A representation-theoretic interpretation of the Schur expansion of two-row genomic Schur functions

TL;DR

This paper proves a conjecture linking the Schur expansion of two-row genomic Schur functions to a representation-theoretic interpretation via $0$-Hecke modules, advancing understanding in algebraic combinatorics.

Contribution

It establishes a representation-theoretic interpretation of the Schur expansion for two-row genomic Schur functions, confirming a conjecture by Kim and Yoo.

Findings

Proves the Kim and Yoo conjecture for two-row cases.

Provides a representation-theoretic framework for genomic Schur functions.

Connects genomic Schur functions to $0$-Hecke modules.

Abstract

Genomic Schur functions were introduced by Pechenik and Yong in connection with the $K$-theory of Grassmannians. Pechenik proved that genomic Schur functions admit a positive expansion in the basis of fundamental quasisymmetric functions and, for partitions with two parts, a positive expansion in the Schur basis. Later, Kim and Yoo constructed $0$-Hecke modules associated with genomic Schur functions and conjectured that the latter expansion admits a representation-theoretic interpretation in terms of $0$-Hecke modules. In this paper, we prove the conjecture of Kim and Yoo, thereby obtaining a representation-theoretic interpretation of the Schur expansion in the two-row case.