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AN IDENTITY OF PARABOLIC KAZHDAN–LUSZTIG POLYNOMIALS ARISING FROM SQUARE-IRREDUCIBLE MODULES

Part of: Representation theory of groups Hopf algebras, quantum groups and related topics Algebraic combinatorics

Published online by Cambridge University Press:  30 April 2019

MAXIM GUREVICH Open the ORCID record for MAXIM GUREVICH [Opens in a new window]
MAXIM GUREVICH*
Affiliation:
Department of Mathematics,National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076
*
e-mail: matmg@nus.edu.sg
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Abstract

We show a precise formula, in the form of a monomial, for certain families of parabolic Kazhdan–Lusztig polynomials of the symmetric group. The proof stems from results of Lapid–Mínguez on irreducibility of products in the Bernstein–Zelevinski ring. By quantizing those results into a statement on quantum groups and their canonical bases, we obtain identities of coefficients of certain transition matrices that relate Kazhdan–Lusztig polynomials to their parabolic analogues. This affirms some basic cases of conjectures raised recently by Lapid.

Keywords

Kazhdan–Lusztig polynomialscanonical basisaffine Hecke algebras

MSC classification

Primary: 16T20: Ring-theoretic aspects of quantum groups
Secondary: 05E10: Combinatorial aspects of representation theory 20C08: Hecke algebras and their representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 1 , February 2021 , pp. 81 - 93
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by A. Henderson

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