Weak Bruhat Interval Modules for Genomic Schur Functions

Abstract

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 m n , we construct an H m (0)-module 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 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.