Weak Bruhat interval modules for genomic Schur functions

Let $\lambda$ be a partition of a positive integer $n$. The genomic Schur function $U_\lambda$ 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 $U_\lambda$ in terms of increasing gapless tableaux. In this paper, for each $1 \le m \le n$, we construct an $H_m(0)$-module $\mathbf{G}_{\lambda;m}$ whose image under the quasisymmetric characteristic is the $m$th degree homogeneous component of $U_\lambda$ 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 $\mathbf{G}_{\lambda;m}$ into a direct sum of weak Bruhat interval modules. Furthermore, we determine the projective cover of each summand of the direct sum decomposition.