On Deposition of the Product of Demazure Atoms and Demazure Characters

TL;DR

This paper investigates Demazure atoms and characters, proving new positivity properties of their products through combinatorial and algebraic methods, and verifies an open conjecture in the field.

Contribution

It establishes the atom-positivity of products involving Demazure atoms and monomials, and provides combinatorial and algebraic proofs for key-positivity properties, including an open conjecture case.

Findings

Proves atom-positivity of product of a monomial and an atom.

Provides combinatorial proof of key-positivity for skyline fillings.

Verifies the first open case of key-positivity conjecture for products of keys.

Abstract

This paper studies the properties of Demazure atoms and characters using linear operators and also tableaux-combinatorics. It proves the atom-positivity property of the product of a dominating monomial and an atom, which was an open problem. Furthermore, it provides a combinatorial proof to the key-positivity property of the product of a dominating monomial and a key using skyline fillings, an algebraic proof to the key-positivity property of the product of a Schur function and a key using linear operator and verifies the first open case for the conjecture of key-positivity of the product of two keys using linear operators and polytopes.