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

For a partition λ of n ∈ N let IλSp be the ideal of R = K[x1,...,xn] generated by all Specht polynomials of shape λ. We assume that char(K) = 0. Then R/I(n-2,2)Sp is Gorenstein, and R/I(d,d,1)Sp 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 (k + 1)-equals ideal, Commun. Math. Phys. 330 (2014) 415-434] already studied minimal free resolutions of R/I(n-d,d)Sp, 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.