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22
A probabilistic proof of the Rogers–Ramanujan identities
 Bull. London Math. Soc
"... The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given, leading to an elementary probabilistic proof of the ..."
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The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given, leading to an elementary probabilistic proof of the RogersRamanujan identities. This is compared with work on the uniform measure. The main case of Bailey’s lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.
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.
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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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.
Practical Cryptanalysis of the Identification Scheme Based on the Isomorphism of Polynomial with One Secret Problem
"... Abstract. This paper presents a practical cryptanalysis of the Identification Scheme proposed by Patarin at Crypto 1996. This scheme relies on the hardness of the Isomorphism of Polynomial with One Secret (IP1S), and enjoys shorter key than many other schemes based on the hardness of a combinatorial ..."
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Abstract. This paper presents a practical cryptanalysis of the Identification Scheme proposed by Patarin at Crypto 1996. This scheme relies on the hardness of the Isomorphism of Polynomial with One Secret (IP1S), and enjoys shorter key than many other schemes based on the hardness of a combinatorial problem (as opposed to numbertheoretic problems). Patarin proposed concrete parameters that have not been broken faster than exhaustive search so far. On the theoretical side, IP1S has been shown to be harder than Graph Isomorphism, which makes it an interesting target. We present two new deterministic algorithms to attack the IP1S problem, and we rigorously analyze their complexity and success probability. We show that they can solve a (big) constant fraction of all the instances of degree two in polynomial time. We verified that our algorithms are very efficient in practice. All the parameters with degree two proposed by Patarin are now broken in a few seconds. The parameters with degree three can be broken in less than a CPUmonth. The identification scheme is thus quite badly broken. 1
Integer Sequences and Matrices Over Finite
, 2006
"... In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences, their generating functions, and examples. Section 2 contains ..."
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In this expository article we collect the integer sequences that count several different types of matrices over finite fields and provide references to the Online Encyclopedia of Integer Sequences (OEIS). Section 1 contains the sequences, their generating functions, and examples. Section 2 contains the proofs of the formulas for the coefficients and the generating functions of those sequences if the proofs are not easily available in the literature. The cycle index for matrices is an essential ingredient in most of the derivations.
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( ..."
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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).
A PROBABILISTIC INTERPRETATION OF THE MACDONALD POLYNOMIALS
, 2012
"... The twoparameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary d ..."
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The twoparameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in the power sum polynomials. The Markov chain has stationary distribution a new twoparameter family of measures on partitions, the inverse of the Macdonald weight (rescaled). The uniform distribution on cycles of permutations and the Ewens sampling formula are special cases. The Markov chain is a version of the auxiliary variables algorithm of statistical physics. Properties of the Macdonald polynomials allow a sharp analysis of the running time. In natural cases, a bounded number of steps suffice for arbitrarily large k.
Practical Cryptanalysis of the Identification Scheme Based on IP1S
, 2010
"... This paper presents a practical cryptanalysis of the Identification Scheme proposed by Patarin at Crypto 1996. This scheme relies on the hardness of the Isomorphism of Polynomial with One Secret (IP1S), and enjoys shorter key than many other schemes based on the hardness of a NPcomplete problem. ..."
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This paper presents a practical cryptanalysis of the Identification Scheme proposed by Patarin at Crypto 1996. This scheme relies on the hardness of the Isomorphism of Polynomial with One Secret (IP1S), and enjoys shorter key than many other schemes based on the hardness of a NPcomplete problem. Patarin proposed concrete challenges that were never broken faster than exhaustive search so far. On the theoretical side, IP1S has been shown to be harder than Graph Isomorphism, which makes it an interesting target. We present two new deterministic algorithms to attack the IP1S problem, and we rigorously analyze their complexity and success probability. We show that they can solve a (big) constant fraction of all the quadratic instances in polynomial time. Our algorithms are also very efficient in practice. All the quadratic challenges proposed by Patarin are now broken in a few seconds. The cubic challenges are broken in less than a month. The identification scheme is thus broken beyond repair.
Asymptotics of Plancherel measures for GL(n,q).
, 806
"... We introduce the Plancherel measures on the sets of partition collections {Λφ}, which parameterize irreducible representations of GL(n,q). Consider the Plancherel random collection {Λφ}. We prove that as n → ∞, the random partitions Λφ converge in finite dimensional distribution to independent rando ..."
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We introduce the Plancherel measures on the sets of partition collections {Λφ}, which parameterize irreducible representations of GL(n,q). Consider the Plancherel random collection {Λφ}. We prove that as n → ∞, the random partitions Λφ converge in finite dimensional distribution to independent random variables. We give explicit formulas for the corresponding limit distributions. 1. Introduction. Let GL(n, q) be the group of all invertible n×n matrices over Fq, where q is a prime power. Denote by Φ the set of all irreducible polynomials over Fq with unit highest coefficient and nonzero constant term. The irreducible representations of GL(n, q) are parameterized by collections