Elements in finite classical groups whose powers have large 1-Eigenspaces

TL;DR

This paper estimates the proportion of certain elements in finite classical groups that have powers with large fixed point subspaces and act irreducibly on their complements, aiding in recognition algorithm complexity analysis.

Contribution

It provides new estimates for element proportions in finite classical groups with specific eigenspace properties, supporting algorithmic recognition methods.

Findings

Proportion estimates for elements with large eigenspaces

Application to complexity analysis of recognition algorithms

Enhanced understanding of element structure in classical groups

Abstract

We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.