Elementary construction of minimal free resolutions of the Specht ideals of shapes $(n-2,2)$ and $(d,d,1)$

Abstract

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 $\lambda$. We assume that ${\rm char}(K)=0$. Then $R/I^{\rm Sp}_{(n-2,2)}$ is Gorenstein, and $R/I^{\rm Sp}_{(d,d,1)}$ is a Cohen-Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Berkesch Zamaere, Griffeth, and Sam had already studied minimal free resolutions of $R/I^{\rm Sp}_{(n-d,d)}$, which are also Cohen-Macaulay, using heighly advanced technique of the representation theory. However we only use the basic theory of Specht modules, and explicitly describe the differential maps.