On the dimension of some modular irreducible representations of the symmetric group
type: Item
instance of: scholarly article
title: On the dimension of some modular irreducible representations of the symmetric group
zbMATH Open document ID: 0855.20010
DOI: 10.1007/BF00398296
published in: Letters in Mathematical Physics
publication date: 1997-01-29T00:00:00Z
review text: For fields of characteristic zero, there is a well known formula for the degree of an irreducible representation of the symmetric group \(S_n\) corresponding to a partition \(\lambda\); namely it is the number of standard tableaux of shape \(\lambda\). The determination of the corresponding formula in characteristic \(p\) has up to now been elusive. This paper provides such a formula for certain partitions \(\lambda\). First, a different, but not new, interpretation of the standard tableaux formula as paths on the set of Young diagrams is given. This interpretation is then modified and applied to the set of \(Y_\ell(p)\) of all Young diagrams corresponding to the partition \(\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)\) of height \(\ell\) such that \(\lambda_1-\lambda_\ell\leq p-\ell\); that is, the degree of the \(p\)-modular representations corresponding to the partition \(\lambda\) is the number of oriented paths from \(\emptyset\) to \(\lambda\) entirely contained in \(Y_\ell(p)\). It is also possible to give a degree formula for an arbitrary partition in terms of paths in a graph. However, in this case, the graph contains multiple edges and in general, so far, it is not possible to compute these multiplicities. The results concerning \(Y_\ell(p)\) have also been proved by \textit{A. S. Kleshchev} [J. Algebra 181, No. 2, 584-592 (1996)].
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 05E10
zbMATH DE Number: 922383
zbMATH Keywords: irreducible representations
zbMATH Keywords: symmetric groups
zbMATH Keywords: standard tableaux
zbMATH Keywords: partitions
zbMATH Keywords: Young diagrams
zbMATH Keywords: \(p\)-modular representations
zbMATH Keywords: degree formula
author: Olivier Mathieu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 922383