Almost Symmetric Schur Functions

TL;DR

This paper introduces almost symmetric Schur functions, a new basis generalizing key polynomials and Schur functions, with combinatorial formulas and positivity results, connecting to representation theory of parabolic subgroups.

Contribution

It defines the almost symmetric Schur functions, provides combinatorial formulas, and proves positivity results, linking them to representation theory and non-symmetric Macdonald functions.

Findings

Established a combinatorial formula for the functions.

Proved positivity of coefficients in monomial and monomial-Schur expansions.

Connected the functions to limits of characters of parabolic subgroup representations.

Abstract

We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur functions. They form a homogeneous basis for the space of almost symmetric functions and are defined using a family of recurrences involving the isobaric divided difference operators and limits of Weyl symmetrization operators. The $s_{(\mu|\lambda)}$ are the $q=t=0$ specialization of the stable limit non-symmetric Macdonald functions $\widetilde{E}_{(\mu|\lambda)}$ defined by the author in previous work. We find a combinatorial formula for these functions simultaneously generalizing well known formulas for the Schur functions and the key polynomials. Further, we prove positivity results for the coefficients of the almost symmetric Schur functions expanded into the monomial basis and into the monomial-Schur basis of the space of almost symmetric functions. The latter positivity result follows after realizing the almost symmetric Schur functions $s_{(\mu|\lambda)}$ as limits of characters of representations of parabolic subgroups in type $GL.$