On the first occurrence of irreducible modular representations of semigroups of all matrices
type: Item
instance of: scholarly article
title: On the first occurrence of irreducible modular representations of semigroups of all matrices
zbMATH Open document ID: 0943.20065
author: Ton That Tri
published in: Vietnam Journal of Mathematics
publication date: 2000-09-04T00:00:00Z
review text: The paper is concerned with irreducible modules over the semigroup algebra \({\mathbf F}_p[M_n]\) of the monoid \(M_n\) of all \(n\times n\) matrices over the prime field \({\mathbf F}_p\). It has been known that every such module embeds into some \(S^d\), with \(d\leq\sum^n_{i=1}(p^i-1)\), where \(S^d\) stands for the subspace of \({\mathbf F}_p[x_1,\dots,x_n]\) consisting of all homogeneous polynomials of degree \(d\), with the natural action of \(M_n\) [see \textit{S. Doty} and \textit{G. Walker}, Math. Proc. Camb. Philos. Soc. 119, No. 2, 231-242 (1996; Zbl 0855.20008)]. A complete list of irreducible \({\mathbf F}_p[M_n]\)-modules \(H_\beta\) was given by the author [in Acta. Math. Vietnam 20, No. 1, 43-53 (1995; Zbl 0886.20039)]. Each \(H_\beta\) is a submodule of some \(S^d\). In this paper it is shown that this occurrence of \(H_\beta\) is the first occurrence of this module as a submodule of \({\mathbf F}_p[x_1,\dots,x_n]\).
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M30
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1369986
zbMATH Keywords: monoids of matrices
zbMATH Keywords: irreducible modules
zbMATH Keywords: semigroup algebras
zbMATH Keywords: homogeneous polynomials
reviewed by: Jan Okniński
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1369986