Kohnert's rule for flagged Schur modules

TL;DR

This paper proves that the characters of flagged Schur modules can be computed using Kohnert's combinatorial algorithm for northwest diagrams, providing new insights into their structure and positivity properties.

Contribution

It establishes a precise condition under which Kohnert's algorithm computes flagged Schur module characters, linking combinatorics with representation theory.

Findings

Kohnert's algorithm correctly computes characters for northwest diagrams.

Characters of flagged Schur modules are nonnegative sums of Demazure characters.

Provides a new representation-theoretic interpretation for Kohnert polynomials.

Abstract

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials.