On the composition factors of the Specht modules
type: Item
instance of: scholarly article
title: On the composition factors of the Specht modules
zbMATH Open document ID: 0548.20007
DOI: 10.1016/0021-8693(84)90213-8
author: G. E. Murphy
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: An important result in the p-modular representation theory of finite symmetric groups says that the composition factors of the Specht module \(S^{\mu}\), \(\mu\) a partition of n, are of the form \(D^{\lambda}\) where \(\lambda\) dominates \(\mu\) (for short: \(\lambda {\underline\triangleright }\mu)\) and both \(\mu\) and \(\lambda\) have the same p-core (for short: \(\mu\sim_ p\lambda)\). The present author extends \(\sim_ p\) to tableaux as follows. \(t\sim_ pt^*\) means that each \(m\leq n\) occupies a node of the same p-residue in t and \(t^*\). Connecting this with G. D. James' concept of ladder he proves for example that in case of \(t_ 1^{\lambda}\) being a ladder tableau and \(t_ 1^{\lambda}\sim t_ k^{\mu}\), then \(\mu {\underline\triangleleft }{\tilde\mu }{\underline\triangleleft }{\tilde\lambda }\) (\({\tilde\mu }\),\({\tilde\lambda }\) the unique p-regular partition which have in each ladder the same number of nodes as \(\mu\), \(\lambda)\). It follows an easy proof of a result of G. D. James which gives the leading element of each row of the decomposition matrix.
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 3873574
zbMATH Keywords: p-modular representation
zbMATH Keywords: finite symmetric groups
zbMATH Keywords: composition factors
zbMATH Keywords: Specht module
zbMATH Keywords: p-core
zbMATH Keywords: ladder
zbMATH Keywords: ladder tableau
zbMATH Keywords: p-regular partition
zbMATH Keywords: decomposition matrix
reviewed by: Adalbert Kerber
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90213-8
OpenAlex ID: W2069653738
cites work: The representation theory of the symmetric groups
cites work: On the decomposition matrices of the symmetric groups. II
cites work: A new construction of Young's seminormal representation of the symmetric groups
cites work: The idempotents of the symmetric group and Nakayama's conjecture
description: scientific article; zbMATH DE number 3873574