Noncommutative Schur functions, switchboards, and Schur positivity

TL;DR

This paper extends noncommutative Schur functions to a broader class of symmetric functions, introduces switchboards as combinatorial tools, and advances understanding of Schur positivity for functions like LLT, Macdonald, and Grothendieck polynomials.

Contribution

It develops the noncommutative Schur functions framework to include new symmetric functions via switchboards, enhancing methods for Schur expansion formulas.

Findings

Extended noncommutative Schur functions to new symmetric functions

Introduced switchboards as combinatorial gadgets for Schur positivity

Provided new combinatorial formulas for LLT polynomial expansions

Abstract

The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a wider class of symmetric functions, explore its strengths and limitations, and obtain new results on Schur positivity. We introduce combinatorial gadgets called switchboards, an adaptation of the D graphs of S. Assaf, and show how symmetric functions associated to them (which include LLT, Macdonald, Stanley, and stable Grothendieck polynomials) fit into the noncommutative Schur functions approach. This extends earlier work by T. Lam, and by C. Greene and the second author, and provides new tools for obtaining combinatorial formulas for Schur expansions of LLT polynomials. This paper can be regarded as a "prequel" to (and, partly, a review of) arXiv:1411.3624, arXiv:1411.3646, and arXiv:1510.00644.