  Let $\lambda$ be a partition of a positive integer $n$. The genomic Schur
function $U_\lambda$ was introduced by Pechenik and 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 then 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.
