A generating function approach to the enumeration of matrices in classical groups over finite fields

Fulman, Jason; Neumann, Peter M.; Praeger, Cheryl E.

doi:10.1090/memo/0830

Cited by 43 publications

(75 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Of special interest is the quantity α(2, P SL); in this case we have 1/2 α(2, P SL) 1 (10) for the same reason as inequality (7). The question again is to close or reduce this gap.…”

Section: Open Questionsmentioning

confidence: 98%

See 1 more Smart Citation

On the Number ofp-Regular Elements in Finite Simple Groups

Babai¹,

Pálfy²,

Saxl³

2009

LMS J. Comput. Math.

View full text Add to dashboard Cite

A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n − 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group A n , this proportion is at least 26/(27 √ n), and for sporadic simple groups, at least 2/29.We also show that for an arbitrary field F , if the simple group S is a quotient of a finite subgroup of GL n (F ) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998). Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.Finally we complement our lower bound results with the following upper bound: for all n 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in P SL(n, q) is less than 3/ √ n.

show abstract

“…Of special interest is the quantity α(2, P SL); in this case we have 1/2 α(2, P SL) 1 (10) for the same reason as inequality (7). The question again is to close or reduce this gap.…”

Section: Open Questionsmentioning

confidence: 98%

“…2 (Guralnick and Lübeck [12]; see also Neumann and Praeger [22] and Fulman, Neumann, and Praeger [10] for certain classical groups).…”

Section: Corollary 13 There Exists a Constant C > 0 Such That For Amentioning

confidence: 99%

On the Number ofp-Regular Elements in Finite Simple Groups

Babai¹,

Pálfy²,

Saxl³

2009

LMS J. Comput. Math.

View full text Add to dashboard Cite

show abstract

“…d , see [5]) or separable matrices (c(t) multiplicity free). There are also less well-known examples.…”

Section: Further Examples Of Ni Setsmentioning

confidence: 99%

“…Often this is described by some property of the minimal polynomial, or combined properties of the characteristic and minimal polynomials. Examples include cyclic matrices, regular matrices and semisimple matrices (estimates for which are found for classical groups in [5,15,26]), primary cyclic matrices, nilpotent matrices and p-abundant matrices in classical groups [21]. Note that some of these example classes are contained in GL(d, q).…”

Section: Further Examples Of Ni Setsmentioning

confidence: 99%

Nilpotent-independent sets and estimation in matrix algebras

Corr

Popiel²,

Praeger

2015

LMS J. Comput. Math.

Self Cite

View full text Add to dashboard Cite

Efficient methods for computing with matrices over finite fields often involve randomised algorithms, where matrices with a certain property are sought via repeated random selection. Complexity analyses for such algorithms require knowledge of the proportion of relevant matrices in the ambient group or algebra. We introduce a method for estimating proportions of families N of elements in the algebra of all d × d matrices over a field of order q, where membership of a matrix in N depends only on its 'invertible part'. The method is based on the availability of estimates for proportions of certain non-singular matrices depending on N , so that existing estimation techniques for non-singular matrices can be used to deal with families containing singular matrices. As an application, we investigate primary cyclic matrices, which are used in the Holt-Rees MEATAXE algorithm for testing irreducibility of matrix algebras.

show abstract

“…From [6,Lemma 1.3.16], N * (1, q) = e and N * (d, q) = 0 for d > 1 odd. Using this together with part (1) of Lemma 3.1 (with w = u 2 /q 2i ) gives that…”

Section: This Is (Qmentioning

confidence: 99%

Generating functions for real character degree sums of finite general linear and unitary groups

Fulman

Vinroot

2014

J Algebr Comb

Self Cite

View full text Add to dashboard Cite

Abstract. We compute generating functions for the sum of the realvalued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or −1. 2010

show abstract

A generating function approach to the enumeration of matrices in classical groups over finite fields

Cited by 43 publications

References 0 publications

On the Number ofp-Regular Elements in Finite Simple Groups

On the Number ofp-Regular Elements in Finite Simple Groups

Nilpotent-independent sets and estimation in matrix algebras

Generating functions for real character degree sums of finite general linear and unitary groups

Contact Info

Product

Resources

About