Cells in certain sets of matrices
type: Item
instance of: scholarly article
title: Cells in certain sets of matrices
zbMATH Open document ID: 0881.17008
author: Jie Du
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1998-02-17T00:00:00Z
review text: It is well-known that for an associative algebra \({\mathcal A}\) over a field \(K\) and a basis \({\mathcal B}\) for \({\mathcal A}\) as a \(K\)-vector space, \({\mathcal B}\) is divided into cells [\textit{G. Lusztig}, Introduction to quantum groups, Birkhäuser, Boston (1993; Zbl 0788.17010)]. In certain nice circumstances, cells are important in the study of representation theory and can be classified combinatorially. In this paper, the author considers two more examples in the case of type \(A\), namely, when \({\mathcal A}\) is a \(q\)-Schur algebra \({\mathcal S}_q (n,r)\) or a modified quantized enveloping algebra \(\dot U\) of type \(A\) and \({\mathcal B}\) is the \(K\)-\(L\) basis of \({\mathcal S}_q (n,r)\) or the canonical basis of \(\dot U\). These bases are indexed by certain sets of matrices. So the cell decomposition of \({\mathcal B}\) induces a cell decomposition of these matrix sets. The author gives a combinatorial description for those two-sided cells. In Section 1, a result of \textit{Curtis Greene} [J. Comb. Theory, Ser. A 20, 69-79 (1979; Zbl 0323.06002)] is first generalized, which associates partitions to finite posets, to a result on a certain matrix semigroup \(M(n)\), namely one associates partitions to the matrices in \(M(n)\). Thus, \(M(n)\) is decomposed into subsets in terms of partitions. The main result is Theorem 2.1, which shows that these subsets are two-sided cells of the \(K\)-\(L\) bases of \(q\)-Schur algebras. This theorem is proved in Section 3. In Section 4 the author applies the result to modified quantum groups \(\dot U\) of type \(A\).
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 1044120
zbMATH Keywords: \(q\)-Schur algebra
zbMATH Keywords: modified quantized enveloping algebra
zbMATH Keywords: cell decomposition
zbMATH Keywords: partitions
zbMATH Keywords: modified quantum groups
reviewed by: Fang Li
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225340
OpenAlex ID: W2001341382
cites work: Q4029510
cites work: \(q\)-Schur algebras, asymptotic forms, and quantum \(SL_ n\)
cites work: Some partitions associated with a partially ordered set
cites work: Instantons and affine algebras. I: The Hilbert scheme and vertex operators
cites work: Q3952291
cites work: Representations of Coxeter groups and Hecke algebras
cites work: Q3737651
cites work: Introduction to quantum groups
cites work: Q3804915
cites work: The Kazhdan-Lusztig cells in certain affine Weyl groups
DOI: 10.2748/TMJ/1178225340
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description: scientific article; zbMATH DE number 1044120