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. II)


Point of post: This post is a continuation of this one.

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March 30, 2011 Posted by | Algebra, Representation Theory | , , , | 1 Comment

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

A Characterization of Real, Complex, and Quaternionic Irreps


Point of post: In this post we derive a result historically attributed to Frobenius and Schur which gives us a characterization to real, complex, and quaternionic irreps based on their admittant characters.

Motivation

In the past we’ve discussed how the set of all irreps are naturally carved up into three subclasses: real, complex, and quaternionic. This analogizes the difference between real, complex, and quaternionic numbers. It turns out that in general it is not, at first glance, clear how to determine from elementary methods whether or not an irrep was real, complex, or quaternionic. Indeed, in our one example of quaternionic irreps the agrument that the irrep in question was, in fact, quaternionic was involved and admittedly convoluted. That said, the theorem we develop in this post shall give us a simple way to determine whether an irrep is real, complex, and conjugate by a simple calculation involving the character of the irrep.

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March 23, 2011 Posted by | Algebra, Representation Theory | , , , , , , | 2 Comments

General Characters and the Uniqueness of Decomposition Into Irreps


Point of post: In this post we use our results about irreducible characters to show that any decomposition of a representation \rho into a direct sum of irreps is unique.  We do this by introducing the notion of a character for a general (not necessarily irreducible) representation.

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February 27, 2011 Posted by | Algebra, Representation Theory | , , , , | 4 Comments

Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. III)


Point of post: This post is a continuation of this one.

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January 18, 2011 Posted by | Algebra, Representation Theory | , , , , | 2 Comments

Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. II)


Point of post: This post is a continuation of this one.

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January 18, 2011 Posted by | Algebra, Representation Theory | , , , , , | 6 Comments