Abstract Nonsense

Crushing one theorem at a time

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


Point of Post: This is a continuation of this post.

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The interesting thing about this ‘dualization’ of \varphi is its relation ship between the original \rho and the aforementioned \mathscr{V}_\chi. Namely:

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Theorem: For any k\in K and \chi\in\text{irr}(A)  one has that \rho_{\widetilde{k}}\left(\mathscr{V}_{\chi}\right)\subseteq \mathscr{V}_{\widehat{\varphi}_k\left(\chi\right)}.

Proof: Let x\in\mathscr{V}_{\chi} be arbitrary, then for any \widetilde{n}\in \widetilde{N} one has

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\begin{aligned}\rho_{\widetilde{n}}\rho_{\widetilde{k}}(x) &= \rho_{\widetilde{k}}\rho_{\widetilde{k}^{-1}\widetilde{n}\widetilde{k}}(x)\\ &= \rho_{\widetilde{k}}\rho_{\widetilde{\varphi_k^{-1}(n)}}(x)\\ &=\rho_{\widetilde{k}}\chi\left(\varphi_k^{-1}(n)\right)(x)\\ &= \rho_{\widetilde{k}}\left(\widehat{\varphi}_k(\chi)\right)(n)x\\ &= \left(\widehat{\varphi}_k(\chi)\right)(n)\rho_{\widetilde{k}}(x)\end{aligned}

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And, since \widetilde{n}\in\widetilde{N} was arbitrary it follows that \rho_{\widetilde{k}}(x) is in \mathscr{V}_{\widehat{\varphi}_k(\chi)} as claimed. Since x\in\mathscr{V}_{\chi} was arbitrary the conclusion follows. \blacksquare

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From this we have the following corollary:

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Corollary: Let \chi_0\in\text{irr}(A) and \mathcal{O}_{\chi_0} the associated orbit under the action of K on \text{irr}(A) induced by \widehat{\varphi}:K\to\text{Aut}(\text{irr}(A)). Then, \displaystyle \bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi} is \rho-invariant.

Proof:  We first show that the space in question is invariant under \rho_{\widetilde{k}} for every k\in K. Let v be an element of \displaystyle \bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi}. Then, there exists v_{\chi}\in\mathscr{V}_{\chi} for every \chi\in\mathcal{O}_{\chi_0} such that \displaystyle v=\sum_{\chi\in\mathcal{O}_{\chi_0}}v_\chi. So, by our last theorem

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\displaystyle \rho_{\widetilde{k}}(x)=\sum_{\chi\in\mathcal{O}_{\chi_0}}\rho_{\widetilde{k}}(v_{\chi})\in \sum_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\widehat{\varphi}_{k}(\chi)}=\bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi}

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Since k\in K was arbitrary the conclusion follows.

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We next show that the space in question is invariant under \rho_{\widetilde{n}} for every n\in N. Indeed, if x is as above then

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\displaystyle \rho_{\widetilde{n}}(x)=\sum_{\chi\in\mathcal{O}_{\chi_0}}\rho_{\widetilde{n}}(v_\chi)=\sum_{\chi\in\mathcal{O}_{\chi_0}}\chi(n)v_\chi\in\sum_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi}=\bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi}

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Since x and n were arbitrary this part of the proof follows.

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To finish the proof we merely note that for any g\in G=A\rtimes_\varphi K one has g=\widetilde{n}\widetilde{k} for some n\in N and k\in K and so for any x in the direct sum in question one has

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\displaystyle \rho_{g}=\rho_{\widetilde{n}\widetilde{k}}(g)=\rho_{\widetilde{n}}\left(\rho_{\widetilde{k}}(x)\right)\in \rho_{\widetilde{n}}\left(\bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi}\right)\subseteq\bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi}

\blacksquare

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From this we conclude as a corollary:

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Corollary: Let \rho:A\rtimes_\varphi K\to\mathcal{U}\left(\mathscr{V}\right) be an irrep. Then, there exists precisely one orbit \mathcal{O} of the action \widehat{\varphi}:K\to\text{Aut}\left(\text{irr}(A)\right) for which \mathscr{V}_\chi\ne\{\bold{0}\} if and only if \chi\in\mathcal{O}.

Proof: We know since

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\displaystyle \bigoplus_{\chi\in\text{irr}(A)}\mathscr{V}_{\chi}=\mathscr{V}

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that there exists some \chi_0\in\text{irr}(A) such that \mathscr{V}_{\chi}\ne\bold{0}. We claim that \mathcal{O}_{\chi_0} is the orbit we seek. Indeed, to see that \mathscr{V}_{\lambda}=\{\bold{0}\} for \lambda\notin\mathcal{O}_{\chi_0} we note that since \displaystyle \bigoplus_{\chi\in\mathcal{O}_{\chi_0}}\mathscr{V}_{\chi} is non-zero and \rho-invariant we may conclude from the irreducibility of \rho that is the full space \mathscr{V}. Thus, one has that \displaystyle \bigoplus_{\chi\in\mathscr{O}_{\lambda}}\mathscr{V}_{\chi}=\{\bold{0}\} and so of course \mathscr{V}_{\lambda}=\{\bold{0}\}. Now, if \chi\in\mathcal{O}_{\chi_0} then \chi=\widehat{\varphi}_{k}(\chi_0) for some k\in K. Thus, by an earlier theorem we have that \rho_{\widehat{k}}\left(\mathscr{V}_{\chi_0}\right)\subseteq\mathscr{V}_{\chi} and so \dim\mathscr{V}_{\chi}\geqslant\dim\mathscr{V}_{\chi_0} and so \mathscr{V}_{\chi}\ne\bold{0}. \blacksquare

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From this it’s easy to see that

Corollary: If \mathcal{O} is the non-zero orbit described in the above proof then \dim\mathscr{V}_{\chi}=\dim\mathscr{V}_{\lambda} for every \chi,\lambda\in\mathcal{O}. In particular, \dim\mathscr{V}=\#\left(\mathcal{O}\right)\dim\mathscr{V}_{\chi}.

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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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