On the Gram determinants of the Specht modules

TL;DR

This paper proves that for Specht modules of symmetric groups with even dimension, the 2-adic valuation of their Gram determinants is also even, confirming a conjecture by Richard Parker.

Contribution

It establishes a new parity property of the 2-adic valuation of Gram determinants for Specht modules, confirming Parker's conjecture in the symmetric group case.

Findings

If the dimension of S^λ is even, then a_λ^(2) is even.

Confirms Parker's conjecture for symmetric groups.

Provides insight into the structure of Gram determinants in representation theory.

Abstract

For every partition $\lambda$ of a positive integer $n$, let $S^{\lambda}$ be the corresponding Specht module of the symmetric group $\mathfrak{S}_n$, and let $\det(\lambda)\in \mathbb Z$ denote the Gram determinant of the canonical bilinear form with respect to the standard basis of $S^{\lambda}$. Writing $\det(\lambda)=m \cdot 2^{a_{\lambda}^{(2)}}$ for integers $a_{\lambda}^{(2)}$ and $m$ with $m$ odd, we show that if the dimension of $S^{\lambda}$ is even, then $a_{\lambda}^{(2)}$ is also even. This confirms a conjecture posed by Richard Parker in the special case of the symmetric groups.