Canonical bases for modified quantumglnand q -Schur algebras

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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Remarks on PBW bases of Ringel–Hall algebras of cyclic quivers

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500541 ◽

2019 ◽

Vol 19 (03) ◽

pp. 2050054

Author(s):

Zhonghua Zhao

Keyword(s):

Canonical Basis ◽

Recursive Formula ◽

Schur Algebras ◽

Canonical Bases ◽

Hall Algebras ◽

Pbw Basis ◽

Loewy Length ◽

Generic Extensions

In this paper, we give a recursive formula for the interesting PBW basis [Formula: see text] of composition subalgebras [Formula: see text] of Ringel–Hall algebras [Formula: see text] of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of [Formula: see text] from the monomial bases [Formula: see text] following [Multiplication formulas and canonical basis for quantum affine, [Formula: see text], Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of [Formula: see text] associated with modules of Loewy length [Formula: see text]. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras.

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Combinatorics of canonical bases revisited: type A

Selecta Mathematica ◽

10.1007/s00029-021-00658-x ◽

2021 ◽

Vol 27 (4) ◽

Author(s):

Volker Genz ◽

Gleb Koshevoy ◽

Bea Schumann

Keyword(s):

Type A ◽

Canonical Bases

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WEAK CANONICAL BASES IN NSOP1 THEORIES

Journal of Symbolic Logic ◽

10.1017/jsl.2021.45 ◽

2021 ◽

pp. 1-29

Author(s):

BYUNGHAN KIM

Keyword(s):

Canonical Bases

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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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