Abstract Nonsense

Crushing one theorem at a time

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.


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

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