Atomic decomposition of characters and crystals
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.
Cite
@article{arxiv.1809.01262,
title = {Atomic decomposition of characters and crystals},
author = {Cedric Lecouvey and Cristian Lenart},
journal= {arXiv preprint arXiv:1809.01262},
year = {2019}
}
Comments
38 pages, 4 figures. Updates to the first version: the main result was extended to type B, so now all classical types are covered; Theorems 5.3 and 5.5 were slightly rephrased; the proof of Lemma 5.4 (1) was rewritten; section 7.3 was added, on the atomic decomposition of stable one-dimensional sums