Abstract
We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated with each classical root system. In finite type A n−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 C n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A 2n−1, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positivity of Lusztig t-analogues associated with zero weight spaces in the irreducible representations of symplectic Lie algebras. We then present three applications of our combinatorial formula. Our methods are expected to extend to the orthogonal types.
By a result of Lascoux, the type A Kostka–Foulkes polynomials also expand positively in terms of the so-called atomic polynomials. We define, in arbitrary type, a combinatorial version of the atomic decomposition, based on the connected components of a modified crystal graph. We prove this property in type A, as well as in types B, C, and D in a stable range for t = 1. We also discuss other cases, applications, and a geometric interpretation. Finally, in classical types, we state the atomic decomposition for stable 1-dimensional sums or, equivalently, for the stable Lusztig t-analogues.
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Notes
- 1.
Here s ν(x) stands for the ordinary Schur function in the variables x 1, …, x n.
- 2.
The factor (1 − t) in (2.1) gives the missing “d i = 1” in type A n−1.
- 3.
Here that the partition λ can have an odd rank.
- 4.
Nevertheless, we will establish that the atomic decomposition also holds in the column shape case, by (12.1).
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Acknowledgements
The second author Cristian Lenart gratefully acknowledges the partial support from the NSF grant DMS–1362627 and the Simons grant #584738. Both authors are grateful to Arthur Lubovsky and Adam Schultze for the computer tests (based on the Sage [28] system) related to this work; they also received support from the NSF grant mentioned above.
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Lecouvey, C., Lenart, C. (2021). Lusztig’s t-Analogue of Weight Multiplicity via Crystals. In: Greenstein, J., Hernandez, D., Misra, K.C., Senesi, P. (eds) Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification. Progress in Mathematics, vol 337. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-63849-8_10
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