Atomic decomposition of characters and crystals

TL;DR

This paper extends the atomic decomposition concept of Kostka-Foulkes polynomials to arbitrary Lie types, providing new combinatorial, geometric insights, and computational methods, with proven results in type A and partial results in other types.

Contribution

It formulates a generalized atomic decomposition for Kostka-Foulkes polynomials across all Lie types, strengthening existing monotonicity results and introducing a combinatorial crystal graph approach.

Findings

Atomic decomposition holds in type A and in types B, C, D in a stable range for t=1.

A combinatorial version of atomic decomposition is developed and proven in type A.

Conjectures are proposed for broader applicability and efficient computation of Kostka-Foulkes polynomials.

Abstract

Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant weight mu in the irreducible representation of highest weight lambda. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K_{lambda,mu}(t). We also define a combinatorial version of the atomic decomposition, as a decomposition of a modified crystal graph. We prove that this stronger version holds in type A (which provides a new, conceptual approach to Lascoux's statement), in types B, C, and D in a stable range for t=1, as well as in some other cases, while we conjecture that it holds more generally. Another conjecture stemming from our work leads to an efficient computation of K_{lambda,mu}(t). We also give a geometric interpretation.