Schur polynomials, banded Toeplitz matrices and Widom's formula

TL;DR

This paper establishes linear recurrence relations for sequences of Schur polynomials and minors of banded Toeplitz matrices, providing explicit roots of the characteristic equations and connecting Widom's formula to Schur polynomial identities.

Contribution

It proves that Schur polynomial sequences satisfy linear recurrences with explicitly given roots and links Widom's determinant formula to Schur polynomial identities.

Findings

Schur polynomial sequences satisfy linear recurrences for large k

Explicit roots of the recurrence characteristic equations are provided

Widom's formula is shown to be a special case of a Schur polynomial identity

Abstract

We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot \mathbf{1}^c)/(\mathbf{\lambda} + k\cdot \mathbf{1}^r)}(x_1,...,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are also valid for certain sequences of minors of banded Toeplitz matrices. In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials.