Cyclotomic đť‘ž-Schur algebras and Schur-Weyl duality

2006 â—˝  
pp. 133-155 â—˝  
Author(s):  
Zongzhu Lin â—˝  
Hebing Rui
Keyword(s):  
Schur Algebras â—˝  
Weyl Duality

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

2021 â—˝  
Author(s):  
Ming Fang â—˝  
Wei Hu â—˝  
Steffen Koenig
Keyword(s):  
Hecke Algebras â—˝  
Symmetric Groups â—˝  
Global Dimension â—˝  
Group Algebras â—˝  
Dominant Dimension â—˝  
Schur Algebras â—˝  
Homological Dimensions â—˝  
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 Schur–Weyl duality for infinitesimal q-Schur algebras sq(2,r)1

Journal of Algebra â—˝  
2008 â—˝  
Vol 320 (3) â—˝  
pp. 1099-1114 â—˝  
Author(s):  
Karin Erdmann â—˝  
Qiang Fu
Keyword(s):  
Schur Algebras â—˝  
Weyl Duality

scholarly journals The Integral Quantum Loop Algebra of $\mathfrak {gl}_{n}$

10.1093/imrn/rnx300 â—˝  
2018 â—˝  
Vol 2019 (20) â—˝  
pp. 6179-6215 â—˝  
Author(s):  
Jie Du â—˝  
Qiang Fu
Keyword(s):  
Canonical Basis â—˝  
Alternative Algebra â—˝  
Integral Form â—˝  
Loop Algebra â—˝  
Algebra Structure â—˝  
Schur Algebras â—˝  
Canonical Bases â—˝  
Weyl Duality â—˝  
Integral Quantum

Abstract We will construct the Lusztig form for the quantum loop algebra of $\mathfrak {gl}_{n}$ by proving the conjecture [4, 3.8.6] and establish partially the Schur–Weyl duality at the integral level in this case. We will also investigate the integral form of the modified quantum affine $\mathfrak {gl}_{n}$ by introducing an affine stabilisation property and will lift the canonical bases from affine quantum Schur algebras to a canonical basis for this integral form. As an application of our theory, we will also discuss the integral form of the modified extended quantum affine $\mathfrak {sl}_{n}$ and construct its canonical basis to provide an alternative algebra structure related to a conjecture of Lusztig in [29, §9.3], which has been already proved in [34].

A REALIZATION OF THE ENVELOPING SUPERALGEBRA

10.1017/nmj.2021.11 â—˝  
2021 â—˝  
pp. 1-36
Author(s):  
JIE DU â—˝  
QIANG FU â—˝  
YANAN LIN
Keyword(s):  
Symmetric Groups â—˝  
Loop Algebra â—˝  
Natural Representation â—˝  
Schur Algebras â—˝  
Tensor Spaces â—˝  
Weyl Duality â—˝  
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.

Schur–Weyl duality and centers of quantum Schur algebras

2021 â—˝  
Vol 225 (5) â—˝  
pp. 106594
Author(s):  
Qiang Fu
Keyword(s):  
Schur Algebras â—˝  
Weyl Duality

scholarly journals Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

2008 â—˝  
Vol 360 (01) â—˝  
pp. 189-214 â—˝  
Author(s):  
Richard Dipper â—˝  
Stephen Doty â—˝  
Jun Hu
Keyword(s):  
Schur Algebras â—˝  
Weyl Duality â—˝  
Brauer Algebras

scholarly journals Exponent Reduction for Projective Schur Algebras

Journal of Algebra â—˝  
2001 â—˝  
Vol 239 (1) â—˝  
pp. 356-364 â—˝  
Author(s):  
Eli Aljadeff â—˝  
Jack Sonn
Keyword(s):  
Schur Algebras

Schur algebras

10.1007/bfb0061705 â—˝  
1974 â—˝  
pp. 4-13
Author(s):  
Toshihiko Yamada
Keyword(s):  
Schur Algebras

scholarly journals A q-Analogue of Derivations on the Tensor Algebra and the q-Schur–Weyl Duality

2015 â—˝  
Vol 105 (10) â—˝  
pp. 1467-1477
Author(s):  
Minoru Itoh
Keyword(s):  
Tensor Algebra â—˝  
Weyl Duality
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