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$q$-Deformation of the Kac-Sylvester tridiagonal matrix


Author: J. F. van Diejen
Journal: Proc. Amer. Math. Soc. 149 (2021), 2291-2304
MSC (2020): Primary 15A18; Secondary 05E05, 15B36, 33D52, 47B36, 65F15
DOI: https://doi.org/10.1090/proc/15413
Published electronically: March 18, 2021
MathSciNet review: 4246783
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Abstract: Upon replacing integers by $q$-integers, one arrives at a natural $q$-deformation of a remarkably ubiquitous tridiagonal integer matrix whose eigenvalues and eigenvectors were first computed by J.J. Sylvester and M. Kac, respectively. The present note computes the eigenvalues and eigenvectors of this novel $q$-deformed Kac-Sylvester matrix through a connection with Macdonald’s hyperoctahedral Hall-Littlewood polynomials. This yields the spectrum as a minimizer of an associated Morse function reminiscent of Stieltjes’ electrostatic potential for the roots of classical orthogonal polynomials.


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Additional Information

J. F. van Diejen
Affiliation: Instituto de Matemáticas, Universidad de Talca, Casilla 747, Talca, Chile
MR Author ID: 306808
ORCID: 0000-0002-5410-8717
Email: diejen@inst-mat.utalca.cl

Keywords: Tridiagonal ($q$-)integer matrices, eigenvalues, eigenvectors, symmetric functions
Received by editor(s): July 10, 2020
Received by editor(s) in revised form: July 17, 2020
Published electronically: March 18, 2021
Additional Notes: Work was supported in part by the Fondo Nacional de Desarrollo Científico y Tecnológico (FONDECYT) Grant # 1210015.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2021 American Mathematical Society