A simplified presentation of Specht modules

TL;DR

This paper simplifies the presentation of Specht modules by reducing the number of generators needed, making the structure more accessible and connecting it to recent related results.

Contribution

It introduces a new, simplified presentation of Specht modules using a single relation per pair of columns, applicable to all Specht modules.

Findings

Reduced the number of generators in Specht module presentations

Established that the new presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation

Unified the presentation approach for all Specht modules

Abstract

Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation.