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

Abstract

For a partition [Formula: see text] of [Formula: see text], let [Formula: see text] be the ideal of [Formula: see text] generated by all Specht polynomials of shape [Formula: see text]. We assume that [Formula: see text]. Then [Formula: see text] is Gorenstein, and [Formula: see text] is a Cohen–Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Zamaere et al. [Jack polynomials as fractional quantum Hall states and the Betti numbers of the [Formula: see text]-equals ideal, Commun. Math. Phys. 330 (2014) 415–434] already studied minimal free resolutions of [Formula: see text], which are also Cohen–Macaulay, using highly advanced technique of the representation theory. However, we only use the basic theory of Specht modules, and explicitly describe the differential maps.