Abstract Nonsense

Crushing one theorem at a time

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.

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May 8, 2011 - Posted by | Algebra, Representation Theory | , , ,

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