# Conjugate Real Classes in General Linear Groups

## TL;DR

This paper characterizes $c$-real elements in general linear groups over fields with involution, showing they correspond exactly to elements representable in some unitary group over the same field.

## Contribution

It establishes a precise equivalence between $c$-reality of elements and their representability in unitary groups over fields with involution.

## Key findings

- $c$-real elements are conjugate to their $c$-inverse
- Characterization holds for all $n
geq 2$
- Connection between $c$-reality and unitary group representations

## Abstract

Let $\F$ be a field with a non-trivial involution $c: \alpha \to \alpha^c$. An element $g \in {\rm GL}_n(\F)$ is called $c$-real if it is conjugate to $(g^c)^{-1}$. We prove that for $n \geq 2$, $g \in {\rm GL}_n(\F)$ is $c$-real if and only if it has a representation in some unitary group of degree $n$ over $\F$.