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 $I_{\lambda }^{\mathrm{Sp}}$ be the ideal of $R=K[x_1,\dots ,x_n]$ generated by all Specht polynomials of shape $\lambda$. We assume that $\mathrm{char}(K)=0$. Then $R/I_{(n-2,2)}^{\mathrm{Sp}}$ is Gorenstein, and $R/I_{(d,d,1)}^{\mathrm{Sp}}$ 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 $R/I_{(n-d,d)}^{\mathrm{Sp}}$, 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.

23 pages. Editorial improvements from Ver.2