The genomic Schur function is fundamental-positive

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 $U_{\lambda }$.

12 pages

References in corpus (6)

• Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams
• Genomic Tableaux
• Set-Valued Skyline Fillings
• A note on passing from a quasi-symmetric function expansion to a Schur function expansion of a symmetric function
• Orbits of Plane Partitions of Exceptional Lie Type
• Puzzles in $K$-homology of Grassmannians

Cited by in corpus (2)

• Asymmetric function theory
• Positivity among P-partition generating functions