A Plactic Algebra Action on Bosonic Particle Configurations

Abstract

We study an action of the plactic algebra on bosonic particle configurations. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations of the special linear Lie algebra \(\mathfrak {s} \mathfrak {l}_{N}\). It turns out that this action factors through a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.

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Acknowledgments

I would like to thank the Max Planck Institute for Mathematics and the Hausdorff Center for Mathematics for excellent research conditions and funding the project through IMPRS/BIGS. I thank Catharina Stroppel for supervising the thesis, and I am grateful to Marcelo Aguiar, Michael Ehrig and Daniel Tubbenhauer for helpful discussions.

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Open Access funding enabled and organized by Projekt DEAL.

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Correspondence to Joanna Meinel.

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Presented by: Peter Littelmann

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Meinel, J. A Plactic Algebra Action on Bosonic Particle Configurations. Algebr Represent Theor (2020). https://doi.org/10.1007/s10468-020-10006-w

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Keywords

  • Plactic algebra
  • Bosonic particle configurations
  • Kashiwara crystals
  • Center
  • Normal form

Mathematics Subject Classification (2010)

  • 05E10 (81R10, 17B37, 20G42)