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 I λ Sp be the ideal of R = K [ x 1 , ... , x n ] 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. [ABSTRACT FROM AUTHOR]