt;2$. We show that any $(\bQ,e)$-restricted $r$-multipartition of $n$ is a Kleshchev multipartition in ${K}_r(n)$; and if $e&gt;1$, then any multi-core $\ulam=(\lam^{(1)},...,\lam^{(r)})$ in ${K}_r(n)$ is a $(\bQ,e)$-restricted $r$-multipartition. As a consequence, we show that if $e=0$ (i.e., $q$ is not a root of unity), then ${K}_r(n)$ coincides with the set of $(\bQ,e)$-restricted $r$-multipartitions of $n$ and also coincides with the set of ladder $r$-multipartitions of $n$.

Corrigendum to “Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras” [J. Pure Appl. Algebra 227(4) (2023) 107281]

2025-05-02
Zhekun He , Jun Hu , Huang Lin

Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules

2025-02-10
Ming Fang , Jun Hu
Abstract Standard modules over quasi-hereditary algebras are by definition certain quotients of projective modules. In this article, we study when they can be realized as submodules of projective modules, a … Abstract Standard modules over quasi-hereditary algebras are by definition certain quotients of projective modules. In this article, we study when they can be realized as submodules of projective modules, a property enjoyed by Verma modules in the BGG category <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="script">𝒪</m:mi> </m:math> {\mathcal{O}} for a finite-dimensional complex semisimple Lie algebra and by quantum Weyl modules over quantized Schur algebras <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>S</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {S(n,r)} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mi>r</m:mi> </m:mrow> </m:math> {n\geq r} .

Relations between Kazhdan-Lusztig bases, Murphy bases and seminormal bases

2025-01-16
Zhekun He , Jun Hu , Yujiao Sun
Let 𝑤↦(𝑃⁡(𝑤),𝑄⁡(𝑤)) be the Robinson-Schensted correspondence between the symmetric group 𝔖𝑛 and the set of pairs of standard tableaux with the same shapes. We show that each Kazhdan-Lusztig basis (KL … Let 𝑤↦(𝑃⁡(𝑤),𝑄⁡(𝑤)) be the Robinson-Schensted correspondence between the symmetric group 𝔖𝑛 and the set of pairs of standard tableaux with the same shapes. We show that each Kazhdan-Lusztig basis (KL basis for short) element 𝐶′𝑤 can be expressed as a linear combination of some 𝑓𝔰⁢𝔱 which satisfies that 𝔰⁢¥⁢𝑃⁢(𝑤)∗, 𝔱⁢¥⁢𝑄⁢(𝑤)∗, where "¥" is the dominance (partial) order between standard tableaux, 𝔲∗ denotes the conjugate of 𝔲 for each standard tableau 𝔲, {𝑓𝔰⁢𝔱|𝔰,𝔱∈Std⁡(𝜆),𝜆⊢𝑛} is the seminormal basis of the Iwahori-Hecke algebra associated to 𝔖𝑛. As a result, we generalize an earlier result of Geck on the relation between the KL basis and the Murphy basis. Similar relations between the twisted KL basis, the dual seminormal basis and the dual Murphy basis are obtained.

On Hecke algebras and $Z$-graded twisting, Shuffling and Zuckerman functors

2024-09-05
Ming Fang , Jun Hu , Yujiao Sun
Let $g$ be a complex semisimple Lie algebra with Weyl group $W$. Let $H(W)$ be the Iwahori-Hecke algebra associated to $W$. For each $w\in W$, let $T_w$ and $C_w$ be … Let $g$ be a complex semisimple Lie algebra with Weyl group $W$. Let $H(W)$ be the Iwahori-Hecke algebra associated to $W$. For each $w\in W$, let $T_w$ and $C_w$ be the corresponding $Z$-graded twisting functor and $Z$-graded shuffling functor respectively. In this paper we present a categorical action of $H(W)$ on the derived category $D^b(O_0^Z)$ of the $Z$-graded BGG category $O_0^Z$ via derived twisting functors as well as a categorical action of $H(W)$ on $D^b(O_0^Z)$ via derived shuffling functors. As applications, we get graded character formulae for $T_sL(x)$ and $C_sL(x)$ for each simple reflection $s$. We describe the graded shifts occurring in the action of the $Z$-graded twisting and shuffling functors on dual Verma modules and simple modules. We also characterize the action of the derived $Z$-graded Zuckerman functors on simple modules.
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On the maximal and minimal degree components of the cocenter of the cyclotomic KLR algebras

2024-01-01
Jun Hu , Lei Shi
Let $\mathscr{R}_\alpha^\Lambda$ be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the ground field $K$ … Let $\mathscr{R}_\alpha^\Lambda$ be the cyclotomic KLR algebra associated to a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$ and polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. Shan, Varagnolo and Vasserot show that, when the ground field $K$ has characteristic $0$, the degree $d$ component of the cocenter $Tr(\mathscr{R}_\alpha^\Lambda)$ is nonzero only if $0\leq d\leq d_{\Lambda,\alpha}$. In this paper we show that this holds true for arbitrary ground field $K$, arbitrary $\mathfrak{g}$ and arbitrary polynomials $\{Q_{ij}(u,v)\}_{i,j\in I}$. We generalize our earlier results on the $K$-linear generators of $Tr(\mathscr{R}_\alpha^\Lambda), Tr(\mathscr{R}_\alpha^\Lambda)_0, Tr(\mathscr{R}_\alpha^\Lambda)_{d_{\Lambda,\alpha}}$ to arbitrary ground field $K$. Moreover, we show that the dimension of the degree $0$ component $Tr(\mathscr{R}_\alpha^\Lambda)_0$ is always equal to $\dim V(\Lambda)_{\Lambda-\alpha}$, where $V(\Lambda)$ is the integrable highest weight $U(\mathfrak{g})$-module with highest weight $\Lambda$, and we obtain a basis for $Tr(\mathscr{R}_\alpha^\Lambda)_0$.

On the minimal elements in conjugacy classes of the complex reflection group G(r,1,n)

2023-12-12
Jun Hu , Xiaolin Shi , Lei Shi

Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

2023-10-14
Jun Hu , Lei Shi
In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra [Formula: see text] associated to an arbitrary symmetrizable Cartan matrix [Formula: see … In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra [Formula: see text] associated to an arbitrary symmetrizable Cartan matrix [Formula: see text], where [Formula: see text] and [Formula: see text]. As applications, we obtain some necessary and sufficient conditions for the KLR idempotent [Formula: see text] (for any [Formula: see text]) to be nonzero in the cyclotomic quiver Hecke algebra [Formula: see text]. We prove several level reduction results which decompose [Formula: see text] into a sum of some products of [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] for each [Formula: see text]. Finally, we construct some explicit monomial bases for the subspaces [Formula: see text] and [Formula: see text] of [Formula: see text], where [Formula: see text] is arbitrary and [Formula: see text] is a certain specific [Formula: see text]-tuple defined in (5.1).
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Graded dimensions and monomial bases for the cyclotomic quiver Hecke superalgebras

2023-08-28
Jun Hu , Lei Shi
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Skew cellularity of the Hecke algebras of type 𝐺(ℓ,𝑝,𝑛)

2023-07-20
Jun Hu , Andrew Mathas , Salim Rostam
This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to … This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis script l comma p comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(\ell ,p,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are graded skew cellular algebras. In the special case when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> this implies that the Hecke algebras of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis script l comma 2 comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(\ell ,2,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis script l comma p comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G(\ell ,p,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

On the Center Conjecture for the Cyclotomic KLR Algebras

2023-05-20
Jun Hu , Huang Lin
Abstract The center conjecture for the cyclotomic KLR algebras $\mathscr{R}_{\beta }^{\Lambda }$ asserts that the center of $\mathscr{R}_{\beta }^{\Lambda }$ consists of symmetric elements in its KLR $x$ and $e(\nu … Abstract The center conjecture for the cyclotomic KLR algebras $\mathscr{R}_{\beta }^{\Lambda }$ asserts that the center of $\mathscr{R}_{\beta }^{\Lambda }$ consists of symmetric elements in its KLR $x$ and $e(\nu )$ generators. In this paper, we show that this conjecture is equivalent to the injectivity of some natural map $\overline{\iota }_{\beta }^{\Lambda ,i}$ from the cocenter of $\mathscr{R}_{\beta }^{\Lambda }$ to the cocenter of $\mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ for all $i\in I$ and $\Lambda \in P^{+}$. We prove that the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is given by multiplication with a center element $z(i,\beta )\in \mathscr{R}_{\beta }^{\Lambda +\Lambda _{i}}$ and we explicitly calculate the element $z(i,\beta )$ in terms of the KLR $x$ and $e(\nu )$ generators. We present explicit monomial bases for certain bi-weight spaces of the defining ideal of $\mathscr{R}_{\beta }^{\Lambda }$. For $\beta =\sum _{j=1}^{n}\alpha _{i_{j}}$ with $\alpha _{i_{1}},\cdots , \alpha _{i_{n}}$ pairwise distinct, we construct an explicit monomial basis of $\mathscr{R}_{\beta }^{\Lambda }$, prove the map $\overline{\iota }_{\beta }^{\Lambda ,i}$ is injective, and thus verify the center conjecture for these $\mathscr{R}_{\beta }^{\Lambda }$.
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Piecewise dominant sequences and the cocenter of the cyclotomic quiver Hecke algebras

2023-03-15
Jun Hu , Lei Shi
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Crystal of Affine $${\widehat {\mathfrak{s}\mathfrak{l}}_\ell}$$ and Modular Branching Rules for Hecke Algebras of Type Dn

2023-03-01
Huang Lin , Jun Hu
Let ℋq(Bn) and ℋq(Dn) denote the Hecke algebras of types Bn and Dn respectively, where q ≠ 1 is the Hecke parameter with quantum characteristic e. We prove that if … Let ℋq(Bn) and ℋq(Dn) denote the Hecke algebras of types Bn and Dn respectively, where q ≠ 1 is the Hecke parameter with quantum characteristic e. We prove that if Dλ is a simple ℋq(B2n)-module which splits as $$D_ + ^\lambda \oplus D_ - ^\lambda $$ upon restriction to ℋq(D2n), then $$D_ + ^\lambda {\downarrow _{{{\cal H}_q}({D_{2n - 1}})}} \cong D_ - ^\lambda {\downarrow _{{{\cal H}_q}({D_{2n - 1}})}}$$ and $$D_ + ^\lambda {\uparrow ^{{{\cal H}_q}({D_{2n + 1}})}} \cong D_ - ^\lambda {\uparrow ^{{{\cal H}_q}({D_{2n + 1}})}}$$ . In particular, we get some multiplicity-free results for certain two-step modular branching rules. We also show that when e = 2ℓ > 2 the highest weight crystal of the irreducible $${\widehat {}_\ell}$$ -module L(Λ0) can be categorified using the simple ℋq (D2n)-modules $$\{D_ + ^{\bf{\lambda}}|{\bf{\lambda}} = ({\lambda ^{(1)}},{\lambda ^{(2)}}) \vdash 2n,{D^{\bf{\lambda}}}{\downarrow _{{{\cal H}_q}({D_{2n}})}} \cong D_ + ^{\bf{\lambda}} \oplus D_ - ^{\bf{\lambda}},n \in \mathbb{N}\} $$ and certain two-step induction and restriction functors. Finally, a complete classification of all the simple blocks of ℋq(Dn) is also obtained.

On radical filtrations of parabolic Verma modules

2023-01-01
Jun Hu , Wei Xiao
In this paper we give a sum formula for the radical filtration of parabolic Verma modules in any (possibly singular) blocks of parabolic BGG category.It can be viewed as a … In this paper we give a sum formula for the radical filtration of parabolic Verma modules in any (possibly singular) blocks of parabolic BGG category.It can be viewed as a generalization of the Jantzen sum formula for Verma modules in the usual BGG category O.The proof makes use of the graded version of parabolic BGG category.Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of parabolic Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given in terms of the Kazhdan-Lusztig polynomials.
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Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras

2022-11-16
Zhekun He , Jun Hu , Huang Lin
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BGG Category <i>𝒪</i> and ℤ-Graded Representation Theory

2022-11-01
Jun Hu

On the seminormal bases and dual seminormal bases of the cyclotomic Hecke algebras of type G(ℓ,1,n)

2022-03-02
Jun Hu , Shixuan Wang

Crystal of affine type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo>ˆ</mml:mo></mml:mrow></mml:mover></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo linebreak="badbreak" linebreakstyle="after">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> and Hecke algebras at a primitive 2ℓth root of unity

2021-09-29
Huang Lin , Jun Hu

Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

2021-09-14
Jun Hu , Zhankui Xiao
In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> … In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a quasi-hereditary algebra with a simple preserving duality and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a faithful tilting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the double centralizer property with respect to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper E n d Subscript upper E n d Sub Subscript upper A Subscript left-parenthesis upper T right-parenthesis Baseline left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>d</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A=End_{End_A(T)}(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript upper K Superscript s y Baseline left-parenthesis m comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mi>K</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mi>y</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">S_K^{sy}(m,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Brauer algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper B Subscript n Baseline left-parenthesis minus 2 m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {B}_n(-2m)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the space of dual partially harmonic tensors under certain condition.
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Matrix units and primitive idempotents for the Hecke algebra of type D

2021-08-04
Jun Hu , Shixuan Wang , Huanmei Sun

Schubert Class and Cyclotomic NilHecke Algebras

2021-07-26
Kai Zhou , Jun Hu
Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of … Let [Formula: see text] and [Formula: see text] be positive integers such that [Formula: see text], and let [Formula: see text] be the Grassmannian which consists of the set of [Formula: see text]-dimensional subspaces of [Formula: see text]. There is a [Formula: see text]-graded algebra isomorphism between the cohomology [Formula: see text] of [Formula: see text] and a natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of the type [Formula: see text] cyclotomic nilHecke algebra [Formula: see text]. We show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class [Formula: see text] coincides with the basis element [Formula: see text] constructed by Hu and Liang by purely algebraic method, where [Formula: see text] with [Formula: see text] for each [Formula: see text], and [Formula: see text] is the [Formula: see text]-multipartition of [Formula: see text] associated to [Formula: see text]. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian [Formula: see text] and the [Formula: see text]'s basis ([Formula: see text] is an [Formula: see text]-multipartition of [Formula: see text] with each component being either [Formula: see text] or empty) of the natural [Formula: see text]-form [Formula: see text] of the [Formula: see text]-graded basic algebra of [Formula: see text] is also obtained. As an application, we obtain a second version of the Giambelli formula for Schubert classes.
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BGG category $\mathcal{O}$ and $\mathbb{Z}$-graded representation theory

2021-01-01
Jun Hu
We give an introduction to the $\mathbb{Z}$-graded representation theory of the BGG category $\mathcal{O}$ of a complex semisimple Lie algebras, with an emphasis on Soergel's combinatorial $\mathbb{V}$ functor, definitions of … We give an introduction to the $\mathbb{Z}$-graded representation theory of the BGG category $\mathcal{O}$ of a complex semisimple Lie algebras, with an emphasis on Soergel's combinatorial $\mathbb{V}$ functor, definitions of $\mathbb{Z}$-graded duality functors and definitions of $\mathbb{Z}$-graded translation functors.

Graded dimensions and monomial bases for the cyclotomic quiver Hecke algebras

2021-01-01
Jun Hu , Lei Shi
In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra $R^\Lambda(\beta)$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, where … In this paper we give a closed formula for the graded dimension of the cyclotomic quiver Hecke algebra $R^\Lambda(\beta)$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, where $\Lambda\in P^+$ and $\beta\in Q_n^+$. As applications, we obtain some {\it necessary and sufficient conditions} for the KLR idempotent $e(\nu)$ (for any $\nu\in I^\beta$) to be nonzero in the cyclotomic quiver Hecke algebra $R^\Lambda(\beta)$. We prove several level reduction results which decomposes $\dim R^\Lambda(\beta)$ into a sum of some products of $\dim R^{\Lambda^i}(\beta_i)$ with $\Lambda=\sum_i\Lambda^i$ and $\beta=\sum_{i}\beta_i$, where $\Lambda^i\in P^+, \beta^i\in Q^+$ for each $i$. We construct some explicit monomial bases for the subspaces $e(\widetilde{\nu})R^\Lambda(\beta)e(\mu)$ and $e(\widetilde{\nu})R^\Lambda(\beta)e(\mu)$ of $R^\Lambda(\beta)$, where $\mu\in I^\beta$ is {\it arbitrary} and $\widetilde{\nu}\in I^\beta$ is a certain specific $n$-tuple (see Section 4).Finally, we use our graded dimension formulae to provide some examples which show that $R^\Lambda(n)$ is in general not graded free over its natural embedded subalgebra $R^\Lambda(m)$ with $m<n$.

Kazhdan-Lusztig left cell preorder and dominance order

2021-01-01
Zhekun He , Jun Hu , Yujiao Sun
Let "$\leq_L$" be the Kazhdan-Lusztig left cell preorder on the symmetric group $S_n$. Let $w\mapsto (P(w),Q(w))$ be the Robinson-Schensted-Knuth correspondence between $S_n$ and the set of standard tableaux with the … Let "$\leq_L$" be the Kazhdan-Lusztig left cell preorder on the symmetric group $S_n$. Let $w\mapsto (P(w),Q(w))$ be the Robinson-Schensted-Knuth correspondence between $S_n$ and the set of standard tableaux with the same shapes. We prove that for any $x,y\in S_n$, $x\leq_L y$ only if $Q(y)\unrhd Q(x)$, where "$\unrhd$" is the dominance (partial) order between standard tableaux. As a byproduct, we generalize an earlier result of Geck by showing that each Kazhdan-Lusztig basis element $C'_w$ can be expressed as a linear combination of some $m_{uv}$ which satisfies that $u\unrhd P(w)^*$, $v\unrhd Q(w)^*$, where $t^*$ denotes the conjugate of $t$ for each standard tableau $t$, $\{m_{st} \mid s,t\in Std(λ),λ\vdash n\}$ is the Murphy basis of the Iwahori-Hecke algebra $H_{v}(S_n)$ associated to $S_n$.

Generalized conformal barycentric extensions of circle maps

2020-01-17
Jun Hu

Jantzen coefficients and radical filtrations for generalized Verma modules

2020-01-01
Jun Hu , Wei Xiao
In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as … In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of generalized Verma modules in any (possibly singular) integral blocks of the parabolic BGG category are also given.

Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

2020-01-01
Jun Hu , Zhankui Xiao
Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $\Lambda^+$ be an indexing set of all simple modules in $A\lmod$. … Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $\Lambda^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $\lambda\in\Lambda^+$, there is an embedding $\Delta(\lambda)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(\lambda)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful tilting $A$-module, then $A$ has the double centralizer property with respect to $T$. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module $T$ over $A$ for which $A=\End_{\End_A(T)}(T)$. We also establish a Schur-Weyl duality between the symplectic Schur algebra $S^{sy}(m,n)$ and $\bb_{n}/\mathfrak{B}_{n}^{(f)}$ on $V^{\otimes n}/V^{\otimes n}\mathfrak{B}_{n}^{(f)}$ when $\cha K>\min\{n-f+m,n\}$, where $V$ is a $2m$-dimensional symplectic space over $K$, $\mathfrak{B}_{n}^{(f)}$ is the two-sided ideal of the Brauer algebra $\bb_{n}(-2m)$ generated by $e_1e_3\cdots e_{2f-1}$ with $1\leq f\leq [\frac{n}{2}]$.
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Modified affine Hecke algebras and quiver Hecke algebras of type A

2019-10-03
Jun Hu , Fang Li
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On some embeddings between the cyclotomic quiver Hecke algebras

2019-06-26
Kai Zhou , Jun Hu
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite index set and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals left-parenthesis a … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finite index set and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals left-parenthesis a Subscript i j Baseline right-parenthesis Subscript i comma j element-of upper I"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">A=(a_{ij})_{i,j\in I}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an arbitrary indecomposable symmetrizable generalized Cartan matrix. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q Superscript plus"> <mml:semantics> <mml:msup> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">Q^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the positive root lattice and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript plus"> <mml:semantics> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">P^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the set of dominant weights. For any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta element-of upper Q Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta \in Q^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda element-of upper P Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>P</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Lambda \in P^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R Subscript beta Superscript normal upper Lamda"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathscr {R}_{\beta }^{\Lambda }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the corresponding cyclotomic quiver Hecke algebra over a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i element-of upper I"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i\in I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, there is a natural unital algebra homomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="iota Subscript beta comma i"> <mml:semantics> <mml:msub> <mml:mi>ι</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\iota _{\beta ,i}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R Subscript beta Superscript normal upper Lamda"> <mml:semantics> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> </mml:msubsup> <mml:annotation encoding="application/x-tex">\mathscr {R}_{\beta }^{\Lambda }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e left-parenthesis beta comma i right-parenthesis script upper R Subscript beta plus alpha Sub Subscript i Superscript normal upper Lamda Baseline e left-parenthesis beta comma i right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">e(\beta ,i)\mathscr {R}_{\beta +\alpha _i}^{\Lambda }e(\beta ,i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In this paper we show that the homomorphism <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="iota Subscript beta Baseline colon equals circled-plus Underscript i element-of upper I Endscripts iota Subscript beta comma i Baseline colon script upper R Subscript beta Superscript normal upper Lamda Baseline right-arrow circled-plus Underscript i element-of upper I Endscripts e left-parenthesis beta comma i right-parenthesis script upper R Subscript beta plus alpha Sub Subscript i Subscript Superscript normal upper Lamda Baseline e left-parenthesis beta comma i right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>ι</mml:mi> <mml:mi>β</mml:mi> </mml:msub> <mml:mo>:=</mml:mo> <mml:munder> <mml:mo>⨁</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>ι</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mo>:</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">→</mml:mo> <mml:munder> <mml:mo>⨁</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> </mml:munder> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msubsup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="script">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>β</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mi>e</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\iota _\beta :=\bigoplus _{i\in I}\iota _{\beta ,i}: \mathscr {R}_{\beta }^{\Lambda }\rightarrow \bigoplus _{i\in I}e(\beta ,i)\mathscr {R}_{\beta +\alpha _i}^{\Lambda }e(\beta ,i)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is always injective unless <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals 0"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta =0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l left-parenthesis normal upper Lamda right-parenthesis equals 0"> <mml:semantics> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell (\Lambda )=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is of finite type and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta equals normal upper Lamda minus w 0 normal upper Lamda"> <mml:semantics> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mi mathvariant="normal">Λ</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta =\Lambda -w_0\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w 0"> <mml:semantics> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">w_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the unique longest element in the finite Weyl group associated to the finite Cartan matrix <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l left-parenthesis normal upper Lamda right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\ell (\Lambda )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the level of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda"> <mml:semantics> <mml:mi mathvariant="normal">Λ</mml:mi> <mml:annotation encoding="application/x-tex">\Lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Integral u-deformed involution modules

2019-05-14
Jun Hu , Yujiao Sun
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Crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ and Hecke algebras at a primitive $2\ell$th root of unity

2019-01-01
Huang Lin , Jun Hu
Let $\ell\in\mathbb{N}$ with $\ell&gt;2$ and $I:=\mathbb{Z}/2\ell\mathbb{Z}$. In this paper we give a new realization of the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ using the modular representation theory of the affine Hecke algebras … Let $\ell\in\mathbb{N}$ with $\ell&gt;2$ and $I:=\mathbb{Z}/2\ell\mathbb{Z}$. In this paper we give a new realization of the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ using the modular representation theory of the affine Hecke algebras $H_n$ of type $A$ and their level two cyclotomic quotients with Hecke parameter being a primitive $2\ell$th root of unity. We realized the Kashiwara operators for the crystal as the functors of taking socle of certain two-steps restriction and of taking head of certain two-steps induction. For any finite dimensional irreducible $H_n$-module $M$, we prove that the irreducible submodules of $\rm{res}_{H_{n-2}}^{H_n}M$ which belong to $\widehat{B}(\infty)$ (Definition 6.1) occur with multiplicity two. The main results generalize the earlier work of Grojnowski and Vazirani on the relations between the crystal of affine $\widehat{\mathfrak{sl}}_{\ell}$ and the affine Hecke algebras of type $A$ at a primitive $\ell$th root of unity.

Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

2018-09-07
Jun Hu , Ngau Lam
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Fayers’ conjecture and the socles of cyclotomic Weyl modules

2018-06-27
Jun Hu , Andrew Mathas
Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> … Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-restricted partitions. We prove an analogue of this result in the very general setting of “Schur pairs”. As an application we show that the socle of a Weyl module of a cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Finally, we prove a cyclotomic analogue of the Carter-Lusztig theorem.
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On the structure of cyclotomic nilHecke algebras

2018-05-01
Jun Hu , Xinfeng Liang
In this paper we study the structure of the cyclotomic nilHecke algebras $\HH_{\ell,n}^{(0)}$, where $\ell,n\in\N$. We construct a monomial basis for $\HH_{\ell,n}^{(0)}$ which verifies a conjecture of Mathas. We show … In this paper we study the structure of the cyclotomic nilHecke algebras $\HH_{\ell,n}^{(0)}$, where $\ell,n\in\N$. We construct a monomial basis for $\HH_{\ell,n}^{(0)}$ which verifies a conjecture of Mathas. We show that the graded basic algebra of $\HH_{\ell,n}^{(0)}$ is commutative and hence isomorphic to the center $Z$ of $\HH_{\ell,n}^{(0)}$. We further prove that $\HH_{\ell,n}^{(0)}$ is isomorphic to the full matrix algebra over $Z$ and construct an explicit basis for the center $Z$. We also construct a complete set of pairwise orthogonal primitive idempotents of $\HH_{\ell,n}^{(0)}$. Finally, we present a new homogeneous symmetrizing form $\Tr$ on $\HH_{\ell,n}^{(0)}$ by explicitly specifying its values on a given homogeneous basis of $\HH_{\ell,n}^{(0)}$ and show that it coincides with Shan--Varagnolo--Vasserot's symmetrizing form $\Tr^{\text{SVV}}$ on $\HH_{\ell,n}^{(0)}$.
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Grassmannian, symmetric functions and cyclotomic nilHecke algebras

2018-04-04
Kai Zhou , Jun Hu
Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell-n}$ be the Grassmannian manifold which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is an $\mathbb{Z}$-graded … Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell-n}$ be the Grassmannian manifold which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is an $\mathbb{Z}$-graded algebra isomorphism between the cohomology $\rm{H}*(\mathbb{G}_{n,\ell-n},\mathbb{Z})$ of $\mathbb{G}_{n,\ell-n}$ and a natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of the type $A$ cyclotomic nilHecke algebra $\mathscr{H}_{\ell,n}^{(0)}= $. In this paper, we show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class $(a_1,\cdots,a_{n})$ coincides with the basis element $b_{\mathbf{\lambda}}$ constructed in an earlier paper by the second author and Xinfeng Liang using purely algebraic method, where $0\leq a_1\leq a_2\leq\cdots\leq a_{n}\leq \ell-n$ with $a_i\in\mathbb{Z}$ for each $i$, $\mathbb{\lambda}$ is the $\ell$-multipartition of $n$ associated to $\(\ell+1-(a_{n}+n), \ell+1-(a_{n-1}+n-1),\cdots,\ell+1-(a_1+1))$. A similar isomorphism between the cohomology $\rm{H}^*(\mathbb{G}_{\ell-n,n},\mathbb{Z})$ of the Grassmannian $\mathbb{G}_{\ell-n,n}$ and the natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of $\mathscr{H}_{\ell,n}^{(0)}$ is also obtained. As applications, we obtain a second version of Giambelli formula for Schubert classes and show that each basis element $z_{\mathbf{\lambda}}$ (constructed in the earlier paper mentioned above) of the center $Z$ of $\mathscr{H}_{\ell,n}^{(0)}$ is equal to the evaluation of a Schur (symmetric) polynomial at $y_1,\cdots,y_n$.

Integral $u$-deformed involution modules

2018-01-01
Jun Hu , Yujiao Sun
Let $(W,S)$ be a Coxeter system and $\ast$ an automorphism of $W$ with order $\leq 2$ and $S^{\ast}=S$. Lusztig and Vogan ([11], [14]) have introduced a $u$-deformed version $M_u$ of … Let $(W,S)$ be a Coxeter system and $\ast$ an automorphism of $W$ with order $\leq 2$ and $S^{\ast}=S$. Lusztig and Vogan ([11], [14]) have introduced a $u$-deformed version $M_u$ of Kottwitz's involution module over the Iwahori-Hecke algebra $\mathscr{H}_{u}(W)$ with Hecke parameter $u^2$, where $u$ is an indeterminate. Lusztig has proved that $M_u$ is isomorphic to the left $\mathscr{H}_{u}(W)$-submodule of ${\hat{\mathscr{H}}}_u$ generated by $X_{\emptyset}:=\sum_{w^*=w\in W}{u^{-\ell(w)}T_w}$, where ${\hat{\mathscr{H}}}_u$ is the vector space consisting of all formal (possibly infinite) sums $\sum_{x\in W}{c_xT_x}$ ($c_x\in\mathbb{Q}(u)$ for each $x$). He speculated that one can extend this by replacing $u$ with any $\lambda\in \mathbb{C}\setminus\{0,1,-1\}$. In this paper, we give a positive answer to his speculation for any $\lambda\in K\setminus\{0,1,-1\}$ and any $W$, where $K$ is an arbitrary ground field.
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Schubert Class and cyclotomic nilHecke algebras

2018-01-01
Kai Zhou , Jun Hu
Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a $\mathbb{Z}$-graded algebra … Let $\ell, n$ be positive integers such that $\ell\geq n$. Let $\mathbb{G}_{n,\ell}$ be the Grassmannian which consists of the set of $n$-dimensional subspaces of $\mathbb{C}^{\ell}$. There is a $\mathbb{Z}$-graded algebra isomorphism between the cohomology $H^*(\mathbb{G}_{n,\ell},\mathbb{Z})$ of $\mathbb{G}_{n,\ell}$ and a natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of the type $A$ cyclotomic nilHecke algebra $H_{\ell,n}^{(0)}=\langle\psi_1,\cdots,\psi_{n-1},y_1,\cdots,y_n\rangle$. In this paper, we show that the isomorphism can be chosen such that the image of each (geometrically defined) Schubert class $(a_1,\cdots,a_{n})$ coincides with the basis element $b_{\mathbf{\lambda}}$ constructed by Jun Hu and Xinfeng Liang by purely algebraic method, where $0\leq a_1\leq a_2\leq\cdots\leq a_{n}\leq \ell-n$ with $a_i\in\mathbb{Z}$ for each $i$, $\mathbf{\lambda}$ is the $\ell$-multipartition of $n$ associated to $(\ell+1-(a_{n}+n), \ell+1-(a_{n-1}+n-1),\cdots,\ell+1-(a_1+1))$. A similar correspondence between the Schubert class basis of the cohomology of the Grassmannian $\mathbb{G}_{\ell-n,\ell}$ and the $b_{\lambda}$'s basis of the natural $\mathbb{Z}$-form $B$ of the $\mathbb{Z}$-graded basic algebra of $H_{\ell,n}^{(0)}$ is also obtained. As an application, we obtain a second version of Giambelli formula for Schubert classes.

Involutions in Weyl group of type F 4

2017-06-03
Jun Hu , Jing Zhang , Yabo Wu

On the centers of cyclotomic quiver Hecke algebras

2017-01-01
Jun Hu
Let $n\in\mathbb{N}$ and $K$ be any field. For any symmetric generalized Cartan matrix $A$, any $β$ in the positive root lattice with height $n$ and any integral dominant weight $Λ$, … Let $n\in\mathbb{N}$ and $K$ be any field. For any symmetric generalized Cartan matrix $A$, any $β$ in the positive root lattice with height $n$ and any integral dominant weight $Λ$, one can associate a quiver Hecke algebras $R_β(K)$ and its cyclotomic quotient $R_β^Λ(K)$ over $K$. It has been conjectured that the natural map from $R_β(K)$ to $R_β^Λ(K)$ maps the center of $R_β(K)$ surjectively onto the center of $R_β^Λ(K)$. A similar conjecture claims that the center of the affine Hecke algebra of type $A$ maps surjectively onto the center of its cyclotomic quotient---the cyclotomic Hecke algebra $H_n^Λ$ of type $G(\ell,1,n)$ over $K$. In this paper, we prove these two conjectures affirmatively. As a consequence, we show that the center of $H_n^Λ$ is stable under base change and it has dimension equal to the number of $\ell$-partitions of $n$. Finally, as a byproduct, we also verify a conjecture of Shan, Varagnolo and Vasserot on the grading structure of the center of $R_β^Λ(K)$.

Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

2017-01-01
Jun Hu , Ngau Lam
We show that for any singular dominant integral weight $\lambda$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category … We show that for any singular dominant integral weight $\lambda$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category $\mathcal{O}_\lambda^{\mathfrak{p}}$ is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra $B$ is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra $B_\lambda^{\mathfrak{p}}$ of the basic projective-injective module of $\mathcal{O}_\lambda^{\mathfrak{p}}$ is a symmetric algebra.

On the generalised Dipper–James–Murphy conjecture in quantum characteristic 2

2016-05-18
Jun Hu , Kai Zhou , Danni Wang

Fayers' conjecture and the socles of cyclotomic Weyl modules

2016-01-01
Jun Hu , Andrew Mathas
Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by $p$-restricted partitions. We prove an analogue of … Gordon James proved that the socle of a Weyl module of a classical Schur algebra is a sum of simple modules labelled by $p$-restricted partitions. We prove an analogue of this result in the very general setting of "Schur pairs". As an application we show that the socle of a Weyl module of a cyclotomic $q$-Schur algebra is a sum of simple modules labelled by Kleshchev multipartitions and we use this result to prove a conjecture of Fayers that leads to an efficient LLT algorithm for the higher level cyclotomic Hecke algebras of type $A$. Finally, we prove a cyclotomic analogue of the Carter-Lusztig theorem.

Modified affine Hecke algebras and quiver Hecke algebras of type $A$

2016-01-01
Jun Hu , Fang Li
We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type $A$. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We … We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type $A$. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type $A$. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of type $A$ holds if and only if the center conjecture for the cyclotomic Hecke algebra of type $A$ holds.

On involutions in Weyl groups

2016-01-01
Jun Hu , Jing Zhang
Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set … Let $(W,S)$ be a Coxeter system and $\ast$ be an automorphism of $W$ with order $\leq 2$ such that $s^{\ast}\in S$ for any $s\in S$. Let $I_{\ast}$ be the set of twisted involutions relative to $\ast$ in $W$. In this paper we consider the case when $\ast=\text{id}$ and study the braid $I_\ast$-transformations between the reduced $I_\ast$-expressions of involutions. If $W$ is the Weyl group of type $B_n$ or $D_n$, we explicitly describe a finite set of basic braid $I_\ast$-transformations for all $n$ simultaneously, and show that any two reduced $I_\ast$-expressions for a given involution can be transformed into each other through a series of basic braid $I_\ast$-transformations. In both cases, these basic braid $I_\ast$-transformations consist of the usual basic braid transformations plus some natural "right end transformations" and plus exactly one extra transformation. The main result generalizes our previous work for the Weyl group of type $A_{n}$.

On involutions in symmetric groups and a conjecture of Lusztig

2015-11-20
Jun Hu , Jing Zhang
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Seminormal forms and cyclotomic quiver Hecke algebras of type A

2015-07-03
Jun Hu , Andrew Mathas
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Quiver Schur algebras for linear quivers

2015-04-13
Jun Hu , Andrew Mathas
We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras R n Λ of type A when e = 0 (the linear quiver) or e > n . … We define a graded quasi-hereditary covering of the cyclotomic quiver Hecke algebras R n Λ of type A when e = 0 (the linear quiver) or e > n . We prove that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When e = 0 , we show that the Khovanov–Lauda–Rouquier grading on the quiver Hecke algebras is compatible with the Koszul grading on the blocks of parabolic category O Λ given by Backelin, building on the work of Beilinson, Ginzburg and Soergel. As a consequence, e = 0 our cyclotomic quiver Schur algebras are Koszul over fields of characteristic zero. Finally, we give an Lascoux–Leclerc–Thibon-like algorithm for computing the graded decomposition numbers of the cyclotomic quiver Schur algebras in characteristic zero.
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On involutions in symmetric groups and a conjecture of Lusztig

2015-01-01
Jun Hu , Jing Zhang
Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown … Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally endowed with a module structure of the Hecke algebra of $(W, S)$. Lusztig has conjectured that this module is isomorphic to the right ideal of the Hecke algebra (with Hecke parameter $u^2$) associated to $(W,S)$ generated by the element $X_{\emptyset}:=\sum_{w^\ast=w}u^{-\ell(w)}T_w$. In this paper we prove this conjecture in the case when $\ast=\text{id}$ and $W$ is the symmetric group on $n$ letters.
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Extending the Dehn quandle to shears and foliations on the torus

2014-01-01
Reza Chamanara , Jun Hu , Joel Zablow
The Dehn quandle, $Q$, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, $\mathbb {T}^2$, we generalize … The Dehn quandle, $Q$, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, $\mathbb {T}^2$, we generalize this to show the existence of a quandle $\hat Q$ extending $Q$ and whose elements a
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SYMMETRIZERS AND ANTISYMMETRIZERS FOR THE BMW-ALGEBRA

2013-02-01
Richard Dipper , Jun Hu , Friederike Stoll
Let n ∈ ℕ and B n (r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that B n (r, q) has … Let n ∈ ℕ and B n (r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that B n (r, q) has two distinct linear representations generated by two central elements of B n (r, q) called the symmetrizer and antisymmetrizer of B n (r, q). These generate for n ≥ 3 the only one-dimensional two sided ideals of B n (r, q) and generalize the corresponding notion for Hecke algebras of type A. The main result, Theorem 3.1, in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of B n (r, q).
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Seminormal forms and cyclotomic quiver Hecke algebras of type $A$

2013-01-01
Jun Hu , Andrew Mathas
This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying … This paper shows that the cyclotomic quiver Hecke algebras of type $A$, and the gradings on these algebras, are intimately related to the classical seminormal forms. We start by classifying all seminormal bases and then give an explicit "integral" closed formula for the Gram determinants of the Specht modules in terms of the combinatorics which utilizes the KLR gradings. We then use seminormal forms to give a deformation of the KLR algebras of type $A$. This makes it possible to study the cyclotomic quiver Hecke algebras in terms of the semisimple representation theory and seminormal forms. As an application we construct a new distinguished graded cellular basis of the cyclotomic KLR algebras of type $A$.
Coauthor Papers Together
Andrew Mathas 13
Huang Lin 6
Kai Zhou 5
Richard Dipper 5
Zhankui Xiao 5
Yujiao Sun 5
Yinhuo Zhang 4
Stephen Doty 4
Lei Shi 4
Zhekun He 4
Friederike Stoll 3
Ming Fang 2
Wei Xiao 2
Ngau Lam 2
Jing Zhang 2
Jianpan Wang 2
Lei Shi 1
Jun Ding 1
Fang Li 1
Fang Li 1
Zhiqiang Li 1
Jing Zhang 1
Reza Chamanara 1
Danni Wang 1
Jing Zhang 1
Shixuan Wang 1
Yichuan Yang 1
Toshiaki Shoji 1
Salim Rostam 1
Joel Zablow 1
Yabo Wu 1
Huanmei Sun 1
Xiandong Wang 1
Lei Shi 1
Xinfeng Liang 1
Shixuan Wang 1
Jun Ding 1
Xiaolin Shi 1
Kaiming Zhao 1

The Representations of Hecke Algebras of Type An

1995-04-01
G.E Murphy

Representations of Hecke Algebras of General Linear Groups

1986-01-01
Richard Dipper , Gordon James

Cyclotomic q–Schur algebras

1998-11-01
Richard Dipper , Gordon James , Andrew Mathas

On the decomposition numbers of the Hecke algebra of $G(m, 1, n)$

1996-01-01
Susumu Ariki
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A Hecke Algebra of (Z/rZ)Sn and Construction of Its Irreducible Representations

1994-07-01
S. Ariki , Kazuhiko Koike

Cellular algebras

1996-12-01
J. J. Graham , G Lehrer

Graded cellular bases for the cyclotomic Khovanov–Lauda–Rouquier algebras of type A

2010-03-18
Jun Hu , Andrew Mathas
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Hecke Algebras of Type <i>B<sub>n</sub> </i> at Roots of Unity

1995-05-01
Richard Dipper , Gordon James , Eugene Murphy

A diagrammatic approach to categorification of quantum groups I

2009-07-28
Mikhail Khovanov , Aaron D. Lauda
To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript q … To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript q Superscript minus Baseline left-parenthesis German g right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> <mml:mo>−<!-- − --></mml:mo> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U^-_q(\mathfrak {g})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {g}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the Kac-Moody Lie algebra associated with the graph.
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Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group

1999-09-07
Andrew Mathas
The Iwahori-Hecke algebra of the symmetric group Cellular algebras The modular representation theory of $\mathcal {H}$ The $q$-Schur algebra The Jantzen sum formula and the blocks of $\mathcal H$ Branching … The Iwahori-Hecke algebra of the symmetric group Cellular algebras The modular representation theory of $\mathcal {H}$ The $q$-Schur algebra The Jantzen sum formula and the blocks of $\mathcal H$ Branching rules, canonical bases and decomposition matrices Appendix A. Finite dimensional algebras over a field Appendix B. Decomposition matrices Appendix C. Elementary divisors of integral Specht modules Index of notation References Index.

Matrix units and generic degrees for the Ariki–Koike algebras

2004-09-18
Andrew Mathas
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Blocks and Idempotents of Hecke Algebras of General Linear Groups

1987-01-01
Richard Dipper , Gordon James

Introduction to Quantum Groups

2010-01-01
G. Lusztig
According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping … According to Drinfeld, a quantum group is the same as a Hopf algebra. This includes as special cases, the algebra of regular functions on an algebraic group and the enveloping algebra of a semisimple

The number of simple modules of the Hecke algebras of type G ( r ,1, n )

2000-03-01
Susumu Ariki , Andrew Mathas

Modular representations of Hecke algebras of type G(p,p,n)

2004-02-24
Jun Hu

Morita equivalences of Ariki-Koike algebras

2002-07-01
Richard Dipper , Andrew Mathas
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Representations of Coxeter groups and Hecke algebras

1979-06-01
David Kazhdan , G. Lusztig

A Morita equivalence theorem for Hecke algebra ℋ q (D n ) when n is even

2002-08-01
Jun Hu

On the Semi-simplicity of the Hecke Algebra of (Z/rZ) ≀ Sn

1994-10-01
S. Ariki

Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras

2009-06-04
Jonathan Brundan , Alexander Kleshchev
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Centers of degenerate cyclotomic Hecke algebras and parabolic category 𝒪

2008-07-29
Jonathan Brundan
We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript d Baseline left-parenthesis l right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo … We prove that the center of each degenerate cyclotomic Hecke algebra associated to the complex reflection group of type<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript d Baseline left-parenthesis l right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>d</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>l</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">B_d(l)</mml:annotation></mml:semantics></mml:math></inline-formula>consists of symmetric polynomials in its commuting generators. The classification of the blocks of the degenerate cyclotomic Hecke algebras is an easy consequence. We then apply Schur-Weyl duality for higher levels to deduce analogous results for parabolic category<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi></mml:mrow><mml:annotation encoding="application/x-tex">\mathcal O</mml:annotation></mml:semantics></mml:math></inline-formula>for the Lie algebra<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l Subscript n Baseline left-parenthesis double-struck upper C right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="fraktur">g</mml:mi><mml:mi mathvariant="fraktur">l</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathfrak {gl}_n(\mathbb {C})</mml:annotation></mml:semantics></mml:math></inline-formula>.

Schur–Weyl duality for higher levels

2008-09-27
Jonathan Brundan , Alexander Kleshchev
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Graded decomposition numbers for cyclotomic Hecke algebras

2009-07-19
Jonathan Brundan , Alexander Kleshchev
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Crystal bases and simple modules for Hecke algebra of type Dn

2003-06-30
Jun Hu
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Blocks of cyclotomic Hecke algebras

2007-06-23
SinĂŠad Lyle , Andrew Mathas

Quantum Weyl Reciprocity and Tilting Modules

1998-07-01
Jie Du , Brian Parshall , Leonard L. Scott

On the classification of simple modules for cyclotomic Hecke algebras of type $G(m,1,n)$ and Kleshchev multipartitions

2001-12-01
Susumu Ariki
We give a proof of a conjecture that Kleshchev multipartitions are those partitions which parametrize non-zero simple modules obtained as factor modules of Specht modules by their own radicals. We give a proof of a conjecture that Kleshchev multipartitions are those partitions which parametrize non-zero simple modules obtained as factor modules of Specht modules by their own radicals.

Polynomial representations of GLn

1981-01-01
James Green
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A characteristic free approach to invariant theory

1976-09-01
Corrado De Concini , Claudio Procesi

Representations of Hecke algebras of type Bn

1992-03-01
Richard Dipper , Gordon James

Representations of Hecke Algebras of Type Dn

1994-10-01
C. A. Pallikaros

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

1996-11-01
Alain Lascoux , Bernard Leclerc , Jean‐Yves Thibon
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2-Kac-Moody algebras

2008-12-30
RaphaĂŤl Rouquier
We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties … We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations.

Cellular bases for the Brauer and Birman–Murakami–Wenzl algebras

2004-09-17
John Enyang

On Algebras Which are Connected with the Semisimple Continuous Groups

1937-10-01
Richard Brauer

Braids, link polynomials and a new algebra

1989-01-01
Joan S. Birman , Hans Wenzl
A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on <inline-formula content-type="math/mathml"> <mml:math … A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones’ algebras and to Brauer’s centralizer algebras.
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Quiver Hecke Algebras and 2-Lie Algebras

2012-05-03
RaphaĂŤl Rouquier
We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, … We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.
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The Jantzen sum formula for cyclotomic 𝑞–Schur algebras

2000-06-14
Gordon James , Andrew Mathas
The cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for … The cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen’s sum formula for the cyclotomic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.

The Kauffman polynomial of links and representation theory

1987-12-01
Jun Murakami

Centralizer Coalgebras, FRT-Construction, and Symplectic Monoids

2001-10-01
Sebastian Oehms

Infinite-Dimensional Lie Algebras

1990-09-20
Victor G. Kač
This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is … This is the third, substantially revised edition of this important monograph. The book is concerned with Kac–Moody algebras, a particular class of infinite-dimensional Lie algebras, and their representations. It is based on courses given over a number of years at MIT and in Paris, and is sufficiently self-contained and detailed to be used for graduate courses. Each chapter begins with a motivating discussion and ends with a collection of exercises, with hints to the more challenging problems.
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Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

2007-09-25
Richard Dipper , Stephen Doty , Jun Hu
In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the … In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field $K$. We show that the natural homomorphism from the Brauer algebra $B_{n}(-2m)$ to the endomorphism algebra of the tensor space $(K^{2m})^{\otimes n}$ as a module over the symplectic similitude group $GSp_{2m}(K)$ (or equivalently, as a module over the symplectic group $Sp_{2m}(K)$) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for $GSp_{2m}(K)$ to the endomorphism algebra of $(K^{2m})^{\otimes n}$ as a module over $B_{n}(-2m)$, is derived as an easy consequence of S. Oehms's results [S. Oehms, J. Algebra (1) 244 (2001), 19–44].
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Cyclotomic Nazarov-Wenzl Algebras

2006-06-01
Susumu Ariki , Andrew Mathas , Hebing Rui
Abstract Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra , in his study of the Brauer algebras. In this paper we study certain “cyclotomic … Abstract Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra , in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r n ( 2n −1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.
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Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra

1990-01-01
G. Lusztig
0.1. An important role in the theory of modular representations is played by certain finite dimensional Hopf algebras u over Fp (the field with p elements, p = prime). Originally, … 0.1. An important role in the theory of modular representations is played by certain finite dimensional Hopf algebras u over Fp (the field with p elements, p = prime). Originally, u was defined (Curtis [3]) as the restricted enveloping algebra of a simple Lie algebra over Fp For our purposes, it will be more convenient to define u as follows. Let us fix an indecomposable positive definite symmetric Cartan matrix
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𝑞-tensor space and 𝑞-Weyl modules

1991-01-01
Richard Dipper , Gordon James
We obtain the irreducible representations of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra, motivated by the fact that these representations give all the … We obtain the irreducible representations of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Schur algebra, motivated by the fact that these representations give all the irreducible representations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G upper L Subscript n Baseline left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G{L_n}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the nondescribing characteristic. The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction.
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A q-analogue of U(g[(N+1)), Hecke algebra, and the Yang-Baxter equation

1986-04-01
Michio Jimbo

On the Structure of Brauer's Centralizer Algebras

1988-07-01
Hans Wenzl

Symmetric Cyclotomic Hecke Algebras

1998-07-01
Gunter Malle , Andrew Mathas

Koszul Duality Patterns in Representation Theory

1996-01-01
Alexander Beilinson , Victor Ginzburg , Wolfgang Soergel
The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper … The aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of Koszul duality to representation theory. The paper consists of three parts relatively independent of each other. The first part gives a reasonably selfcontained introduction to Koszul rings and Koszul duality. Koszul rings are certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-graded rings with particularly nice homological properties which involve a kind of duality. Thus, to a Koszul ring one associates naturally the dual Koszul ring. The second part is devoted to an application to representation theory of semisimple Lie algebras. We show that the block of the Bernstein-Gelfand-Gelfand category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that corresponds to any fixed central character is governed by the Koszul ring. Moreover, the dual of that ring governs a certain subcategory of the category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> again. This generalizes the selfduality theorem conjectured by Beilinson and Ginsburg in 1986 and proved by Soergel in 1990. In the third part we study certain categories of mixed perverse sheaves on a variety stratified by affine linear spaces. We provide a general criterion for such a category to be governed by a Koszul ring. In the flag variety case this reduces to the setup of part two. In the more general case of affine flag manifolds and affine Grassmannians the criterion should yield interesting results about representations of quantum groups and affine Lie algebras.
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Representation Theory of a Hecke Algebra of G(r, p, n)

1995-10-01
S. Ariki