The Smith normal form of a specialized Jacobi–Trudi matrix

Abstract

Let JTλ be the Jacobi–Trudi matrix corresponding to the partition λ, so detJTλ is the Schur function sλ in the variables x1,x2,…. Set x1=⋯=xn=1 and all other xi=0. Then the entries of JTλ become polynomials in n of the form (n+j−1j). We determine the Smith normal form over the ring Q[n] of this specialization of JTλ. The proof carries over to the specialization xi=qi−1 for 1≤i≤n and xi=0 for i>n, where we set qn=y and work over the ring Q(q)[y].