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Mathematics > Representation Theory

arXiv:2207.08446 (math)
[Submitted on 18 Jul 2022 (v1), last revised 21 Jan 2023 (this version, v3)]

Title:Symplectic cacti, virtualization and Berenstein-Kirillov groups

Authors:Olga Azenhas, Mojdeh Tarighat Feller, Jacinta Torres
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Abstract:We explicitly realize an internal action of the symplectic cactus group, recently defined by Halacheva for any complex, reductive, finite-dimensional Lie algebra, on crystals of Kashiwara-Nakashima tableaux. Our methods include a symplectic version of jeu de taquin due to Sheats and Lecouvey, symplectic reversal, and virtualization due to Baker. As an application, we define and study a symplectic version of the Berenstein-Kirillov group and show that it is a quotient of the symplectic cactus group. In addition two relations for symplectic Berenstein-Kirillov group are given that do not follow from the defining relations of the symplectic cactus group.
Comments: 71 pages, comments welcome
Subjects: Representation Theory (math.RT); Combinatorics (math.CO)
Cite as: arXiv:2207.08446 [math.RT]
  (or arXiv:2207.08446v3 [math.RT] for this version)
  https://doi.org/10.48550/arXiv.2207.08446

Submission history

From: Jacinta Torres [view email]
[v1] Mon, 18 Jul 2022 08:57:40 UTC (63 KB)
[v2] Tue, 6 Sep 2022 12:27:30 UTC (65 KB)
[v3] Sat, 21 Jan 2023 01:54:20 UTC (66 KB)
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