$q$-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
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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