On representation types of \(q\)-Schur algebras
type: Item
instance of: scholarly article
title: On representation types of \(q\)-Schur algebras
zbMATH Open document ID: 0795.16010
DOI: 10.1016/0022-4049(93)90163-N
published in: Journal of Pure and Applied Algebra
publication date: 1993-05-16T00:00:00Z
review text: The main results of this paper concern the representation type of Schur algebras and their generalizations called \(q\)-Schur algebras. Let \(S_ k(n,r)\) be a Schur algebra, where \(k\) is an algebraically closed field of prime characteristic \(p>0\) and \(n\geq r\). The first result states that \(S_ k(n,r)\) is representation-finite (that is, there are only finitely many indecomposable non-isomorphic modules over \(S_ k(n,r)\)) if and only if \(r<2p\) and, in this case, the relations for \(S_ k(n,r)\) are generated by elements of degree 2 (i.e., the algebra is quadratic). The representation-infinite algebras can be divided into two classes (tame or wild) according to the hardness of the study of their category of finitely generated modules. The author shows in this paper that representation-infinite Schur algebras \(S_ k(n,r)\), \(r\leq n\), are always wild. Now let \(S_ k(n,r,q)\), \(r\leq n\), be a \(q\)-Schur algebra, where \(k\) is an algebraically closed field of arbitrary characteristic and let \(m\) be the minimal number such that \(1+q+\cdots+ q^{m-1}\) is equal to zero. The author has shown that if \(r<2m\), then \(S_ k(n,r,q)\) is representation-finite and quadratic.
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16S15
zbMATH DE Number: 166164
zbMATH Keywords: representation type
zbMATH Keywords: Schur algebras
zbMATH Keywords: \(q\)-Schur algebras
zbMATH Keywords: relations
zbMATH Keywords: representation-infinite algebras
zbMATH Keywords: category of finitely generated modules
zbMATH Keywords: representation-infinite Schur algebras
author: Changchang Xi
reviewed by: Flávio U. Coelho
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 166164