Specht modules for partitions (k,3,3) have no one-dimensional summand

Dodge, Craig J.; Van Vlack, Elijah

doi:10.2140/involve.2023.16.659

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Specht modules for partitions $(k,3,3)$ have no one-dimensional summand

Craig J. Dodge and Elijah Van Vlack

Vol. 16 (2023), No. 4, 659–672

DOI: 10.2140/involve.2023.16.659

Abstract

We consider the Specht module graphs for partitions of the form $(n - 6, 3, 3)$ and determine the parity of the number of paths on these directed graphs. For values of $n \equiv 3, 7 mod 8$ the partition is Lucas perfect, so the associated Specht module will have a one-dimensional summand exactly when the number of paths on the graph is odd. By establishing equivalence classes on the set of paths and comparing to smaller partitions, we are able to demonstrate that none of these Specht modules have a one-dimensional summand.

Keywords

Specht modules, standard tableau, symmetric group

Mathematical Subject Classification

Primary: 20C20, 20C30

Milestones

Received: 10 June 2022

Revised: 27 September 2022

Accepted: 27 September 2022

Published: 31 October 2023

Communicated by Ken Ono

Authors

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

Elijah Van Vlack
Department of Mathematics Allegheny College Meadville, PA United States

Vol. 16, No. 4, 2023