𝑞-Deformation of the Kac-Sylvester tridiagonal matrix

van Diejen, J.

doi:10.1090/proc/15413

$q$-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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