Abstract Nonsense

Crushing one theorem at a time

Representation Theory: A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces (Pt. II)


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

 

Conversely, let \rho:G\to\mathscr{U}\left(\mathscr{V}\right) be a complex or quaternionic irreducible \mathbb{C}-representation and suppose that \{\bold{0}\}<\mathscr{W}<\mathscr{V}\oplus\mathscr{V} is \left(\rho\oplus\text{Conj}_\rho^J,\widetilde{J}\right)-invariant. Let then \mathscr{W}_2=\mathscr{W}\cap\left(\{\bold{0}\}\times\mathscr{V}\right) if \pi_2:\mathscr{W}_2\to\mathscr{V}:(0,w)\mapsto w we claim that \pi_2\left(\mathscr{W}_2\right) is \text{Conj}_\rho^J-invariant. Indeed, let g\in G and w\in\pi_2\left(\mathscr{W}_2\right) be arbitrary, then (0,w)\in\mathscr{W}_2 and so by assumption, since \mathscr{W} is \rho\oplus\text{Conj}_\rho^J-invariant, we have that \left(\rho(g)\oplus\text{Conj}_\rho^J(g)\right)(0,w)\in\mathscr{W} but evidently it’s also a member of \{\bold{0}\}\times\mathscr{V} since \left(\rho\oplus\text{Conj}_\rho^J\right)(0,w)=\left(0,\left(\text{Conj}_\rho^J(g)\right)(w)\right). It then follows that \left(0,\left(\text{Conj}_\rho^J(g)\right)(w)\right)\in\mathscr{W}_2 and thus \left(\text{Conj}_\rho^J(g)\right)(w)\in\pi_2\left(\mathscr{W}_2\right) as desired. It follows then by assumption (since \rho is an irrep and thus \text{Conj}_\rho^J is an irrep) that \pi_2\left(\mathscr{W}_2\right)=\{\bold{0}\} or \pi_2\left(\mathscr{W}_2\right)=\mathscr{V}–or equivalently \mathscr{W}_2=\{(0,0)\} or \mathscr{W}_2=\{\bold{0}\}\times\mathscr{V}. Suppose that the latter is true, then let v\in\mathscr{V} be arbitrary, then by assumption \left(0,J(v)\right)\in\{\bold{0}\}\times\mathscr{V}=\mathscr{W}_2\subseteq\mathscr{W} and since \mathscr{W} is \widetilde{J}-invariant we have that \widetilde{J}\left(0,J(v)\right)=\left(J(J(v)),J(0)\right)=(v,0)\in\mathscr{W}. Thus, since v\in\mathscr{V} was arbitrary we have that \mathscr{V}\times\{\bold{0}\}\subseteq\mathscr{W}. Thus, we must have (since both of the following are contained in the subspace \mathscr{W}) that \mathscr{V}\times\{\bold{0}\}+\{\bold{0}\}\times\mathscr{V}=\mathscr{V}\oplus\mathscr{V}\subseteq\mathscr{W} so that \mathscr{W}=\mathscr{V}\oplus\mathscr{V} contradictory to assumption. Thus, we may assume that \mathscr{W}_2=\{(0,0)\}. Using the exact same methodology we can show that \left\{w:(w,0)\in\mathscr{W}\right\} is a \rho-invariant subspace and thus by the same reasoning we must have that \mathscr{W}\cap\left(\mathscr{V}\times\{\bold{0}\}\right) is either \{(0,0)\} or \mathscr{V}\times\{\bold{0}\}. We clearly must have though that it is the latter.

 

 

Thus, for every v\in\mathscr{V} we know that there exists some w\in\mathscr{V} such that (v,w)\in\mathscr{W}. Moreover, we know that w\in\mathscr{V} is the only element of \mathscr{V} which appears as a second coordinate after v. Indeed, if (v,u)\in\mathscr{W} then (v,w)-(v,u)=(0,w-u)\in\mathscr{W} but by the first part of the above paragraph we know that w-u=\bold{0} or w=u. Thus, we can unambiguously define the map A:\mathscr{V}\to\mathscr{V} by A(v) is the unique element of \mathscr{V} such that \left(v,A(v)\right)\in\mathscr{W}. Note that evidently A is linear since (\alpha v,\alpha A(v))+(\beta w,\beta A(w))=\left(\alpha v+\beta w,\alpha A(v)+\beta A(w)\right) and so by definition A\left(\alpha v+\beta w\right)=\alpha A(v)+\beta A(w). Note though that \ker A is \rho-invariant since if v\in\ker A then (v,0)\in\mathscr{V} and so \left(\rho_g\oplus\text{Conj}_\rho^J(g)\right)\left(v,0\right)=\left(\rho_g(v),0\right)\in\mathscr{W} so that A\left(\rho_g(v)\right)=0 or \rho_g(v)\in\ker A. But, evidently \ker A\ne\mathscr{V} otherwise \mathscr{W}=\left\{(v,A(v)):v\in\mathscr{V}\right\}=\left\{(v,0):v\in\mathscr{V}\right\} but a quick check shows then that \mathscr{W} is not \widetilde{J}-invariant in this case. Thus, \ker A=\{\bold{0}\} and thus \text{im} A=\mathscr{V}. Note though that for any g\in G and v\in\mathscr{V}

 

\left(\rho_g\oplus\text{Conj}_\rho^J(g)\right)\left(v,A(v)\right)=\left(\rho_g(v),J\rho_g(v)JA(v)\right)

 

so that  (since \mathscr{W} is \rho\oplus\text{Conj}_\rho^J-invariant) A\rho_g=J\rho_g(v)JA or A\rho_gA^{-1}=J\rho_g J for every g\in G–and by an earlier theorem we may conclude that \rho is self-conjugate. Thus, there exists some unitary U\in\mathcal{U}\left(\mathscr{V}\right) such that UJ\rho_g JU^{-1}=\rho_g for every g\in G. Thus, we have that

 

J\rho_g J=A\rho_gA^{-1}=AUJ\rho_gJU^{-1}A^{-1}

 

for every g\in G and thus (since \rho and so \text{Conj}_\rho^J is an irrep) by Schur’s lemma we may conclude that AU=\alpha\mathbf{1} and so A=\alpha U^{-1}. Note though that since \mathscr{W} is \widetilde{J}-invariant one has that for every v\in\mathscr{V}

 

\widetilde{J}\left(v,A(v)\right)=\left(JA(v),J(v)\right)

 

so that AJA=J. Or using the fact that A=\alpha U^{-1} we have that (\alpha U^{-1})J(\alpha U^{-1})=|\alpha|^2 U^{-1}JU^{-1}=J so that |\alpha|=1 and so A itself is unitary. Finally, let K=JA=A^{-1}J then K is antilinear, antiunitary, and K^2=KK=JAA^{1}J=J^2=\mathbf{1} and so K is a complex conjugate. Note though that

 

K\rho_g K=A^{-1}J\rho_g J=A^{-1}\left(A\rho_g A^{-1}\right)A=\rho_g

 

for every g\in G. Thus, by one of our previous characterizations of real irreps we may conclude that \rho is real–contradictory to assumption. The conclusion follows. \blacksquare

 

 

References:

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

 

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April 2, 2011 - Posted by | Uncategorized

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