Semistandard Tableaux for Demazure Characters (Key Polynomials) and Their Atoms

TL;DR

This paper provides combinatorial descriptions of tableaux contributing to Demazure characters (key polynomials) and their atoms, extending classical Schur function theory with new tableau characterizations.

Contribution

It introduces two new tableau descriptions for Demazure characters and atoms, enhancing understanding of their combinatorial structure.

Findings

Descriptions of tableaux for key polynomials and atoms

Decomposition of Schur functions into atoms

Characterization of tableaux with specific key properties

Abstract

The Schur function indexed by a partition lambda with at most n parts is the sum of the weight monomials for the Young tableaux of shape lambda. Let pi be an n-permutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by pi and lambda. (These polynomials are the characters of the Demazure modules for GL(n).) The "atom" indexed by pi is the sum of weight monomials of the tableaux whose right keys are the "key" tableau for pi. Schur functions and key polynomials can be decomposed into sums of atoms. We also describe the tableaux that contribute to an atom, the tableaux that have a left key equal to a given key, and the tableaux that have a left key bounded below by a given key.