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Unitary representations of type B rational Cherednik algebras and crystal combinatorics

Part of: Representation theory of rings and algebras Representation theory of groups Algebraic combinatorics Lie algebras and Lie superalgebras Rings and algebras arising under various constructions

Published online by Cambridge University Press:  29 September 2021

Emily Norton Open the ORCID record for Emily Norton [Opens in a new window]
Emily Norton*
Affiliation:
School of Mathematics, Statistics and Actuarial Science, Sibson Building, Parkwood Road, University of Kent, Canterbury, Kent CT2 7FS
*
e-mail: E.Norton@kent.ac.uk
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Abstract

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.

Keywords

rational Cherednik algebraunitary representationFock spaceaffine Lie algebra crystalcombinatorial representation theory

MSC classification

Primary: 16G99: None of the above, but in this section
Secondary: 05E10: Combinatorial aspects of representation theory 17B37: Quantum groups (quantized enveloping algebras) and related deformations 20C08: Hecke algebras and their representations 16S80: Deformations of rings

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 1 , February 2023 , pp. 140 - 169
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

Part of this work was carried out under the auspices of the grant SFB-TRR 195.

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