The eigenvalue distribution of a random unipotent matrix in its representation on lines

The eigenvalue distribution of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed are of the unit circle and offer an approach to other asymptotics. For the case of all unipotent matrices, the proof gives a probabilistic interpretation to identities of Macdonald from symmetric function theory. For the case of upper triangular matrices over a finite held, connections between symmetric function theory and a probabilistic growth algorithm of Borodin and Kirillov emerge. (C) 2000 Academic Press.