Computations in Classical Groups

TL;DR

This thesis develops algorithms for classical groups to compute invariants like the spinor norm and similitude characters, and analyzes the structure of z-classes in unitary groups over certain fields, with applications to finite fields.

Contribution

It introduces algorithms for symplectic and orthogonal similitude groups and establishes finiteness and enumeration results for z-classes in unitary groups over specific fields.

Findings

Algorithms for computing spinor norm and similitude characters.

Finiteness of z-classes in unitary groups over certain fields.

Counting z-classes in unitary groups over finite fields for large q.

Abstract

In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we get similitude character for symplectic and split orthogonal similitude groups, as a byproduct of our algorithms. Consider a perfect field k with odd characteristics, which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field k_0 has the property that there are only finitely many field extensions of any finite degree. In this thesis, we prove that the number of z-classes in the unitary group defined over k_0 is finite. Eventually, we count the number of z-classes in the unitary group over a finite field F_q and prove that this number is same as that of the general linear group over F_q when q is large enough.