On symplectic transformations

TL;DR

This paper reviews T.A. Springer's 1951 thesis on classifying conjugacy classes in symplectic groups over fields with characteristic not 2, highlighting the invariants used for classification.

Contribution

It provides a detailed analysis of the invariants characterizing conjugacy classes in symplectic groups, including polynomials, Hermitian forms, and quadratic forms.

Findings

Conjugacy classes are characterized by invariants including polynomials and forms.

Classification extends the general linear group case to symplectic groups.

The invariants involve Hermitian and quadratic forms over fields of characteristic not 2.

Abstract

This is an English translation, prepared by Wilberd van der Kallen, of the 1951 thesis by T.A. Springer. In this thesis Springer studied the classification of conjugacy classes in the symplectic group $Sp_n(K)$, where the commutative field $K$ is of characteristic different from 2. He finds that each conjugacy class is characterized by a system of invariants. These invariants are first of all -- as in the case of the general linear group -- irreducible polynomials and systems of non-negative integers, but secondly also equivalence classes of certain Hermitian forms and of certain quadratic forms. Original Dutch title: Over symplectische transformaties.