Abstract Nonsense

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

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

Invariant Subspaces and Subrepresentations

Let $\mathscr{V}$ be a finite-dimensional pre-Hilbert space and $G$ a finite group. Let $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ be a representation. Then, we call $\mathscr{W}\leqslant \mathscr{V}$ invariant under $\rho$ if $\mathscr{W}$ is invariant under $\rho_g$ in the normal sense for each $g\in G$. It’s clear then that the map

$\omega:G\to\mathcal{U}\left(\mathscr{W}\right):g\mapsto \left(\rho_g\right)_{\mid\mathscr{W}}$

is a representation of $G$ on $\mathscr{W}$. Such a representation is called a subrepresentation and is denoted $\rho_{\mid\mathscr{W}}$ or $\rho\upharpoonright \mathscr{W}$. Our first result concerning invariant subspaces and subrepresentations is a fundamental one. Namely:

Theorem: Let $\mathscr{V}$ be a finite dimensional pre-Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ a representation of $G$ on $\mathscr{V}$. Suppose that $\mathscr{W}\leqslant \mathscr{V}$ is invariant under $\rho$ then so is

$\mathscr{W}^{\perp}=\left\{v\in\mathscr{V}:\langle v,w\rangle=0\text{ for all }w\in\mathscr{W}\right\}$

Moreover, $\rho\simeq \rho_{\mid\mathscr{W}}\oplus\rho_{\mid\mathscr{W}^{\perp}}$.

Proof: Fix $x\in\mathscr{W}^{\perp}$, and let $w\in\mathscr{W}$ and $g\in G$ be arbitrary. We note then that

\displaystyle \begin{aligned}\left\langle \rho_g(x),w\right\rangle &= \left\langle x,\rho_g^\ast(w)\right\rangle\\ &= \left\langle x,\rho_g^{-1}(w)\right\rangle\\ &= \left\langle x,\rho_{g^{-1}}(w)\right\rangle\\ &= 0\end{aligned}

and since $w,g$ were arbitrary it follows that $\rho_g(x)\in\mathscr{W}^{\perp}$ for every $g\in G$. But, by the arbitrariness of $x$ we may conclude that $\rho_g(x)\in\mathscr{W}^{\perp}$ for every $g\in G$ and $x\in \mathscr{W}^{\perp}$. Said differently $\mathscr{W}^{\perp}$ is invariant under $\rho$.

Now, to prove that $\rho\simeq \rho_{\mid\mathscr{W}}\oplus\rho_{\mid\mathscr{W}^{\perp}}$ we define

$U:\mathscr{V}\to\mathscr{W}\oplus\mathscr{W}^{\perp}:w+w'\mapsto (w,w')$

where $w+w'$ is the unique representation of every element of $\mathscr{V}$ as the sum of an element of $\mathscr{W}$ and an element of $\mathscr{W}^{\perp}$ (this comes from the elementary fact that $\mathscr{V}=\mathscr{W}\oplus\mathscr{W}^{\perp}$ where now the $\oplus$ denotes the internal direct sum). This is evidently a unitary map since (recalling that $\langle w_1,w'_2\rangle=\langle w'_1,w_2\rangle=0$ by definition)

\begin{aligned}\left\langle U\left(w_1+w'_1\right),U\left(w_2+w'_2\right)\right\rangle &= \left\langle (w_1,w'_1),(w_2,w'_2)\right\rangle\\ &= \left\langle w_1,w_2\right\rangle +\left\langle w_1,w'_2\right\rangle+\left\langle w'_1,w_2\right\rangle+\left\langle w'_1,w'_2\right\rangle\\ &= \left\langle w_1+w'_1,w_2+w'_2\right\rangle\end{aligned}

We then note that for every $g\in G$ and $w+w'\in\mathscr{V}$ we have that

\begin{aligned}U\rho_g(w+w') &=U(\rho_g(w)+\rho_g(w'))=U(\left(\rho_g\right)_{\mid\mathscr{W}}(w)+\left(\rho_g\right)_{\mid\mathscr{W}^{\perp}}(w')\\ &=\left(\left(\rho_g\right)_{\mid\mathscr{W}}(w),\left(\rho_{g}\right)_{\mid\mathscr{W}^{\perp}}(w')\right)\\ &=\left(\left(\rho_g\right)_{\mid\mathscr{W}}\oplus\left(\rho_g\right)_{\mid\mathscr{W}^{\perp}}\right)U\left(w+w'\right)\\ &= \left(\rho_{\mid\mathscr{W}}\oplus\rho_{\mid\mathscr{W}^{\perp}}\right)_gU\left(w,w'\right)\end{aligned}

from where it follows that $\rho\simeq\rho_{\mid\mathscr{W}}\oplus\rho_{\mid\mathscr{W}^{\perp}}$. $\blacksquare$

We also have somewhat of a converse of this theorem:

Theorem: Let $\mathscr{V}$ be a finite dimensional pre-Hilbert space and $G$ a finite group. Then, if $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ is a representation of $G$ on $\mathscr{V}$ such that $\rho\simeq \omega_1\oplus\omega_2$ where

$\omega_1:G\to\mathcal{U}\left(\mathscr{W}\right)$

and

$\omega_2:G\to\mathcal{U}\left(\mathscr{U}\right)$

then if $U:\mathscr{W}\oplus\mathscr{U}\to\mathscr{V}$ is the guaranteed unitary isomorphism such that $U^{-1}\rho_gU=\left(\omega_1\oplus\omega_2\right)_g$ for every $g\in G$ then $U\left(\mathscr{W}\times\{e\}\right)$ is invariant under $\rho$.

Proof: Let $U((w,e))\in U\left(\mathscr{W}\times\{e\}\right)$ then we have that $\rho_g(U(w,e))=U\left(\left(\omega_1\oplus\omega_2\right)_g(w,e)\right)\in U\left(\mathscr{W}\times\{e\}\right)$. Since $g\in G$ and $U((w,e))\in U\left(\mathscr{W}\times\{e\}\right)$ were arbitrary the conclusion follows. $\blacksquare$

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 -

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