The Smith Normal Form of a Specialized Jacobi-Trudi Matrix

Abstract

Let ${\mathrm{JT}}_{\lambda }$ be the Jacobi-Trudi matrix corresponding to the partition $\lambda$, so $\det {\mathrm{JT}}_{\lambda }$ is the Schur function $s_{\lambda }$ in the variables $x_1,x_2,\dots$. Set $x_1=\cdots =x_n=1$ and all other $x_i=0$. Then the entries of ${\mathrm{JT}}_{\lambda }$ become polynomials in $n$ of the form $\left(\genfrac{}{}{0ex}{}{n+j-1}{j}\right)$. We determine the Smith normal form over the ring $\mathbb{Q}[n]$ of this specialization of ${\mathrm{JT}}_{\lambda }$. The proof carries over to the specialization $x_i=q^{i-1}$ for $1\le i\le n$ and $x_i=0$ for $i>n$, where we set $q^n=y$ and work over the ring $\mathbb{Q}(q)[y]$.

5 pages, 2 figures