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.
@article{10_37236_12896,
author = {Young-Hun Kim and Semin Yoo},
title = {Weak {Bruhat} interval modules for genomic {Schur} functions},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12896},
zbl = {1560.20013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12896/}
}
TY - JOUR
AU - Young-Hun Kim
AU - Semin Yoo
TI - Weak Bruhat interval modules for genomic Schur functions
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12896/
DO - 10.37236/12896
ID - 10_37236_12896
ER -
%0 Journal Article
%A Young-Hun Kim
%A Semin Yoo
%T Weak Bruhat interval modules for genomic Schur functions
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12896/
%R 10.37236/12896
%F 10_37236_12896
Young-Hun Kim; Semin Yoo. Weak Bruhat interval modules for genomic Schur functions. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12896
