Noncommutative Schur functions, switchboards, and Schur positivity

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.