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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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Cyclotomic 𝑞-Schur algebras and Schur-Weyl duality

Representations of Algebraic Groups, Quantum Groups, and Lie Algebras - Contemporary Mathematics ◽

10.1090/conm/413/07844 ◽

2006 ◽

pp. 133-155 ◽

Cited By ~ 2

Author(s):

Zongzhu Lin ◽

Hebing Rui

Keyword(s):

Schur Algebras ◽

Weyl Duality

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

International Mathematics Research Notices ◽

10.1093/imrn/rnx300 ◽

2018 ◽

Vol 2019 (20) ◽

pp. 6179-6215 ◽

Cited By ~ 2

Author(s):

Jie Du ◽

Qiang Fu

Keyword(s):

Canonical Basis ◽

Alternative Algebra ◽

Integral Form ◽

Loop Algebra ◽

Algebra Structure ◽

Schur Algebras ◽

Canonical Bases ◽

Weyl Duality ◽

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].

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A REALIZATION OF THE ENVELOPING SUPERALGEBRA

Nagoya Mathematical Journal ◽

10.1017/nmj.2021.11 ◽

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.

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