The Smith Normal Form of a Specialized Jacobi-Trudi Matrix

TL;DR

This paper determines the Smith normal form of a specialized Jacobi-Trudi matrix, revealing its structure over certain polynomial rings and extending results to q-analogues, with implications for symmetric functions.

Contribution

It provides the first explicit computation of the Smith normal form for a class of specialized Jacobi-Trudi matrices over polynomial rings.

Findings

Smith normal form over $\\mathbb{Q}[n]$ is explicitly determined.

Results extend to q-analogues with $x_i=q^{i-1}$.

The approach applies to specializations involving $x_i=0$ for $i>n$.

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 ${n+j-1\choose j}$. 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 $1\leq i\leq n$ and $x_i=0$ for $i>n$, where we set $q^n=y$ and work over the ring $\mathbb{Q}(q)[y]$.