Abstract Nonsense

Representation Theory: Definitions and Basics

Point of post: In this post I will start my discussion of the representation theory of finite groups from a simplistic point of view

Motivation

The beginnings of representation theory can be traced all the way back to the likes of Gauss, Lagrange, et al. From then it’s spread like wildfire to the point where one would be hard-pressed to find any subject in modern mathematics which doesn’t benefit substantially from some aspect of representation theory. What is representation theory, this mystical subject? Representation theory is just another instance of a common technique in mathematics. This technique can be badly verbalized as “simplification through transfer”. In essence, one takes a complex object/structure and associates with it another object/structure which is more tractable. For example, given the $2$-sphere $\mathbb{S}^2$ it is possible to extract by purely topological means books upon books of information: it’s compact, connected, path connected, second countable, etc. But, it happens that $\mathbb{S}^2$, such an ostensibly simple structure, does not yield all of its secrets to purely topological methods. For example, using all of these properties it’s impossible (in any practical sense) to decide whether $\mathbb{S}^1\approx \mathbb{T}^2$ (where $\mathbb{T}^2$ is the torus). Out of this was born algebraic topology, where we learned that we could tell much about a topological space $X$ by associating to it certain groups: homotopy groups and homology groups for example. In short, we turned damn near intractable questions in point-set topology into questions about groups.

Representation theory lives in the same vein. We aim to take complicated problems in group theory and transform them magically into questions in the tamer field of linear algebra. Put explicitly, a representation of a group $G$ is a homomorphism $\rho:G\to\text{GL}\left(\mathscr{V}\right)$ where $\text{GL}\left(\mathscr{V}\right)$ is the group of invertible linear transformations on some complex vector space $\mathscr{V}$, thought of as a group under the usual multiplication (composition). With this in mind, let’s begin

Representation Definition

Let $G$ be a finite group and $\mathscr{V}$ be a finite-dimensional complex vector space. Then a representation of $G$ on $\mathscr{V}$ is a homomorphism

$\rho:G\to \text{GL}\left(\mathscr{V}\right)$

where $\text{GL}\left(\mathscr{V}\right)$ is defined, as usual, to be the set of invertible endomorphisms on $\mathscr{V}$ under composition. Often when dealing explicitly with images of group elements we will tend to write $\rho_g$ (or whatever symbol we use for the representation with) in lieu of $\rho(g)$. We define the degree of $\rho$ to be the dimension of $\mathscr{V}$, or symbolically $\deg \rho=\dim_{\mathbb{C}}\mathscr{V}$. We call $\mathscr{V}$ to be a representation space of $\mathscr{V}$, or more commonly (indulging in somewhat of a notational inaccuracy) a representation of $\mathscr{V}$.

Unitary Representations

Recall that an inner product $\langle \cdot,\cdot\rangle$ on a complex vector space $\mathscr{V}$ of finite dimension is defined to be map $\langle\cdot,\cdot\rangle:\mathscr{V}\times\mathscr{V}\to\mathbb{C}$ such that the following axioms hold:

\begin{aligned}&\mathbf{(1)}\quad \langle x,y\rangle=\overline{\langle y,x\rangle}\\ &\mathbf{(2)}\quad \left\langle \alpha x+\beta x',y\right\rangle=\alpha\left\langle x,y\right\rangle+\beta\langle x',y\rangle\\ &\mathbf{(3)}\quad \left\langle x,\alpha y+\beta y'\right\rangle=\overline{\alpha}\langle x,y\rangle+\overline{\beta}\langle x,y'\rangle\\ &\mathbf{(4)}\quad \langle x,x\rangle\geqslant0\text{ and }\langle x,x\rangle=0\text{ iff }x=\bold{0}\end{aligned}

(note that these axioms aren’t logically independent, see here for more information). We call a complex vector space $\mathscr{V}$ with a designated inner product an inner product space or a pre-Hilbert space. Assuming you already have a fair background in the theory of inner product spaces (once I actually get to blogging about them with my concurrently running lin. alg. series I’ll link to them) we just remind the reader of a few key facts and definitions. Let $\left(\mathscr{V},\langle \cdot,\cdot\rangle_1\right)$ and $\left(\mathscr{W},\langle \cdot,\cdot\rangle_2\right)$ be two pre-Hilbert spaces. Then, $T\in\text{Hom}\left(\mathscr{V},\mathscr{W}\right)$ is called unitary if for all $u,v\in\mathscr{V}$ we have

$\langle u,v\rangle_1=\left\langle T(u),T(v)\right\rangle_2$

We reserve the term unitary endomorphism when both the domain and codomain of the endomorphism not only coincide as vector spaces but also as pre-Hilbert spaces. We denote the set of all unitary endomorphisms on a pre-Hilbert space $\left(\mathscr{V},\langle\cdot,\cdot\rangle\right)$ by $\mathcal{U}\left(\mathscr{V}\right)$. We first note that for any pre-Hilbert space $\mathscr{V}$ one has that $\mathcal{U}\left(\mathscr{V}\right)\leqslant \text{GL}\left(\mathscr{V}\right)$. Indeed, it’s clear that $\mathcal{U}\left(\mathscr{V}\right)\subseteq\text{GL}\left(\mathscr{V}\right)$ since invertibility is equivalent to injectivity for endomorphisms on finite-dimensional spaces and for any $T\in\mathcal{U}\left(\mathscr{V}\right)$ and distinct $u,v\in\mathscr{V}$ we have that

$0\ne \langle u-v,u-v\rangle=\left\langle T(u-v),T(u-v)\right\rangle=\left\langle T(u)-T(v),T(u)-T(v)\right\rangle$

and thus $T(u)-T(v)\ne 0$ and thus $T(u)\ne T(v)$, from where injectivity and thus invertibility follow. The fact that $\mathcal{U}\left(\mathscr{V}\right)$ is a subgroup follows from the fact that if $T,T'\in\mathcal{U}\left(\mathscr{V}\right)$ then evidently for any $u,v\in\mathscr{V}$ we have that

$\left\langle TT'(u),TT'(v)\right\rangle=\left\langle T(u),T(v)\right\rangle=\left\langle u,v\right\rangle$

and if $T\in\mathcal{U}\left(\mathscr{V}\right)$ then $T^{-1}\in\mathcal{U}\left(\mathscr{V}\right)$ since

$\left\langle T^{-1}(u),T^{-1}(v)\right\rangle=\left\langle TT^{-1}(u),TT^{-1}(v)\right\rangle=\langle u,v\rangle$

With this in mind we define a homomorphism $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ where $G$ is a finite group and $\mathscr{V}$ a finite-dimensional complex pre-Hilbert space a unitary representation of $G$ in $\mathscr{V}$.

Our first theorem regarding representations is that given any representation $\rho:G\to\text{GL}\left(\mathscr{V}\right)$ where $\left(\mathscr{V},\langle\cdot,\cdot\rangle_0\right)$ is a pre-Hilbert space then we can canonically define an inner product $\langle\cdot,\cdot\rangle$ on $\mathscr{V}$ for which $\rho_g$ is unitary under this new inner product. More formally:

Theorem: Let $\left(\mathscr{V},\langle\cdot,\cdot\rangle_0\right)$ be a pre-Hilbert space and $\rho:G\to\text{GL}\left(\mathscr{V}\right)$ a representation then

$\displaystyle \langle u,v\rangle=\frac{1}{|G|}\sum_{g\in G}\left\langle \rho_g(u),\rho_g(v)\right\rangle_0$

is an inner product on $\mathscr{V}$ for which $\rho_h$ is a unitary endomorphism on $\left(\mathscr{V},\langle \cdot,\cdot\rangle\right)$ for each $h\in G$.

Proof: It’s clear that $\langle\cdot,\cdot\rangle$ is, in fact, an inner product on $\mathscr{V}$. To see that each of the $\rho_h$ are unitary it suffices to note that for every $u,v\in\mathscr{V}$ we have that

\displaystyle \begin{aligned}\langle \rho_g(u),\rho_g(v)\rangle &= \frac{1}{|G|}\sum_{g\in G}\left\langle \rho_g\left(\rho_h(u)\right),\rho_g\left(\rho_h(v)\right)\right\rangle_0\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle \rho_{gh}(u),\rho_{gh}(v)\right\rangle_0\\ &= \frac{1}{|G|}\sum_{g\in G}\left\langle \rho_g(u),\rho_g(v)\right\rangle_0\\ &= \langle u,v\rangle\end{aligned}

from where the conclusion follows. $\blacksquare$

With this in mind we now make the convention that when we say “representation” it’s always meant to be interpreted as “unitary representation”. The above theorems says that there is really no loss is making this restriction since using the above one can show for any homomorphism $\rho:G\to\text{GL}\left(\mathscr{V}\right)$ where $G$ is finite and $\mathscr{V}$ is some finite-dimensional complex pre-Hilbert space there exists some $T\in\text{GL}\left(\mathscr{V}\right)$  such that $T\rho_g T^{-1}\in\mathcal{U}\left(\mathscr{V}\right)$ for all $g\in G$.

References:

1.Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. Linear Representations of Finite Groups. New York: Springer-Verlag, 1977. Print.

January 18, 2011 - Posted by | Algebra, Representation Theory

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3. […] has the usual inner product ,denoted just when no confusion will arise, given […]

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4. Some typos to be fixed:

Motivation
“we could tell must” to “we could tell much”
“poin-set” to “point-set”
“Put explicit” to “Put explicitly”

Unitary Representation
inner product mapping “V x V -> V” changed so the map is to the complex numbers

Comment by Stephen Tashiro | September 19, 2011 | Reply

• Stephen,

Thank you very much for the corrections! They have been changed!

Best,
Alex

Comment by Alex Youcis | September 19, 2011 | Reply

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