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  McKay matrices for finite-dimensional Hopf algebras

Benkart, G., Biswal, R., Kirkman, E., Nguyen, V. C., & Zhu, J. (2022). McKay matrices for finite-dimensional Hopf algebras. Canadian Journal of Mathematics, 74(3), 686-731. doi:10.4153/S0008414X21000067.

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 Creators:
Benkart, Georgia, Author
Biswal, Rekha1, Author           
Kirkman, Ellen, Author
Nguyen, Van C., Author
Zhu, Jieru, Author
Affiliations:
1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Mathematics, Rings and Algebras, Representation Theory
 Abstract: For a finite-dimensional Hopf algebra $A$, the McKay matrix $M_V$ of an
$A$-module $V$ encodes the relations for tensoring the simple $A$-modules with
$V$. We prove results about the eigenvalues and the right and left
(generalized) eigenvectors of $M_V$ by relating them to characters. We show how
the projective McKay matrix $Q_V$ obtained by tensoring the projective
indecomposable modules of $A$ with $V$ is related to the McKay matrix of the
dual module of $V$. We illustrate these results for the Drinfeld double $D_n$
of the Taft algebra by deriving expressions for the eigenvalues and
eigenvectors of $M_V$ and $Q_V$ in terms of several kinds of Chebyshev
polynomials. For the matrix $N_V$ that encodes the fusion rules for tensoring
$V$ with a basis of projective indecomposable $D_n$-modules for the image of
the Cartan map, we show that the eigenvalues and eigenvectors also have such
Chebyshev expressions.

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Language(s): eng - English
 Dates: 2022
 Publication Status: Published in print
 Pages: 46
 Publishing info: -
 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 2007.05510
DOI: 10.4153/S0008414X21000067
 Degree: -

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Title: Canadian Journal of Mathematics
Source Genre: Journal
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Publ. Info: Cambridge University Press
Pages: - Volume / Issue: 74 (3) Sequence Number: - Start / End Page: 686 - 731 Identifier: -