Generalized Schur function determinants using the Bazin identity

TL;DR

This paper introduces new determinant identities for Schur functions using Bazin's 1851 identity, extending classical formulas and proving a conjecture for factorial Schur functions.

Contribution

It derives novel determinant identities for Macdonald's Schur variations via Bazin's identity and generalizes the Hamel-Goulden formula, including a proof of a conjecture.

Findings

Derived determinant identities for Macdonald's Schur functions

Proved a conjecture for factorial Schur functions by Morales, Pak, and Panova

Generalized the Hamel-Goulden formula and its converse

Abstract

In the literature there are several determinant formulas for Schur functions: the Jacobi-Trudi formula, the dual Jacobi-Trudi formula, the Giambelli formula, the Lascoux-Pragacz formula, and the Hamel-Goulden formula, where the Hamel-Goulden formula implies the others. In this paper we use an identity proved by Bazin in 1851 to derive determinant identities involving Macdonald's 9th variation of Schur functions. As an application we prove a determinant identity for factorial Schur functions conjectured by Morales, Pak, and Panova. We also obtain a generalization of the Hamel-Goulden formula, which contains a result of Jin, and prove a converse of the Hamel-Goulden theorem and its generalization.