The Primitive Spaces of Complete Intersections and Kostka Numbers

Let R be the polynomial ring over a field in n variables. Let \(A=R/(x_1^{d_1}, \ldots , x_n^{d_n})\) be the Artinian monomial complete intersection with generator degrees \(\{d_j\}\) , \(d_1\ge \cdots \ge d_n \ge 2\) and socle degree m. Let \(H(A)=(h_0, h_1, \ldots , h_m)\) be the h-vector of A. We prove that the positive integers that appear in the difference sequence \(\Delta (H(A))=(h_0, h_1-h_0, \ldots )\) are Kostka numbers \(K_{\lambda \mu }\) , where \(\lambda \) is a two rowed partition \(\lambda =(m-k,k) \vdash m\) and \(\mu = (d_1-1, \ldots , d_n-1) \vdash m\) , \(m=d_1+ \cdots + d_n-n\) .