On the conjugacy classes in the unitary, symplectic and orthogonal groups

Abstract

This paper is concerned with two mani problems:(a) the determination of the conjugacy classes in the finite-dimensional unitary, symplectic and orthogonal groups over division rings or fields;(b) the determination of the equivalence classes of non-degenerate sesquilinear forms on finite-dimensional vector spaces.

Fuchsian groups, finite simple groups and representation varieties

“…Now D is a product of at most m 1 + 1 simple factors, with x 2 embedded almost freely in each, so it is sufficient to prove (14) for each simple factor of D. Let E be a simple factor, with natural module of dimension d. On this module the unipotent element x 2 acts as ((J m 2 ) l , (J 1 ) u ), where J i is a unipotent Jordan block of size i, and we have d = lm 2 + u with u ≤ t. The dimension of C E (x 2 ) can be read off from [50]: assuming E is not symplectic or orthogonal in characteristic 2, dim C E (x 2 ) is as follows:…”

Existence of an invariant form under a linear map

“…Suppose that the linear map T admits an invariant non-degenerate hermitian, resp skew-hermitian, form H. Then the necessary condition follows from existing literatures, for example see [Wal63,Ser87,Ser08].…”

“…Suppose T : V → V is a linear map that admits an invariant hermitian or skew-hermitian form H. Then canonical forms for T is known in the literature, for example, see [Wal63]. This provides the necessary condition in the above theorem.…”

On the Prime Graph of Simple Groups

“…In order to handle the cases labelled (b) -(f) in Table 2, we need information on the orders of centralisers of elements of prime order in finite symplectic and orthogonal groups. In [15], Wall provides detailed information on the conjugacy classes in finite classical groups, but we prefer to use an alternative description that is more suited to our specific needs. Let G = PSp n (q) be a symplectic group over F q , where q = p f and p is a prime.…”