Finite affine groups: cycle indices, Hall-Littlewood polynomials, and probabilistic algorithms (Q1604369)

This paper is a major contribution to the study of random matrices over a finite field. The author is studying conjugacy classes in the group of invertible affine transformations over a finite field. This problem leads to a natural probability measure on the set of all partitions of all positive integers. Using symmetric function theory and Markov chains, Fulman analyzes this measure leading to interesting and nontrivial enumerative results. He derives cycle index generating functions and computes the asymptotic distributions (and convergence rates) for various types of elements in the affine group (as the dimension increases) -- in particular, separable, cyclic and semisimple elements are considered. These results are interesting and will have many applications (for example, in studying derangements). Interesting connections with Rogers-Ramanujan identities and Hall-Littlewood polynomials are made. We refer the reader to the article for details and statements of the results.