Identities for Schur-type determinants and pfaffians

TL;DR

This paper presents new polynomial identities for determinants and pfaffians, along with tableau formulas for difference operators and enriched Schubert polynomials in Types A and C, advancing algebraic combinatorics.

Contribution

It introduces simple formulas for certain determinants and pfaffians, including interpolating expressions, and provides tableau formulas for difference operators and enriched Schubert polynomials.

Findings

New polynomial identities for determinants and pfaffians.

Tableau formulas for difference operators in Types A and C.

Explicit tableau formulas for enriched Schubert polynomials.

Abstract

We give a simple formula for some determinants, and an analogous formula for pfaffians, both of which are polynomial identities. The second involve some expressions that interpolate between determinants and pfaffians. We give several tableau formulas for difference operators in Types A and C, as well as other operators that appear in the "enriched", or "back-stable" Schubert polynomials. There are also tableau formulas for the enriched Schubert polynomials for vexillary and $321$-avoiding permutations.