Random matrix theory, the exceptional Lie groups and<i>L</i>-functions

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 here extended 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 14dimensional 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 7-dimensional representation of G2.The random matrix calculations extend to all exceptional Lie groups.

Locations

• Journal of Physics A Mathematical and General - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View