Elementary construction of the minimal free resolution of the Specht ideal of shape $(n-d,d)$

TL;DR

This paper provides an elementary construction of the minimal free resolution, including differential maps, for Specht ideals of shape $(n-d,d)$, complementing and simplifying previous advanced representation-theoretic approaches.

Contribution

It introduces an elementary method to construct the differential maps of the minimal free resolution of Specht ideals of shape $(n-d,d)$, simplifying prior complex techniques.

Findings

Constructed differential maps for the minimal free resolution.

Provided an elementary proof of existing results.

Enhanced understanding of Specht ideals of shape $(n-d,d)$.

Abstract

Let $K$ be a field with ${\rm char}(K)=0$. For a partition $\lambda$ of $n \in {\mathbb N}$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$. These ideals have been studied from several points of view (and under several names). Using advanced tools of the representation theory, Berkesch Zamaere et al [BGS]. constructed a minimal free resolution of $I^{\rm Sp}_{(n-d,d)}$ except differential maps. The present paper constructs the differential maps, and also gives an elementary proof of the result of [BGS].