# Abstract Nonsense

## *-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

January 21, 2011 -

1. […] Point of post: This is a continuation of this post […]

Pingback by Representation Theory: *-representations of the Group Algebra « Abstract Nonsense | January 21, 2011 | Reply

2. […] be a finite group and the group algebra. Then, if is a pre-Hilbert space and   a  -representation we call  -invariant (or invariant when is evident) if is invariant under for each . We then, […]

Pingback by Representation Theory: Schur’s Lemma (*-representation Form) « Abstract Nonsense | January 27, 2011 | Reply

3. […] group and be an irreducible character of . Let then be any irrep which admits as its character. Recall that we may then define the induced -representation by . Our first claim is that is irreducible. […]

Pingback by Representation Theory: A ‘Lemma’ « Abstract Nonsense | March 10, 2011 | Reply

4. […] We begin by considering the usual extension of to given […]

Pingback by Different Formula For the Character of an Induced Representation « Abstract Nonsense | April 26, 2011 | Reply