# Abstract Nonsense

## The Irreps of the Product of Finitely Many Finite Groups

Point of post: In this post we shall discuss how one can find the set of all irreps, up to equivalence, of a group of the form $G_1\times\cdots\times G_n$ given the irreps of $G_k$ for $k\in[n]$.

Motivation

We’ve developed quite an extensive theory regarding how to find the irreps of a finite group $G$ and how to relate those irreps to one another, as well as their characters.The question remains though if there is a natural way that the irreps of $G$ relate naturally to constructions based on $G$. In particular, in this post we are interested in determining the relationships between the irreps (in particular the characters) of the product $G\times H$ of two groups $G$ and $H$ given the knowledge of the characters of $G$ and $H$. So for example, it’s easy (as we’ve shown) to construct the character table for $S_3$ and it’s equally easy to construct the character table for $\mathbb{Z}_2$. A next logical step would be to combine them and find the character table for $S_3\times\mathbb{Z}_2$. It would be nice if one would have to not go tromping through all the extra work to create this (larger) character table having gone through the (admittedly small amount of work) to construct the ones for $S_3$ and $\mathbb{Z}_2$. In this post we shall show that our greatest wish is true–we can’t just easily get some characters of $S_3\times\mathbb{Z}_2$ from those of $S_3$ and $\mathbb{Z}_2$ but we can easily get all of them.

Finding The Irreducible Characters of the Finite Product of Finitely Many Groups

Let $G$ and $H$ be groups, and $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ and $\psi:H\to\mathcal{U}\left(\mathscr{W}\right)$ be representations on $G$ and $H$ respectively. We can define a new representation, called the tensor product of $\rho$ and $\psi$, on $G\times H$ by

$\text{ }$

$\rho\boxtimes\psi:G\times H\to\mathscr{U}\left(\mathscr{V}\otimes\mathscr{W}\right)$

by $(\rho\boxtimes\psi)(g,h)=\rho(g)\otimes\psi(h)$ where $\mathscr{V}\otimes\mathscr{W}$ is the tensor product of vector spaces and $\rho(g)\otimes\psi(h)$ is the tensor product of linear transformations (compare with the tensor product of representations given one group). This is indeed a representation since the tensor product of two transformations is unitary and it’s a homomorphism since

$\text{ }$

\begin{aligned}\left(\rho\boxtimes\psi\right)\left((g,h)(u,v)\right) &=\left(\rho\boxtimes\psi\right)\left(gu,hv\right)\\ &=\left(\rho(gu)\right)\otimes\left(\psi(hv)\right)\\ &=\left(\rho(g)\rho(u)\right)\otimes\left(\psi(h)\psi(v)\right)\\ &=\left(\rho(g)\otimes\psi(h)\right)\left(\rho(u)\otimes\psi(v)\right)\\ &=\left(\rho\boxtimes\psi\right)(g,h)\left(\rho\boxtimes\psi\right)(u,v)\end{aligned}

The interesting difference this type of tensor representation and the one we previously discussed is that the tensor product of two irreps of $G$ and $H$ is an irrep on $G\times H$. The fascinating thing is that the set of all such tensor product of representations constitutes all of the irreps. Indeed

Theorem: Let $G$ and $H$ be finite groups and for each $\alpha\in\widehat{G}$ and each $\beta\in\widehat{H}$ choose a representative $\rho^{(\alpha)}\in\alpha$ and $\psi^{(\beta)}\in\beta$. Then, for every such $\alpha,\beta$ one has that $\rho^{(\alpha)}\boxtimes\psi^{(\beta)}$ is an irreducible representation for $G\times H$ and $\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\cong\rho^{(\gamma)}\boxtimes\psi^{(\gamma)}$ if and only if $(\alpha,\beta)=(\gamma,\delta)$. Moreover

$\text{ }$

$\widehat{G\times H}=\left\{\left[\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right]:\alpha\in\widehat{G}\text{ and }\beta\in\widehat{H}\right\}$

$\text{ }$

Proof: Let $\alpha\in\widehat{G}$ and $\beta\in\widehat{H}$ be arbitrary. To prove that $\rho^{(\alpha)}\boxtimes\psi^{(\beta)}$ is an irrep for $G\times H$ we note that evidently $\text{tr}\left(\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right)=\chi^{(\alpha)}\chi^{(\beta)}$ and so

$\text{ }$

\displaystyle \begin{aligned}\left\langle \text{tr}\left(\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right),\text{tr}\left(\rho^{(\alpha)}\boxtimes \psi^{(\beta)}\right)\right\rangle &= \frac{1}{|G\times H|}\sum_{(g,h)\in G\times H}\left|\text{tr}\left(\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right)\right|^2\\ &= \frac{1}{|G\times H|}\sum_{(g,h)\in G\times H}\left|\chi^{(\alpha)}(g)\right|^2\left|\chi^{(\beta)}(h)\right|^2\\ &= \frac{1}{|G||H|}\sum_{g\in G}\sum_{h\in H}\left|\chi^{(\alpha)}(g)\right|^2\left|\chi^{(\beta)}(h)\right|^2\\ &= \left(\frac{1}{|G|}\sum_{g\in G}\left|\chi^{(\alpha)}(g)\right|^2\right)\left(\frac{1}{|H|}\sum_{h\in H}\left|\chi^{(\beta)}(h)\right|^2\right)\\ &= 1\cdot 1\\ &=1\end{aligned}

and thus by our alternate characterization of irreducibility we may conclude that $\rho^{(\alpha)}\boxtimes\psi^{(\beta)}$ is an irrep of $G\times H$.

To prove our second we actually prove the stronger claim that

$\text{ }$

$\left\langle\text{tr}\left(\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right),\text{tr}\left(\rho^{(\gamma)}\boxtimes\psi^{(\delta)}\right)\right\rangle=\delta_{\alpha,\gamma}\delta_{\beta,\delta}$

Indeed,

\displaystyle \begin{aligned}\left\langle \text{tr}\left(\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right),\text{tr}\left(\rho^{(\gamma)}\boxtimes\psi^{(\delta)}\right)\right\rangle &= \frac{1}{|G\times H|}\sum_{(g,h)\in G\times H}\text{tr}\left(\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right)\overline{\text{tr}\left(\rho^{(\gamma)}\boxtimes\psi^{(\delta)}\right)}\\ &= \frac{1}{|G\times H|}\sum_{(g,h)\in G\times H}\chi^{(\alpha)}(g)\chi^{(\beta)}(h)\overline{\chi^{(\gamma)}(g)}\overline{\chi^{(\delta)}(h)}\\ &= \frac{1}{|G||H|}\sum_{g\in G}\sum_{h\in H}\chi^{(\alpha)}(g)\overline{\chi^{(\gamma)}(g)}\chi^{(\beta)}(h)\overline{\chi^{(\delta)}(h)}\\ &= \left(\frac{1}{|G|}\sum_{g\in G}\chi^{(\alpha)}(g)\overline{\chi^{(\gamma)}(g)}\right)\left(\frac{1}{|H|}\sum_{h\in H}\chi^{(\beta)}(h)\overline{\chi^{(\delta)}(h)}\right)\\ &= \delta_{\alpha,\gamma}\delta_{\beta,\delta}\end{aligned}

To prove the last claim we note that we’ve defined a map $f:\widehat{G}\times\widehat{H}\to\widehat{G\times H}$ by $(\alpha,\beta)\mapsto \left[\rho^{(\alpha)}\boxtimes\psi^{(\beta)}\right]$ which by the second part of our theorem is an injection. But, by a previous theorem we have that $\#\left(\widehat{G}\times\widehat{H}\right)=\#\left(\widehat{G\times H}\right)$ and thus by a simple set theoretic fact we may conclude that $f$ is also a surjection from where the conclusion follows. $\blacksquare$

References:

1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.

April 11, 2011 -

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

Pingback by Representation Theory: The Irreps of the Product of Finitely Many Finite Groups (Pt. II) « Abstract Nonsense | April 11, 2011 | Reply

2. […] put our newly developed theory to […]

Pingback by Representation Theory: The Character Table of S_3xZ_3 « Abstract Nonsense | April 11, 2011 | Reply

3. […] of the full group if we knew the dual group of each of the product factors. In this post we show, once again, that everything carries over just as we would hope and that the product of the dual groups is […]

Pingback by Representation Theory: The Dual Group of the Product is the Product of the Dual Groups « Abstract Nonsense | April 12, 2011 | Reply

4. […] finds such a deomposition everything becomes easy. Namely, we know from our theorems regarding the irreps of the products of groups that every irrep of , up to equivalence, looks like where is an irrep of . But, since is […]

Pingback by Representation Theory: Irreps of an Abelian Group « Abstract Nonsense | April 16, 2011 | Reply

5. […] a character. Moreover, let be the trivial character for . Consider then the character , which we know is a character for . That said, it’s clear by definition that and that from where it […]

Pingback by University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2004) « Abstract Nonsense | May 6, 2011 | Reply

6. […] constructions based on and . Probably the most important that we’ve so far discussed is the relationship between irreps and and the irreps of their direct product . We continue in this vein and discuss […]

Pingback by Representation Theory of Semidirect Products: The Preliminaries (Pt. I) « Abstract Nonsense | May 8, 2011 | Reply