Jacobi--Trudi formulas for flagged refined dual stable Grothendieck polynomials

TL;DR

This paper derives Jacobi--Trudi formulas for flagged refined dual stable Grothendieck polynomials, expanding the understanding of their structure and confirming a conjecture, with implications for symmetric functions and combinatorics.

Contribution

It provides Jacobi--Trudi formulas for flagged refined dual stable Grothendieck polynomials, resolving a conjecture and generalizing previous results.

Findings

Jacobi--Trudi formulas for flagged refined dual stable Grothendieck polynomials

Resolution of Grinberg's conjecture

Generalization of Iwao and Amanov--Yeliussizov's results

Abstract

Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and upper bounds are given for the entries of each row or column. In this paper Jacobi--Trudi-type formulas for flagged refined dual stable Grothendieck polynomials are proved using plethystic substitution. This resolves a conjecture of Grinberg and generalizes a result by Iwao and Amanov--Yeliussizov.