HOMOMORPHISMS AND HIGHER EXTENSIONS FOR SCHUR ALGEBRAS AND SYMMETRIC GROUPS

This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

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Evaluation of Nonsymmetric Macdonald Superpolynomials at Special Points

Symmetry ◽

10.3390/sym13050779 ◽

2021 ◽

Vol 13 (5) ◽

pp. 779

Author(s):

Charles F. Dunkl

Keyword(s):

Preceding Paper ◽

Hecke Algebra ◽

Type A ◽

Macdonald Polynomials ◽

Special Points ◽

Two Parameters

In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type A (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to define nonsymmetric Macdonald superpolynomials. These polynomials depend on two parameters q,t and are defined by means of a Yang–Baxter graph. The present paper determines the values of a subclass of the polynomials at the special points 1,t,t2,… or 1,t−1,t−2,…. The arguments use induction on the degree and computations with products of generators of the Hecke algebra. The resulting formulas involve q,t-hook products. Evaluations are also found for Macdonald superpolynomials having restricted symmetry and antisymmetry properties.

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Dominant and global dimension of blocks of quantised Schur algebras

Mathematische Zeitschrift ◽

10.1007/s00209-021-02792-w ◽

2021 ◽

Author(s):

Ming Fang ◽

Wei Hu ◽

Steffen Koenig

Keyword(s):

Hecke Algebras ◽

Symmetric Groups ◽

Global Dimension ◽

Group Algebras ◽

Dominant Dimension ◽

Schur Algebras ◽

Homological Dimensions ◽

Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

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Jones' trace function on the Hecke algebra of symmetric groups

Banach Center Publications ◽

10.4064/-26-2-425-432 ◽

1990 ◽

Vol 26 (2) ◽

pp. 425-432

Author(s):

T. A. Springer

Keyword(s):

Symmetric Groups ◽

Hecke Algebra ◽

Trace Function

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Fixed-point functors for symmetric groups and Schur algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2004.04.014 ◽

2004 ◽

Vol 280 (1) ◽

pp. 295-312 ◽

Cited By ~ 5

Author(s):

David J. Hemmer

Keyword(s):

Fixed Point ◽

Symmetric Groups ◽

Schur Algebras

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On Schur algebras, Ringel duality and symmetric groups

Journal of Pure and Applied Algebra ◽

10.1016/s0022-4049(01)00071-8 ◽

2002 ◽

Vol 169 (2-3) ◽

pp. 175-199 ◽

Cited By ~ 6

Author(s):

K. Erdmann ◽

A. Henke

Keyword(s):

Symmetric Groups ◽

Schur Algebras

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Young modules for symmetric groups

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700002846 ◽

2001 ◽

Vol 71 (2) ◽

pp. 201-210 ◽

Cited By ~ 23

Author(s):

Karin Erdmann

Keyword(s):

Representation Theory ◽

Finite Groups ◽

Symmetric Groups ◽

General Linear ◽

Linear Groups ◽

Schur Algebras ◽

General Linear Groups ◽

Green Correspondence

AbstractLet K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

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