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Pattern avoidance seen in multiplicities of maximal weights of affine Lie algebra representations


Authors: Shunsuke Tsuchioka and Masaki Watanabe
Journal: Proc. Amer. Math. Soc. 146 (2018), 15-28
MSC (2010): Primary 17B67; Secondary 05A05
DOI: https://doi.org/10.1090/proc/13597
Published electronically: September 28, 2017
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Abstract: We prove that the multiplicities of certain maximal weights of $ \mathfrak{g}(A^{(1)}_{n})$-modules are counted by pattern avoidance on words. This proves and generalizes a conjecture of Jayne-Misra. We also prove similar phenomena in types $ A^{(2)}_{2n}$ and $ D^{(2)}_{n+1}$. Both proofs are applications of Kashiwara's crystal theory.


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Shunsuke Tsuchioka
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan
Email: tshun@kurims.kyoto-u.ac.jp

Masaki Watanabe
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan
Email: mwata@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/proc/13597
Keywords: Weight multiplicities, affine Lie algebras, pattern avoidance, maximal weights, Kashiwara's crystal, Littelmann's path model, RSK correspondence, plane partitions, orbit Lie algebras, quantum binomial coefficients, categorification, Hecke algebras, symmetric groups, modular representation theory, Mullineux involution
Received by editor(s): November 10, 2015
Received by editor(s) in revised form: October 12, 2016
Published electronically: September 28, 2017
Additional Notes: The first author was supported in part by JSPS Kakenhi Grants 26800005.
Communicated by: Kailash C. Misra
Article copyright: © Copyright 2017 American Mathematical Society

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