Abstract Nonsense

Crushing one theorem at a time

A Bijection Between A Subset of the Complex Reps of a Finite Group and the Real Reps (Pt. I)


Point of post: In this post we describe a certain subset of the set of all \mathbb{C}-representations of a finite group G and show that this subset is in bijective correspondence with the set of all \mathbb{R}-representations of G. Moreover, we shall show that this bijection naturally restricts to a subset of the set of all \mathbb{C}-reps of G to the the \mathbb{R}-irreps of G.

Motivation

In our last post we discussed the notion of \mathbb{R}representations for a finite group G. Naturally our first desire would be to see if we could, in some way, connect \mathbb{R}representations of G to the \mathbb{C}-representations  which has held the center of our attention for so long. We begin this process in this post by showing that there is a natural place for which these \mathbb{R}-representations occur. Namely, we shall see that every \mathbb{C}-representation \rho:G\to\mathcal{U}\left(\mathscr{V}\right) for which there is a complex conjugate J for which J\rho(g)J=\rho(g) for every g\in G naturally admits a \mathbb{R}-representation \rho_{\Re}. We shall show then that in fact the reverse is true–namely that for every \mathbb{R}-representation \psi there is a natural way to produce a \mathbb{C}-representation \rho such that \rho_{\Re}=\psi. Moreover, we’ll show that if \rho:G\to\mathcal{U}\left(\mathscr{V}\right) is a \mathbb{C}-representation satisfying the aforementioned conditions and the added condition that there does not exist \mathscr{W}\leqslant \mathscr{V} such that J\left(\mathscr{W}\right)=\mathscr{W} and \mathscr{W} is \rho-invariant then we shall see that \rho_{\Re} is an irreducible \mathbb{R}-representation.

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March 29, 2011 Posted by | Algebra, Representation Theory | , , , , | 3 Comments