Upper bounds of dual flagged Weyl characters

TL;DR

This paper proves a conjecture that the dual flagged Weyl character reaches its upper bound precisely when the associated diagram avoids a specific subdiagram, linking combinatorial diagram properties to algebraic bounds.

Contribution

It provides a proof confirming that the dual flagged Weyl character attains its upper bound exactly when the diagram avoids a certain subdiagram.

Findings

Proof of the conjecture relating diagram avoidance to upper bounds

Characterization of when dual flagged Weyl characters reach their bounds

Connection between diagram combinatorics and algebraic properties

Abstract

For a subset $D$ of boxes in an $n\times n$ square grid, let $\chi_{D}(x)$ denote the dual character of the flagged Weyl module associated to $D$. It is known that $\chi_{D}(x)$ specifies to a Schubert polynomial (resp., a key polynomial) in the case when $D$ is the Rothe diagram of a permutation (resp., the skyline diagram of a composition). One can naturally define a lower and an upper bound of $\chi_{D}(x)$. M{\'e}sz{\'a}ros, St. Dizier and Tanjaya conjectured that $\chi_{D}(x)$ attains the upper bound if and only if $D$ avoids a certain subdiagram. We provide a proof of this conjecture.