On the classification of Specht modules with one-dimensional summands

Collins, Aubrey Piper; Dodge, Craig J.

doi:10.2140/involve.2019.12.1399

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On the classification of Specht modules with one-dimensional summands

Aubrey Piper Collins and Craig J. Dodge

Vol. 12 (2019), No. 8, 1399–1413

DOI: 10.2140/involve.2019.12.1399

Abstract

This paper extends a result of James to a combinatorial condition on partitions for the corresponding Specht module to have a summand isomorphic to the unique one-dimensional $F Σ$ -module over fields of characteristic 2. The work makes use of a recursively defined condition to reprove a result of Murphy and prove a new result for self-conjugate partitions. Finally we present a Python script which utilizes this work to test Specht modules for a one-dimensional summand.

Keywords

representation theory, group theory, symmetric groups, Specht modules, decomposable

Mathematical Subject Classification 2010

Primary: 20C20, 20C30

Milestones

Received: 20 May 2019

Revised: 29 July 2019

Accepted: 29 August 2019

Published: 25 October 2019

Communicated by Kenneth S. Berenhaut

Authors

Aubrey Piper Collins
Department of Mathematics Allegheny College Meadville, PA United States

Craig J. Dodge
Department of Mathematics Allegheny College Meadville, PA United States

Vol. 12, No. 8, 2019