The genomic Schur function is fundamental-positive

TL;DR

This paper proves that the genomic Schur function is fundamentally positive in the basis of fundamental quasisymmetric functions, providing a new combinatorial formula and insights into its structure.

Contribution

It establishes the fundamental-positivity of the genomic Schur function and introduces a finite combinatorial formula for its expansion.

Findings

$U_\lambda$ is symmetric but not Schur-positive

Provides a positive combinatorial formula in terms of gapless increasing tableaux

Yields a compact combinatorial formula for the Schur expansion of $U_\lambda$

Abstract

In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant $K$-theory of Grassmannians. We then studied the genomic Schur function $U_\lambda$, a generating function for such tableaux, showing that it is non-trivially a symmetric function, although generally not Schur-positive. Here we show that $U_\lambda$ is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, moreover, the first finite expression for $U_\lambda$. Combined with work of A. Garsia and J. Remmel, this yields a compact combinatorial (but necessarily non-positive) formula for the Schur expansion of $U_\lambda$.