IngentaConnect Random matrix theory, the exceptional Lie groups and L-functions

Authors: Keating J.P.¹; Linden N.¹; Rudnick Z.²

Source: Journal of Physics A: Mathematical and General, Volume 36, Number 12, 2003 , pp. 2933-2944(12)

Publisher: Institute of Physics Publishing

Abstract:

There has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number-theoretical L-functions. One example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of L-functions. These connections are extended here to non-classical groups. We focus on an explicit example: the exceptional Lie group G₂. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G₂, defined with respect to the uniform invariant (Haar) measure, are calculated using two of the Macdonald constant term identities. A one-parameter family of L-functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the seven-dimensional representation of G₂. The random matrix calculations extend to all exceptional Lie groups.

Language: English

Document Type: Miscellaneous

Affiliations: 1: School of Mathematics, University of Bristol, Bristol BS8 1TW, UK 2: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

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