Elementary construction of the minimal free resolution of the Specht ideal 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].