On eigenvalues related to finite Poincaré planes
type: Item
instance of: scholarly article
title: On eigenvalues related to finite Poincaré planes
zbMATH Open document ID: 0916.11065
author: J. H. Kuang
published in: Finite Fields and their Applications
publication date: 1999-07-08T00:00:00Z
review text: Suppose that \(F\) is the finite field with \(q\) elements and \(q\) is odd. The finite (Poincaré) upper half plane in the sense of \textit{A. Terras} [Fourier analysis on finite groups and applications, Cambridge Univ. Press (1999)] can be identified with the quotient \(G/K\), where \(G=GL(2,F)\) and \(K=\{\left(\begin{smallmatrix} a&b\delta\\b&a\end{smallmatrix}\right)\in G\}\), where \(\delta\) is a fixed non-square in \(F\). In an earlier paper [\textit{J. Kuang}, Proc. Am. Math. Soc. 123, 3615-3622 (1995; Zbl 0851.11074)], the author identified the adjacency operators of the finite upper half plane graphs defined in Terras [loc. cit., p. 312] with elements of the Hecke algebra \(H(G,K)\) and thus the eigenvalues of the adjacency matrices of finite upper half plane graphs are obtained from spherical irreducible representations of \(G\) [see also Terras, loc. cit., p. 352]. Here some evidence is given that the spectra of these adjacency matrices have a limiting distribution which is semi-circular as \(q\) goes to infinity.
Mathematics Subject Classification ID: 11T30
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 1199974
zbMATH Keywords: finite Poincaré planes
zbMATH Keywords: eigenvalues
zbMATH Keywords: adjacency matrices
zbMATH Keywords: spectra
zbMATH Keywords: limiting distribution
reviewed by: Audrey A. Terras
MaRDI profile type: Publication
full work available at URL: ffta.1996.0178
OpenAlex ID: W2034896778
cites work: Q3138326
cites work: Q3139646
cites work: Q4314355
cites work: Estimates for Soto-Andrade sums.
cites work: A Natural Orthogonal Basis of Eigenfunctions of the Hecke Algebra Acting on Cayley Graphs
cites work: Ramanujan graphs
cites work: Complex Representations of 𝐺𝐿(2,𝐾) for Finite Fields 𝐾
cites work: Q4280543
cites work: Q4273629
DOI: 10.1006/FFTA.1996.0178
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description: scientific article; zbMATH DE number 1199974