Symmetric Groups and Schur Algebras

1998 ◽  
pp. 91-102 ◽  
Author(s):  
Gordon James
Keyword(s):  
Symmetric Groups ◽  
Schur Algebras

scholarly journals Dominant and global dimension of blocks of quantised Schur algebras

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).

scholarly journals On Schur algebras, Ringel duality and symmetric groups

2002 ◽  
Vol 169 (2-3) ◽  
pp. 175-199 ◽  
Author(s):  
K. Erdmann ◽  
A. Henke
Keyword(s):  
Symmetric Groups ◽  
Schur Algebras

scholarly journals Young modules for symmetric groups

2001 ◽  
Vol 71 (2) ◽  
pp. 201-210 ◽  
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.

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

2021 ◽  
pp. 1-36
Author(s):  
JIE DU ◽  
QIANG FU ◽  
YANAN LIN
Keyword(s):  
Symmetric Groups ◽  
Loop Algebra ◽  
Natural Representation ◽  
Schur Algebras ◽  
Tensor Spaces ◽  
Weyl Duality ◽  
Geometric Setting

Abstract In [2], Beilinson–Lusztig–MacPherson (BLM) gave a beautiful realization for quantum $\mathfrak {gl}_n$ via a geometric setting of quantum Schur algebras. We introduce the notion of affine Schur superalgebras and use them as a bridge to link the structure and representations of the universal enveloping superalgebra ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ of the loop algebra $\widehat {\mathfrak {gl}}_{m|n}$ of ${\mathfrak {gl}}_{m|n}$ with those of affine symmetric groups ${\widehat {{\mathfrak S}}_{r}}$ . Then, we give a BLM type realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ via affine Schur superalgebras. The first application of the realization of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ is to determine the action of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ on tensor spaces of the natural representation of $\widehat {\mathfrak {gl}}_{m|n}$ . These results in epimorphisms from $\;{\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ to affine Schur superalgebras so that the bridging relation between representations of ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ and ${\widehat {{\mathfrak S}}_{r}}$ is established. As a second application, we construct a Kostant type $\mathbb Z$ -form for ${\mathcal U}_{\mathbb Q}(\widehat {\mathfrak {gl}}_{m|n})$ whose images under the epimorphisms above are exactly the integral affine Schur superalgebras. In this way, we obtain essentially the super affine Schur–Weyl duality in arbitrary characteristics.

scholarly journals HOMOMORPHISMS AND HIGHER EXTENSIONS FOR SCHUR ALGEBRAS AND SYMMETRIC GROUPS

2005 ◽  
Vol 04 (06) ◽  
pp. 645-670 ◽  
Author(s):  
ANTON COX ◽  
ALISON PARKER
Keyword(s):  
Symmetric Groups ◽  
Hecke Algebra ◽  
Type A ◽  
Schur Algebras ◽  
Ext 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.

STRATIFYING SYSTEMS, FILTRATION MULTIPLICITIES AND SYMMETRIC GROUPS

2005 ◽  
Vol 04 (05) ◽  
pp. 551-555 ◽  
Author(s):  
KARIN ERDMANN
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Schur Algebras

We show that the theorem by Hemmer and Nakano, on uniqueness of Specht filtration multiplicities, can be proved working entirely with representations of symmetric groups, or Hecke algebras. Furthermore, we give a new proof that Schur algebras are quasi-hereditary provided the characteristic of the field is at least 5. Our tools are some more general results on stratifying systems.

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

2004 ◽  
Vol 7 (1) ◽  
pp. 67-99 ◽  
Author(s):  
Stephen R. Doty ◽  
Karin Erdmann ◽  
Daniel K. Nakano
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Schur Algebras
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