On the Gram determinants of the Specht modules

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.