Weak Bruhat interval modules for genomic Schur functions

Abstract

Let $\hat{I}»$ be a partition of a positive integer $n$. The genomic Schur function $U_{\hat{I}}»$ was introduced by Pechenik--Yong in the context of the $K$-theory of Grassmannians. Recently, Pechenik provided a positive combinatorial formula for the fundamental quasisymmetric expansion of $U_{\hat{I}}»$ in terms of increasing gapless tableaux. In this paper, for each $1\le m\le n$, we construct an $H_m(0)$-module ${\mathbf{G}}_{\hat{I}»;m}$ whose image under the quasisymmetric characteristic is the $m$th degree homogeneous component of $U_{\hat{I}}»$ by defining an $H_m(0)$-action on increasing gapless tableaux. We provide a method to assign a permutation to each increasing gapless tableau, and use this assignment to decompose ${\mathbf{G}}_{\hat{I}»;m}$ into a direct sum of weak Bruhat interval modules. Furthermore, we determine the projective cover of each summand of the direct sum decomposition.

48 pages; to appear in Electronic Journal of Combinatorics