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17
Random matrix theory over finite fields
 Bull. Amer. Math. Soc. (N.S
"... Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with sym ..."
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Abstract. The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need. Then we describe a probabilistic picture of conjugacy classes of the finite classical groups. Connections are made with symmetric function theory, Markov chains, RogersRamanujan type identities, potential theory, and various measures on partitions.
Affine shuffles, shuffles with cuts, the Whitehouse model, and patience sorting
 Journal of Algebra
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Conjugacy class properties of the extension of gl(n; Fq
 Journal of Algebra
, 2004
"... Abstract. Letting τ denote the inverse transpose automorphism of GL(n, q), a formula is obtained for the number of g in GL(n, q) so that gg τ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg τ ..."
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Cited by 8 (1 self)
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Abstract. Letting τ denote the inverse transpose automorphism of GL(n, q), a formula is obtained for the number of g in GL(n, q) so that gg τ is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg τ behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n, q) and of symmetric function theory, including a new identity for HallLittlewood polynomials. We obtain information about random elements of finite symplectic groups in even characteristic, and explicit bounds for the number of conjugacy classes and centralizer sizes in the extension of GL(n, q) generated by the inverse transpose automorphism. We give a second approach to these results using the theory of bilinear forms over a field. The results in this paper are key tools in forthcoming work of the authors on derangements in actions of almost simple groups, and we give a few examples in this direction. 1.
Probabilistic group theory
 C. R. Math. Acad. Sci. Canada
, 2002
"... This survey discusses three aspects of the ways in which probability has been applied to the theory of finite groups: probabilistic statements about groups; construction of randomized algorithms in computational group theory; and application of probabilistic methods to prove deterministic theorems i ..."
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This survey discusses three aspects of the ways in which probability has been applied to the theory of finite groups: probabilistic statements about groups; construction of randomized algorithms in computational group theory; and application of probabilistic methods to prove deterministic theorems in group theory. It concludes with a brief summary of related results for infinite groups. Cet article donne un apercu sur trois aspects des façons dont la probabilité est appliquée à la théorie des groupes finis: les faits probabilistiques des groupes; la construction d’algorithmes aléatoires dans la computation; et l’application des moyens probabilistiques pour obtenir les theorems déterministiques dans la théorie des groupes. On termine avec un bref sommaire de resultats se rapportant aux groupes infinis.
Bounds on the number and sizes of conjugacy classes in finite Chevalley groups
"... Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the f ..."
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Cited by 4 (1 self)
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Abstract. We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the BostonShalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (the remaining cases are settled in two other papers of the authors and applications are given in a third). 1.
Eigenvalues of random matrices over finite fields, unpublished
, 1999
"... We determine the limiting distribution of the number of eigenvalues of a random n×n matrix over Fq as n → ∞. We show that the q → ∞ limit of this distribution is Poisson with mean 1. The main tool is a theorem proved here on asymptotic independence for events defined by conjugacy class data arising ..."
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Cited by 3 (2 self)
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We determine the limiting distribution of the number of eigenvalues of a random n×n matrix over Fq as n → ∞. We show that the q → ∞ limit of this distribution is Poisson with mean 1. The main tool is a theorem proved here on asymptotic independence for events defined by conjugacy class data arising from distinct irreducible polynomials. The proof of this theorem uses the cycle index for matrices developed by Kung and Stong. Several examples are given to illustrate the use of the asymptotic independence theorem. These include previous results about derangements and about cyclic and semisimple matrices. 1 The Cycle Index for Mn(q) For a prime power q let Fq denote the field with q elements. Let Mn(q) be the space of n × n matrices over Fq and let GLn(q) be the group of invertible matrices. To define the cycle index for Mn(q) we recall the rational canonical form of a matrix (see, for example, [8]). Each matrix α ∈ Mn(q) gives rise to a finite dimensional Fq[z]module isomorphic to a direct sum k⊕ i=1 li⊕ j=1 Fq[z]/(φ λi,j i) where φ1,..., φk are distinct monic irreducible polynomials; for each i, λi,1 ≥ λi,2 ≥ · · · ≥ λi,li is a partition of ni = j λi,j. If α is n × n, then n = i ni deg φi = i,j λi,j deg φi. Let λi denote the partition of ni given by the λi,j and define λi  = ni. The conjugacy class of α in Mn(q) is determined by the data consisting of the finite list of distinct monic irreducible polynomials φ1,..., φk and the partitions λ1,..., λk. We introduce an indeterminate xφ,λ for each pair of a monic irreducible polynomial φ and a partition λ of some nonnegative integer. The conjugacy class data for α is encoded into the product xφ1,λ1... xφk,λk. We can write this more economically if we define λφ(α) to be the (possibly empty) partition corresponding to φ in the rational canonical form for α. We regard the empty partition as the
Cellini’s descent algebra and semisimple conjugacy classes of finite groups of Lie type
 Studies in Advanced Mathematics 29
"... By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measur ..."
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By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini which uses the affine Weyl group. This conjecture is confirmed in special cases such as type C odd characteristic and the identity conjugacy class in type A. Models of card shuffling, old and new, arise naturally. Type A shuffles lead to interesting number theory involving Ramanujan sums. It is shown that a proof of our conjecture in type C even characteristic would give an alternate solution to a problem in dynamical systems. An idea is offered for how, at least in type A, to associate to a semisimple conjugacy class an element of the Weyl group, refining the map to conjugacy classes. This is confirmed for the simplest nontrivial example. Key words: card shuffling, hyperplane arrangement, conjugacy class, descent algebra, dynamical systems. 1
A Probabilistic Approach Toward Conjugacy Classes In The Finite General Linear And Unitary Groups
"... The conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms for growing random partitions according to these measures are obtained. These algorithms are applied to prove gr ..."
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The conjugacy classes of the finite general linear and unitary groups are used to define probability measures on the set of all partitions of all natural numbers. Probabilistic algorithms for growing random partitions according to these measures are obtained. These algorithms are applied to prove group theoretic results which are typically proved by techniques such as character theory and Moebius inversion. Among the theorems studied are Steinberg's count of unipotent elements, Rudvalis' and Shindoda's work on the fixed space of a random matrix, and Lusztig's count of nilpotent matrices of a given rank. Generalizations of these algorithms based on Macdonald's symmetric functions are given.
POWER SERIES COEFFICIENTS FOR PROBABILITIES IN FINITE CLASSICAL GROUPS
, 2005
"... Abstract. It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in q −1. Moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equ ..."
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Abstract. It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in q −1. Moreover it is proved that the coefficients of the limiting probabilities in the general linear and unitary cases are equal modulo 2. The rate of stabilization of the finite dimensional coefficients as the dimension increases is discussed. 1.