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On the Gram determinants of the Specht modules

Linda Hoyer

Abstract

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

On the Gram determinants of the Specht modules

Abstract

For every partition of a positive integer , let be the corresponding Specht module of the symmetric group , and let denote the Gram determinant of the canonical bilinear form with respect to the standard basis of . Writing for integers and with odd, we show that if the dimension of is even, then is also even. This confirms a conjecture posed by Richard Parker in the special case of the symmetric groups.

Paper Structure

This paper contains 3 sections, 14 theorems, 63 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['Main-Thm-Symmetric']}

Key Result

Theorem 1.1

Let $\lambda$ be an even partition of $n$. Then $a_{\lambda}^{(2)}$ is even.

Figures (3)

  • Figure 1: Hook diagram of $(7,4,3)$, with the hook of the cell $c=(1,2)$ colored gray.
  • Figure 2: Hook diagrams of $(2,1^{(5)})$ and $(3,3,1)$.
  • Figure 3: Hook diagrams of $(6,3,2)$ and $(6,4,1)$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Example 2.1
  • Proposition 2.2
  • Example 2.3
  • Lemma 2.4
  • Corollary 2.5
  • Definition 2.6
  • Corollary 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 30 more