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 $\mathscr{R}_{\beta }^{\Lambda }$.
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.
We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type C. This quiver provides an efficient method to obtain …
We first investigate a connected quiver consisting of all dominant maximal weights for an integrable highest weight module in affine type C. This quiver provides an efficient method to obtain all dominant maximal weights. Then, we completely determine the representation type of cyclotomic Khovanov-Lauda-Rouquier algebras of arbitrary level in affine type C, by using the quiver we construct. We also determine the Morita equivalence classes and graded decomposition matrices of certain representation-finite and tame cyclotomic KLR algebras.
Let $\ell\in\mathbb{N}$ with $\ell>2$ 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 $\ell>2$ 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.
In this paper,we continuously explore the equivalent design of the non-standard dices.We prove that there are exactly two pairs of type one,two pairs type two and three pairs of type …
In this paper,we continuously explore the equivalent design of the non-standard dices.We prove that there are exactly two pairs of type one,two pairs type two and three pairs of type three non-standard dices which are equivalent in throw effect to a pair of standard cubic dice respectively.
We apply Lusztig’s theory of cells and asymptotic algebras to the Iwahori–Hecke algebra of a finite Weyl group extended by a group of graph automorphisms. This yields general results about …
We apply Lusztig’s theory of cells and asymptotic algebras to the Iwahori–Hecke algebra of a finite Weyl group extended by a group of graph automorphisms. This yields general results about splitting fields (extending earlier results by Digne–Michel) and decomposition matrices (generalizing earlier results by the author). Our main application is to establish an explicit formula for the number of simple modules in type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript n"> <mml:semantics> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">D_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (except in characteristic <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>), using the known results about type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> due to Dipper, James, and Murphy and Ariki and Mathas.
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.
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.
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.
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>.
0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s …
0.1. Let H,o be an affine Hecke algebra with parameter v0 E C* assumed to be of infinite order. (The basis elements Ts E H,o corresponding to simple reflections s satisfy (Ts + l)(Ts v2c(s)) = 0, where C(S) E N depend on s and are subject only to c(s) = c(s') whenever s, s are conjugate in the affine Weyl group.) Such Hecke algebras appear naturally in the representation theory of semisimple p-adic groups, and understanding their representation theory is a question of considerable interest. Consider the special where c(s) is independent of s and the coroots generate a direct summand. In this special case, the question above has been studied in [1] and a classification of the simple modules was obtained. The approach of [1] was based on equivariant K-theory. This approach can be attempted in the general case (some indications are given in [5, 0.3]), but there appear to be some serious difficulties in carrying it out.
In [BK], Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke–Clifford superalgebras and its finite-dimensional “cyclotomic” quotients are controlled by the Lie theory of …
In [BK], Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke–Clifford superalgebras and its finite-dimensional “cyclotomic” quotients are controlled by the Lie theory of type A_{2l}^{(2)} when the quantum parameter q is a primitive (2l + 1) -th root of unity. We show that similar theorems hold when q is a primitive 4l -th root of unity by replacing the Lie theory of type A_{2l}^{(2)} with that of D_l^{(2)} .
We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety.As …
We realize the crystal associated to the quantized enveloping algebras with a symmetric generalized Cartan matrix as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety.As a by-product, we give a counterexample to the conjecture of Kazhdan-Lusztig on the irreducibility of the characteristic variety of the intersection cohomology sheaves associated with the Schubert cells of type A and also to the similar problem asked by Lusztig on the characteristic variety of the perverse sheaves corresponding to canonical bases.
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), 7â20]. The separated case was completed in [J. Hu, J. Algebra 274 (2004), 446â490]. Furthermore, we use Naito and Sagakiâs work [S. Naito & D. Sagaki, J. Algebra 251, (2002) 461â474] on LakshmibaiâSeshadri 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 $G(2,2,n)$).
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.
Abstract We prove that, for simple modules $M$ and $N$ over a quantum affine algebra, their tensor product $M\otimes N$ has a simple head and a simple socle if $M\otimes …
Abstract We prove that, for simple modules $M$ and $N$ over a quantum affine algebra, their tensor product $M\otimes N$ has a simple head and a simple socle if $M\otimes M$ is simple. A similar result is proved for the convolution product of simple modules over quiver Hecke algebras.
We give an Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{\Lambda _0}(\beta )$ of affine type $D^{(2)}_{\ell +1}$, which tells their representation type. If $R^{\Lambda _0}(\beta )$ is …
We give an Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{\Lambda _0}(\beta )$ of affine type $D^{(2)}_{\ell +1}$, which tells their representation type. If $R^{\Lambda _0}(\beta )$ is not of wild representation type, we may compute its stable Auslander-Reiten quiver.
We give a graded dimension formula described in terms of combinatorics of Young diagrams and a simple criterion to determine the representation type for the finite quiver Hecke algebras of …
We give a graded dimension formula described in terms of combinatorics of Young diagrams and a simple criterion to determine the representation type for the finite quiver Hecke algebras of type $C_{\ell}^{(1)}$.
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.
We categorify one-half of the quantum group associated to an arbitrary Cartan datum.
We categorify one-half of the quantum group associated to an arbitrary Cartan datum.
In this paper we derive a closed formula for the $(\mathbb{Z}\times\mathbb{Z}_2)$-graded dimension of the cyclotomic quiver Hecke superalgebra $R^\Lambda(\beta)$ associated to an {\it arbitrary} Cartan superdatum $(A,P,\Pi,\Pi^\vee)$, polynomials $(Q_{i,j}({\rm x}_1,{\rm …
In this paper we derive a closed formula for the $(\mathbb{Z}\times\mathbb{Z}_2)$-graded dimension of the cyclotomic quiver Hecke superalgebra $R^\Lambda(\beta)$ associated to an {\it arbitrary} Cartan superdatum $(A,P,\Pi,\Pi^\vee)$, polynomials $(Q_{i,j}({\rm x}_1,{\rm x}_2))_{i,j\in I}$, $\beta\in Q_n^+$ and $\Lambda\in P^+$. As applications, we obtain a necessary and sufficient condition for which $e(\nu)\neq 0$ in $R^\Lambda(\beta)$. We construct an explicit monomial basis for the bi-weight space $e(\widetilde{\nu})R^\Lambda({\beta})e(\widetilde{\nu})$, where $\widetilde{\nu}$ is a certain specific $n$-tuple defined in (1.4). In particular, this gives rise to a monomial basis for the cyclotomic odd nilHecke algebra. Finally, we consider the case when $\beta=\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}$ with $\alpha_1,\cdots,\alpha_n$ distinct. We construct an explicit monomial basis of $R^\Lambda(\beta)$ and show that it is indecomposable in this case.
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 $m<n$.
We define and study sl 2 -categorifications on abelian categories.We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection.We construct …
We define and study sl 2 -categorifications on abelian categories.We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection.We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this implies Broué's abelian defect group conjecture for symmetric groups.We give similar results for general linear groups over finite fields.The constructions extend to cyclotomic Hecke algebras.We also construct categorifications for category O of gl n (C) and for rational representations of general linear groups over Fp , where we deduce that two blocks corresponding to weights with the same stabilizer under the dot action of the affine Weyl group have equivalent derived (and homotopy) categories, as conjectured by Rickard.
Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This paper undertakes a systematic study of these algebras culminating in the construction of homogeneous affine cellular bases in finite or …
Weighted KLRW algebras are diagram algebras generalizing KLR algebras. This paper undertakes a systematic study of these algebras culminating in the construction of homogeneous affine cellular bases in finite or affine types A and C, which immediately gives cellular bases for the cyclotomic quotients of these algebras. In addition, we construct subdivision homomorphisms that relate weighted KLRW algebras for different quivers.