Regular semisimple elements and involutions in finite general linear groups of odd characteristic. (Q2845724)

The basic question being considered in this paper is the following: given two conjugate involutions, \(x\) and \(x'\), in \(\mathrm{GL}(n,q)\), what is the probability that their product, \(xx'\), is regular semisimple?NEWLINENEWLINE The significance of the question lies in its application to computational group theory, where the computation of involution centralizers is used to distinguish between different finite groups of Lie type.NEWLINENEWLINE To answer the question, one observes first that, by a simple dimension argument, either \(x\) or \(-x\) must lie in the conjugacy class \(\mathcal C(V)\) consisting of involutions with a fixed point subspace of dimension \(\lfloor\frac{n}{2}\rfloor\). This observation translates the basic question at hand to the problem of computing NEWLINE\[NEWLINE\iota(n,q):=\frac{|\{(x,x')\in\mathcal C(V)\times\mathcal C(V)\mid xx'\text{ is regular semisimple}\}|}{|\mathcal C(V)|^2}.NEWLINE\]NEWLINE If \(n\) is odd, the same dimension considerations imply that \(xx'\) has a fixed point subspace \(W\) of dimension \(1\) such that \(x|_W=x'|_W\). This allows one to calculate \(\iota(n,q)\) when \(n\) is odd using the result of one's calculations in the even-dimensional case.NEWLINENEWLINE To handle the even-dimensional case, the authors make use of the fact that if an element \(y\in\mathrm{GL}(n,q)\) is a product of two involutions \(x\) and \(x'\), then \(y^x=y^{-1}\), i.e. \(y\) is a `strongly real' element of \(\mathrm{GL}(n,q)\), a property that is equivalent to the characteristic polynomial of \(y\) being self-conjugate. Now one proceeds by analysing the primary decomposition of \(V=(\mathbb F_q)^n\) as an \(\mathbb F_q\langle y\rangle\)-module.NEWLINENEWLINE The asymptotic results presented in this paper are expressed in terms of the quantity NEWLINE\[NEWLINE\Phi(x)=\lim_{j\to\infty}\prod_{k=1}^j(1-x^{-k}),NEWLINE\]NEWLINE which is defined for \(x>1\). In particular one has NEWLINE\[NEWLINE\begin{aligned} (1-q^{-1})^8\leq\lim_{d\to\infty}\iota(2d,q)&=(1-q^{-1})^2\Phi(q)^3\leq (1-q^{-1})^5,\\ (1-q^{-1})^7\leq\lim_{d\to\infty}\iota(2d+1,q)&=(1-q^{-1})\Phi(q)^3\leq (1-q^{-1})^4.\end{aligned}NEWLINE\]