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.

arXiv comment23 pages. Editorial improvements from Ver.2

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