Abstract Nonsense

Crushing one theorem at a time

*-representations of the Group Algebra


Point of post: In this post we discuss the notion of \ast-representations on the group algebra \mathcal{A}\left(G\right) of a finite group G.

Motivation

In our last few posts we’ve been discussing the notion of the group algebra \mathcal{A}\left(G\right) over a finite group and parsing some of it’s complicated definitions to get some relatively routine results. In this post we’ll show (at least start to) why the group algebra in all its  at-first-glance complexity is worth considering in our study of group representations.

\ast-representations

Let G be a finite group and \mathcal{A}\left(G\right) the group algebra on G. If \left(\mathscr{V},\langle\cdot,\cdot\rangle\right) is a pre-Hilbert space then a mapping \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right) is called a \ast-representation of \mathcal{A}\left(G\right) if the following conditions hold for every a,b\in\mathcal{A}\left(G\right)

 

\begin{aligned}&\mathbf{(1)}\quad \varrho\left(a+b\right)=\varrho(a)+\varrho(b)\\ &\mathbf{(2)}\quad \varrho\left(a\ast b\right)=\varrho(a)\varrho(b)\\ &\mathbf{(3)}\quad \varrho\left(a^\ast\right)=\varrho(a)^\ast\\ &\mathbf{(4)}\quad \varrho\left(\delta_e\right)=\mathbf{1}\end{aligned}

 

 

Notice that conditions \mathbf{(1)},\mathbf{(2)} and \mathbf{(3)} say that \varrho is an associative unital algebra embedding into \text{End}\left(\mathscr{V}\right).

Our first theorem gives us the first clue how group representations of G on \mathscr{V} and \ast-representations of \mathcal{A}\left(G\right) on \mathscr{V} are intimately linked. In particular:

Theorem: Let G be a finite group, \mathscr{V} a pre-Hilbert space and \rho:G\to\mathcal{U}\left(\mathscr{V}\right) a representation of G on \mathscr{V}. Then, the mapping


\displaystyle \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right):a\mapsto \sum_{g\in G}a(g)\rho_g

 

is a \ast-representation of \mathcal{A}\left(G\right) on \mathscr{V}.

Proof: Assume first that \varrho:\mathcal{A}\left(G\right)\to\text{End}\left(\mathscr{V}\right) is given by

\displaystyle \varrho(a)=\sum_{g\in G}a(g)\rho_g

 

for some representation \rho:G\to\mathcal{U}\left(\mathscr{V}\right). We claim that \varrho is a \ast-representation of \mathcal{A}\left(G\right) on \text{End}\left(\mathscr{V}\right). Indeed, to see that \varrho satisfies \mathbf{(1)} we must merely note that for any a,b\in\mathcal{A}\left(G\right) we have that

\displaystyle \begin{aligned}\varrho\left(a+b\right) &=\sum_{g\in G}\left(a+b\right)(g)\rho_g\\ &=\sum_{g\in G}\left(a(g)+b(g)\right)\rho_g\\ &=\sum_{g\in G}a(g)\rho_g+\sum_{g\in G}b(g)\rho_g\\ &=\varrho(a)+\varrho(b)\end{aligned}

 

To see that \varrho satisfies \mathbf{(2)} we merely note that for any a,b\in\mathcal{A}\left(G\right) we have that

\displaystyle \begin{aligned}\varrho\left(a\ast b\right) &= \sum_{g\in G}\left(a\ast b\right)(g)\rho_g\\ &= \sum_{g\in G}\left(\sum_{k\in G}a\left(gk^{-1}\right)b(k)\right)\rho_g\\ &= \sum_{g\in G}\sum_{k\in G}a\left(gk k^{-1}\right)b(k)\rho_{gk}\\ &= \sum_{g\in G}\sum_{k\in G}a(g)b(k)\rho_g\rho_k\\ &= \left(\sum_{g\in G}a(g)\rho_g\right)\left(\sum_{k\in G}b(k)\rho_k\right)\\ &= \varrho\left(a\right)\varrho\left(b\right)\end{aligned}

 

to see that \varrho satisfies \mathbf{(3)} note that

\displaystyle \begin{aligned}\varrho\left(a^\ast\right) &= \sum_{g\in G}a^\ast(g)\rho_g\\ &= \sum_{g\in G}\overline{a\left(g^{-1}\right)}\rho_g\\ &= \sum_{g\in G}\overline{a\left(g^{-1}\right)}\rho_{g^{-1}}^\ast\\ &=\sum_{g\in G}\left(a\left(g^{-1}\right)\rho_{g^{-1}}\right)^\ast\\ &= \left(\sum_{g\in G}a\left(g^{-1}\right)\rho_{g^{-1}}\right)^\ast\\ &= \left(\sum_{g\in G}a(g)\rho_g\right)^\ast\\ &= \varrho(a)^\ast\end{aligned}

 

Lastly to prove \mathbf{(4)} we merely note that

\displaystyle \varrho\left(\delta_e\right)=\sum_{g\in G}\delta_e(g)\rho_g=\rho_e=\mathbf{1}

 

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 21, 2011 - Posted by | Algebra, Representation Theory | , , ,

4 Comments »

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