Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur-Weyl duality is the key for an …
Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur-Weyl duality is the key for an equivalence between both categories.
Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur-Weyl duality is the key for an …
Polynomial representations of general linear groups and modules over Schur algebras are compared. We work over an arbitrary commutative ring and show that Schur-Weyl duality is the key for an equivalence between both categories.
In this paper we investigate which Sheffer polynomials can be represented as moments of convolution semigroups of probability measures. We also obtain general integral representations for shift‐invariant operators and for …
In this paper we investigate which Sheffer polynomials can be represented as moments of convolution semigroups of probability measures. We also obtain general integral representations for shift‐invariant operators and for umbral operators. As a corollary, we obtain new proofs for representation theorems for Sheffer polynomials due to Sheffer and Thorne.
We give a polynomial basis of each irreducible representation of the Hecke algebra, as well as an adjoint basis. Decompositions in these bases are obtained by mere specializations.
We give a polynomial basis of each irreducible representation of the Hecke algebra, as well as an adjoint basis. Decompositions in these bases are obtained by mere specializations.
A derivation for the kernel of the irreducible representation T (λ) of the general linear group GL n (C) is given. This is then applied to the problem of determining …
A derivation for the kernel of the irreducible representation T (λ) of the general linear group GL n (C) is given. This is then applied to the problem of determining necessary and sufficient conditions under which T (λ)(A) = T (λ)(B), where A and B are linear transformations, not necessarily invertible. Finally, conditions are obtained under which normality of T (λ)(A) implies normality of A.
It is shown that the polynomial bases for representations of a semisimple Lie algebra are just the various terms of typical concomitants of the Lie algebra. Consequently, the construction of …
It is shown that the polynomial bases for representations of a semisimple Lie algebra are just the various terms of typical concomitants of the Lie algebra. Consequently, the construction of polynomial bases reduces to a problem in the theory of invariants.
viii, 50 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
viii, 50 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number.
We develop a modular version of a super analogue of Schur's duality by means of supergroups, rather than Lie superalgebras, in preparation for a geometric analogue.
We develop a modular version of a super analogue of Schur's duality by means of supergroups, rather than Lie superalgebras, in preparation for a geometric analogue.
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.
Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing …
Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.
We introduce the alternating Schur algebra AS F (n, d) as the commutant of the action of the alternating group A d on the d-fold tensor power of an n-dimensional …
We introduce the alternating Schur algebra AS F (n, d) as the commutant of the action of the alternating group A d on the d-fold tensor power of an n-dimensional F -vector space.When F has characteristic different from 2, we give a basis of AS F (n, d) in terms of bipartite graphs, and a graphical interpretation of the structure constants.We introduce the abstract Koszul duality functor on modules for the even part of any Z/2Z-graded algebra.The algebra AS F (n, d) is Z/2Z-graded, having the classical Schur algebra S F (n, d) as its even part.This leads to an approach to Koszul duality for S F (n, d)-modules that is amenable to combinatorial methods.We characterize the category of AS F (n, d)-modules in terms of S F (n, d)-modules and their Koszul duals.We use the graphical basis of AS F (n, d) to study the dependence of the behavior of derived Koszul duality on n and d.
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper Z With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\hat {\mathbb {Z}}_p</mml:annotation></mml:semantics></mml:math></inline-formula>be the ring of<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>-adic integers. We prove in the …
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper Z With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\hat {\mathbb {Z}}_p</mml:annotation></mml:semantics></mml:math></inline-formula>be the ring of<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>-adic integers. We prove in the present paper that the category of polynomial functors from finitely generated free abelian groups to<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove double-struck upper Z With caret Subscript p"><mml:semantics><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mover><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mo stretchy="false">^</mml:mo></mml:mover></mml:mrow><mml:mi>p</mml:mi></mml:msub><mml:annotation encoding="application/x-tex">\hat {\mathbb {Z}}_p</mml:annotation></mml:semantics></mml:math></inline-formula>-modules of degree at most<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>is equivalent to the category of modules over a particularly well understood ring, called Green order. This case was conjectured by Yuri Drozd.
Work of Clifford, Munn and Ponizovskiĭ parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations …
Work of Clifford, Munn and Ponizovskiĭ parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing $0$-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskiĭ result based on a lemma of J. A. Green, which allows us to circumvent the theory of $0$-simple semigroups. A novelty of this approach is that it works over any base ring.
Abstract A sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard …
Abstract A sequence of finite-dimensional quotients of affine Hecke algebras is studied. Each element of the sequence is constructed so as to have a weight space labelling scheme for Specht⁄standard modules. As in the weight space formalism of algebraic Lie theory, there is an action of an affine reflection group on this weight space that fixes the set of labelling weights. A linkage principle is proved in each case. Further, it is shown that the simplest non-trivial example may essentially be identified with the blob algebra (a physically motivated quasihereditary algebra whose representation theory is very well understood by Lie-theory-like methods). An extended role is hence proposed for Soergel's tilting algorithm, away from its algebraic Lie theory underpinning, in determining the simple content of standard modules for these algebras. This role is explicitly verified in the blob algebra case. A tensor space representation of the blob algebra is constructed, as a candidate for a full tilting module (subsequently proven to be so in a paper by Martin and Ryom-Hansen), further evidencing the extended utility of Lie-theoretic methods. Possible generalisations of this representation to other elements of the sequence are discussed.
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary …
This paper reports some advances in the study of the symplectic blob algebra. We find a presentation for this algebra. We find a minimal poset for this as a quasi-hereditary algebra. We discuss how to reduce the number of parameters defining the algebra from 6 to 4 (or even 3) without loss of representation theoretic generality. We then find some non-semisimple specialisations by calculating Gram determinants for certain cell modules (or standard modules) using the good parametrisation defined. We finish by considering some quotients of specialisations of the symplectic blob algebra which are isomorphic to Temperley--Lieb algebras of type $A$.
Abstract In this paper, the Brauer trees are completed for the sporadic simple Lyons group Ly in characteristics 37 and 67. The results are obtained using tools from computational representation …
Abstract In this paper, the Brauer trees are completed for the sporadic simple Lyons group Ly in characteristics 37 and 67. The results are obtained using tools from computational representation theory—in particular, a new condensation technique—and with the assistance of the computer algebra systems MeatAxe and GAP.
We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative …
We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.
The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested …
The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for $GL_n(k)$. This paper is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology. Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for $GL_n(k)$, these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of $GL_n(k)$-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be self-dual trivial source modules.
Let $m, n\in{\mathbb N}$. In this paper we study the right permutation action of the symmetric group ${\mathfrak S}_{2n}$ on the set of all the Brauer $n$-diagrams. A new basis …
Let $m, n\in{\mathbb N}$. In this paper we study the right permutation action of the symmetric group ${\mathfrak S}_{2n}$ on the set of all the Brauer $n$-diagrams. A new basis for the free ${\mathbb Z}$-module ${\mathfrak B}_n$ spanned by these Brauer $n$-diagrams is constructed, which yields Specht filtrations for ${\mathfrak B}_n$. For any $2m$-dimensional vector space $V$ over a field of arbitrary characteristic, we give an explicit and characteristic free description of the annihilator of the $n$-tensor space $V^{\otimes n}$ in the Brauer algebra ${\mathfrak B}_n(-2m)$. In particular, we show that it is a ${\mathfrak S}_{2n}$-submodule of ${\mathfrak B}_n(-2m)$.
We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these …
We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola's results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao's algebra, which is related to the Mantaci-Reutenauer descent algebra.
Work of Clifford, Munn and Ponizovski{\uı} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations …
Work of Clifford, Munn and Ponizovski{\uı} parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovski{\uı} result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.
We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, the case $l=1$ recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group …
We extend Schur-Weyl duality to an arbitrary level $l \geq 1$, the case $l=1$ recovering the classical duality between the symmetric and general linear groups. In general, the symmetric group is replaced by the degenerate cyclotomic Hecke algebra over $\C$ parametrized by a dominant weight of level $l$ for the root system of type $A_\infty$. As an application, we prove that the degenerate analogue of the quasi-hereditary cover of the cyclotomic Hecke algebra constructed by Dipper, James and Mathas is Morita equivalent to certain blocks of parabolic category $\mathcal{O}$ for the general linear Lie algebra.
The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences …
The recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor. The method is hence adapted to treat sequences that are not necessarily towers by inclusion, such as symplectic blob algebras (diagram algebra quotients of the type-$\hati{C}$ Hecke algebras). By carefully reviewing the diagram algebra construction, we find a new set of functors interrelating module categories of ordinary blob algebras (diagram algebra quotients of the type-${B}$ Hecke algebras) at {\em different} values of the algebra parameters. We show that these functors generalise to determine the structure of symplectic blob algebras, and hence of certain two-boundary Temperley-Lieb algebras arising in Statistical Mechanics. We identify the diagram basis with a cellular basis for each symplectic blob algebra, and prove that these algebras are quasihereditary over a field for almost all parameter choices, and generically semisimple. (That is, we give bases for all cell and standard modules.)
Motivated by representation theory we exhibit an interior structure to Catalan sequences and many generalisations thereof. Certain of these coincide with well known (but heretofore isolated) structures. The remainder are …
Motivated by representation theory we exhibit an interior structure to Catalan sequences and many generalisations thereof. Certain of these coincide with well known (but heretofore isolated) structures. The remainder are new.
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.
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.
Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for …
Let p be a prime number. We compute the Yoneda extension algebra of $GL_2$ over an algebraically closed field of characteristic p by developing a theory of Koszul duality for a certain class of 2-functors, one of which controls the category of rational representations of $GL_2$ over such a field.
The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of …
The affine Schur algebra $\widetilde{S}(n,r)$ (of type A) over a field $K$ is defined to be the endomorphism algebra of the tensor space over the extended affine Weyl group of type $A_{r-1}$. By the affine Schur-Weyl duality it is isomorphic to the image of the representation map of the $\mathcal{U}(\hat{\mathfrak{gl}}_{n})$ action on the tensor space when $K$ is the field of complex numbers. We show that $\widetilde{S}(n,r)$ can be defined in another two equivalent ways. Namely, it is the image of the representation map of the semigroup algebra $K\widetilde{GL}_{n,a}$ (defined in Section \ref{S:semigroups}) action on the tensor space and it equals to the 'dual' of a certain formal coalgebra related to this semigroup. By these approaches we can show many relations between different Schur algebras and affine Schur algebras and reprove one side of the affine Schur-Weyl duality.
We show that the projective module $P$ over a cellular algebra is injective if and only if the socle of $P$ coincides with the top of $P$, and this is …
We show that the projective module $P$ over a cellular algebra is injective if and only if the socle of $P$ coincides with the top of $P$, and this is also equivalent to the condition that the $m$th socle layer of $P$ is isomorphic to the $m$th radical layer of $P$ for each positive integer $m$. This eases the process of determining the Loewy series of the projective-injective modules over cellular algebras.