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Article Dans Une Revue International Mathematics Research Notices Année : 2020

COMBINATORICS OF GENERALIZED EXPONENTS

Résumé

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type An−1, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type Cn, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A2n−1, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated to zero weight spaces in the irreducible representations of symplectic Lie algebras. We also prove a related conjecture of the first author as an application, and discuss some implications to relating two type C branching rules. Our methods are expected to extend to the orthogonal types.
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Dates et versions

hal-01560815 , version 1 (12-07-2017)

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Cédric Lecouvey, Cristian Lenart. COMBINATORICS OF GENERALIZED EXPONENTS. International Mathematics Research Notices, 2020, 2020 (16), pp.4942-4992. ⟨10.1093/imrn/rny157⟩. ⟨hal-01560815⟩
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