Lusztig’s t-Analogue of Weight Multiplicity via Crystals

Lecouvey, Cédric; Lenart, Cristian

doi:10.1007/978-3-030-63849-8_10

Interactions of Quantum Affine Algebras With Cluster Algebras, Current Algebras and Categorification

2020

DOI: 10.1007/978-3-030-63849-8_10

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Lusztig’s t-Analogue of Weight Multiplicity via Crystals

Cédric Lecouvey

Cristian Lenart

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2021

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Atomic decomposition of characters and crystals

Lecouvey

Lenart

2021

Advances in Mathematics

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Lascoux stated that the type A Kostka-Foulkes polynomials K λ,µ (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 µ in the irreducible representation of highest weight λ. We formulate the atomic decomposition in arbitrary type, and view it as a strengthening of the monotonicity of K λ,µ (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 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 λ,µ (t). We also give a geometric interpretation.

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Atomic decomposition of characters and crystals

Lecouvey

Lenart

2021

Advances in Mathematics

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Lusztig’s t-Analogue of Weight Multiplicity via Crystals

Cited by 1 publication

References 24 publications

Atomic decomposition of characters and crystals

Atomic decomposition of characters and crystals

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