A theorem on power series with applications to classical groups over finite fields

For some of the classical groups over finite fields it is possible to express the proportion of eigenvalue-free matrices in terms of generating functions. We prove a theorem on the monotonicity of the coefficients of powers of power series and apply this to the generating functions of the general linear, symplectic and orthogonal groups. This proves a conjecture on the monotonicity of the proportions of eigenvalue-free elements in these groups.

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ON INVOLUTIONS AND INDICATORS OF FINITE ORTHOGONAL GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s144678871800006x ◽

2018 ◽

Vol 105 (3) ◽

pp. 380-416 ◽

Cited By ~ 1

Author(s):

GREGORY K. TAYLOR ◽

C. RYAN VINROOT

Keyword(s):

Asymptotic Behavior ◽

Simple Group ◽

Generating Functions ◽

Finite Simple Group ◽

Point Of View ◽

Character Degree ◽

Simple Groups ◽

Finite Simple Groups ◽

Classical Groups ◽

Orthogonal Groups

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

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Binomial subsampling

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1622 ◽

2006 ◽

Vol 462 (2068) ◽

pp. 1181-1195 ◽

Cited By ~ 12

Author(s):

Carsten Wiuf ◽

Michael P.H Stumpf

Keyword(s):

Mathematical Biology ◽

Power Series ◽

Generating Functions ◽

Binomial Distribution ◽

Random Variable ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

The Family ◽

Necessary And Sufficient ◽

Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X , p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X , p ) to mean that given X = x , Z is a draw from the binomial distribution Bi( x , p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

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