# Abstract Nonsense

## Subrepresentations, Direct Sum of Representations, and Irreducible Representations

Point of post: In this post we discuss the notion of subrepresentations and irreducible representations, i.e. irreps.

Motivation

This is the post which inevitably happens when discussing any new subject. It is the post where one defines the “simplest” substructures of which all other structures are “built” out of. In essence, we will discuss the concept of irreducible representations (irreps) which are the indivisible parts out of which all representations are built. To discuss this concept though we are bound by formality to build up enough machinery to state precisely what “indivisible” and “built up” means. This entails the topics of subrepresentations and the direct sum of representations which, unsurprisingly, are precisely what they sound like.

Direct Sum of Representations

Let $\mathscr{V}$ and $\mathscr{U}$ be finte-dimensional pre-Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. We can define an inner product on $\mathscr{V}\oplus\mathscr{U}$ by

$\left\langle (v,u),(v',u')\right\rangle=\langle v,v'\rangle_1+\langle u,u'\rangle_2\quad\mathbf{(1)}$

it’s trivial that this does define an inner product on $\mathscr{V}\oplus\mathscr{U}$.  With this definition of inner product we can define the direct product of the pre-Hilbert spaces $\left(\mathscr{V},\langle\cdot,\cdot\rangle\right)$ and $\left(\mathscr{U},\langle\cdot,\cdot\rangle_2\right)$ to be the vector space $\mathscr{V}\oplus\mathscr{U}$ with the inner product given in $\mathbf{(1)}$. We define, pursuant to earlier definitions, the direct sum of $T_1\in\text{End}\left(\mathscr{V}\right)$ and $T_2\in\mathscr{U}$ to be the map

$T_1\oplus T_2:\mathscr{V}\oplus\mathscr{U}\to\mathscr{V}\oplus\mathscr{U}:(v,u)\mapsto \left(T_1(v),T_2(u)\right)$

We note then that if $T_1\in\mathcal{U}\left(\mathscr{V}\right)$ and $T_2\in\mathcal{U}\left(\mathscr{U}\right)$ then $T_1\oplus T_2\in\mathcal{U}\left(\mathscr{V}\oplus\mathscr{U}\right)$. Indeed, for any $\left(v,u\right),(v',u')\in\mathscr{V}\oplus\mathscr{U}$ we have that

\displaystyle \begin{aligned}\left\langle \left(T_1\oplus T_2\right)\left(v,u\right),\left(T_1\oplus T_2\right)\left(v',u'\right)\right\rangle &= \left\langle T_1(u),T_1(u')\right\rangle_1+\left\langle T_2(v),T_2(v')\right\rangle_2\\ &= \langle u,u'\rangle_1+\langle v,v'\rangle_2\\ &= \left\langle (v,u),\left(v',u'\right)\right\rangle\end{aligned}

Thus, it makes sense to define the direct sum of the representations $\rho:G\to \mathcal{U}\left(\mathscr{V}\right)$ and $\rho':G\to\mathcal{U}\left(\mathscr{U}\right)$, denoted $\rho\oplus\rho'$, to be the map

$\rho\oplus\rho':G\to\mathcal{U}\left(\mathscr{V}\oplus\mathscr{U}\right):g\mapsto \rho_g\oplus\rho'_g$

The only possible uncertainty about the above definition is whether $\rho\oplus\rho'$ is a homomorphism. But, this follows clearly from the simple calculation

$\left(\rho\oplus\rho'\right)(gg')= \rho_{gg'}\oplus \rho_{gg'}= \left(\rho_{g}\rho_{g'}\right)\oplus\left(\rho'_g\rho'_{g'}\right)=\left(\rho_g\oplus\rho_{g'}\right)\left(\rho'_g\oplus\rho'_{g;}\right)$

where we’ve used the easy to verify fact that in general

$\left(T_1\oplus T_2\right)\left(T_1'\oplus T_2'\right)=T_1T_1'\oplus T_2 T_2'$

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 -