Results 1  10
of
15
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 ..."
Abstract

Cited by 22 (6 self)
 Add to MetaCart
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 module, and patience sorting
 J. Algebra
"... Using representation theoretic work on the Whitehouse module, a formula is obtained for the cycle structure of a riffle shuffle followed by a cut. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Type A affine shuffles are compared with riffl ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Using representation theoretic work on the Whitehouse module, a formula is obtained for the cycle structure of a riffle shuffle followed by a cut. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group Sn are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter q satisfies gcd(n, q −1) = 1, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices.
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 ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
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.
Cellini's descent algebra and semisimple conjugacy classes of finite groups of Lie type
"... 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 ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
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. This conjecture is confirmed for type C in odd characteristic, and for type A 2 . Connections with old and new card shuffling models are indicated. 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. 1 Introduction Let \Pi = fff 1 ; \Delta \Delta \Delta ; ff r g be a set of simple roots for a root system of a finite Coxeter group W of rank...
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 ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
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.
The Expected Order of a Random Unitary Matrix (Preliminary Version)
, 2008
"... Let U(n, q) be the group consisting of those invertible matrices A = (ai,j) 1≤i,j≤n whose inverse is the conjugate transpose with respect to the involution c ↦ → c q of the finite field Fq2. ∑ In other words, the i, j’th en. Let µn = Order(A) be the average of try of A −1 is a q j,i 1 U(n,q) A∈U( ..."
Abstract
 Add to MetaCart
Let U(n, q) be the group consisting of those invertible matrices A = (ai,j) 1≤i,j≤n whose inverse is the conjugate transpose with respect to the involution c ↦ → c q of the finite field Fq2. ∑ In other words, the i, j’th en. Let µn = Order(A) be the average of try of A −1 is a q j,i 1 U(n,q) A∈U(n,q) the orders of the elements in this finite group. We prove the following conjecture of Fulman: for any fixed q, as n → ∞, log µn = n log(q) − log n + oq(log n).
Affine shuffles, shuffles with cuts, and patience sorting
, 1999
"... Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group Sn are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter q satisfies gcd(n, q−1) = 1, the induced meas ..."
Abstract
 Add to MetaCart
Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group Sn are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter q satisfies gcd(n, q−1) = 1, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness. Generating functions for the first pile size in patience sorting from decks with repeated values are derived. This relates to random matrices. Key words: card shuffling, conjugacy class, sorting, random matrix, cycle structure. 1