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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
Abstract

We consider the Specht module graphs for partitions of the form (n6,3,3) and determine the parity of the number of paths on these directed graphs. For values of n 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