Abstract Nonsense

Representation Theory of Semidirect Products: The Preliminaries (Pt. I)

Point of Post: In this post we begin our study of how representation theory interacts with semidirect products.

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Motivation

As of now we have spent a considerable amount of our efforts considering how given the information (relevant to representation theory that is) about two groups $G$ and $H$ what can we say about the representation theory about certain constructions based on $G$ and $H$. Probably the most important that we’ve so far discussed is the relationship between irreps $G$ and $H$ and the irreps of their direct product $G\times H$. We continue in this vein and discuss the representation theory of $G\rtimes_\varphi H$, the semidirect product in the particular case where $G$ is abelian.

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Semidirect Product Theory

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Let $G=A\rtimes_\varphi K$ where $A$ is abelian. We recall that we can identify $A$ with $\widetilde{A}=A\times\{e\}\unlhd A\rtimes_\varphi K$ and $\widetilde{K}=\{0\}\times K\leqslant A\rtimes_\varphi K$. If $a\in A$ we write $\widetilde{a}\in\widetilde{A}$ and similarly for $k\in K$. Consider then a representation $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$. We next let $\psi=\rho_{\mid \widetilde{A}}$ and let, for each $\chi\in\text{irr}(A)$

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$\mathscr{V}_\chi=\left\{v\in\mathscr{V}:\psi_{\widetilde{a}}(v)=\chi(a)v\text{ for all }\widetilde{a}\in\widetilde{A}\right\}$

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Our first theorem says:

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Theorem: Let $\psi, G$, and $\mathscr{V}^{(\alpha)}$ be as above. Then, $\mathscr{V}$ is equal to $\displaystyle \bigoplus_{\chi\in\text{irr}(G)}V_\chi$.

Proof: Since this quality is clearly invariant under equivalence it suffices to assume that $\psi$ is actually equal to

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$\displaystyle \bigoplus_{\alpha\in\widehat{\widetilde{A}}}m^{(\alpha)}\rho^{(\alpha)}:\widetilde{A}\to \mathcal{U}\left(\bigoplus_{\alpha\in\widehat{\widetilde{A}}}m^{(\alpha)}\;\mathscr{W}^{(\alpha)}\right)$

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Thus, we want to prove that, letting $\mathscr{W}$ be the representation space of the above representation,

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$\displaystyle \mathscr{W}=\bigoplus_{\chi\in\text{irr}(G)}\mathscr{W}_\chi$

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What we claim though is that

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$\displaystyle \mathscr{W}_{\chi^{(\alpha_0)}}=\widetilde{m^{(\alpha_0)}\mathscr{W}^{(\alpha_0)}}$

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for each $\alpha_0$ in $\widehat{\widetilde{A}}$.Indeed, it’s clear that $\widetilde{m^{(\alpha_0)}\mathscr{W}^{(\alpha_0)}}\subseteq\mathscr{W}_{\chi^{(\alpha_0)}}$ since by definition for each $(v^{(\alpha)})\in \widetilde{m^{(\alpha_0)}\mathscr{W}^{(\alpha_0)}}$ and each $a$ in $\widetilde{A}$ we have that

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$\displaystyle \left(\bigoplus_{\alpha\in\widehat{\widetilde{A}}}m^{(\alpha)}\rho^{(\alpha)}(a)\right)(v^{(\alpha)})=\left(\left(\rho^{(\alpha)}(a)\right)(v^{(\alpha)})\right)=\left(\chi^{(\alpha)}(a)v^{(\alpha)}\right)=\chi^{(\alpha_0)}(a)(v^{(\alpha)})$

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where we’ve used the fact that  any of the coordinates not corresponding to $\alpha_0$ is zero. Conversely, if $(v^{(\alpha)})$ is such that one of its coordinates, say $v^{(\beta)}$, is non-zero then since $\chi^{(\beta)}\ne\chi^{(\alpha_0)}$ we may choose some $a\in\widetilde{A}$ such that $\chi^{(\alpha_0)}(a)\ne\chi^{(\beta)}(a)$ it’s easy to see that

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$\displaystyle \left(\bigoplus_{\alpha\in\widehat{\widetilde{A}}}m^{(\alpha)}\rho^{(\alpha)}(a)\right)v^{(\alpha)})=(\chi^{(\alpha)}(a)v^{(\alpha)})\ne \chi^{(\alpha_0)}(a)(v^{(\alpha)})$

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from where the conclusion follows. The entire theorem then follows since

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$\displaystyle \bigoplus_{\alpha\in\widehat{\widetilde{A}}}m^{(\alpha)}\mathscr{W}^{(\alpha)}=\bigoplus_{\alpha\in\widehat{\widetilde{A}}}\widetilde{m^{(\alpha)}\mathscr{W}^{(\alpha)}}$

$\blacksquare$

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References:

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