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

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

2002-10-30

arxiv.org/abs/math-ph/0210058v1

Physics

Mathematical Physics

14 pages

Scientific paper

10.1088/0305-4470/36/12/305

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 G_2. The value distributions for characteristic polynomials associated with the 7- and 14-dimensional representations of G_2, 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 G_2. The random matrix calculations extend to all exceptional Lie groups

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