A non-commutative generalization of $k$-Schur functions

TL;DR

This paper develops non-commutative versions of $k$-Schur functions, providing explicit expansion formulas and drawing parallels to existing conjectures in the commutative setting.

Contribution

It introduces non-commutative analogues of $k$-Schur functions and derives explicit formulas for their expansions with one and two parameters.

Findings

Explicit formulas for non-commutative $k$-Schur function expansions

Analogies to conjectures in the commutative case

New framework for non-commutative symmetric functions

Abstract

We introduce non-commutative analogues of $k$-Schur functions of Lapointe-Lascoux and Morse. We give an explicit formulas for the expansions of non-commutive functions with one and two parameters in terms of these new functions. These results are similar to the conjectures existing in the commutative case.