A new presentation for Specht modules with distinct parts

TL;DR

The paper addresses efficient presentations of Specht modules for partitions whose conjugate parts are distinct, via a restricted Garnir-relations approach. It introduces a linear operator on the space of column tabloids that enables an eigenspace decomposition and proves a main result: $M^{\lambda} / G^{\lambda,\max} \cong S^{\lambda}$ when $\lambda^*$ has distinct parts, with $G^{\lambda,\max}$ generated by $g^t_{c,l_{c+1}}$, optionally restricting to column-strict tableaux. The key technique reduces the general case to the two-column shapes by analyzing the operator on $V_{n,m}=M^{2^m1^{n-m}}$ and identifying $\ker \varphi \cong S^{2^m1^{n-m}}$ via eigenstructure, thereby establishing the two-column instance and implying the full result. This work extends prior staircase-shape results, corrects a prior proof in the literature, and provides a compact presentation framework (via minimal Garnir sets) for a broad class of Specht modules.

Abstract

We obtain a new presentation for Specht modules whose conjugate shapes have strictly decreasing parts by introducing a linear operator on the space generated by column tabloids. The generators of the presentation are column tabloids and the relations form a proper subset of the Garnir relations of Fulton. The results in this paper extend earlier results of the authors and Stanley on Specht modules of staircase shape.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Remark 2.6
• Corollary 2.7