The down operator and expansions of near rectangular k-Schur functions

TL;DR

This paper demonstrates that the down operator induces a derivation on a specific algebra, enabling the verification of a conjecture about the expansion of near rectangular k-Schur functions and providing a combinatorial interpretation of related coefficients.

Contribution

It introduces a new derivation induced by the down operator and verifies a conjecture on k-Schur function expansions in the affine nilCoxeter algebra.

Findings

The down operator acts as a derivation on the affine Fomin-Stanley subalgebra.

Confirmed the conjecture on the expansion of near rectangular k-Schur functions.

Provided a combinatorial interpretation of k-Littlewood--Richardson coefficients.

Abstract

We prove that the Lam-Shimozono "down operator" on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of "near rectangles" in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood--Richardson coefficients.