Abstract Nonsense

Crushing one theorem at a time

A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces

Point of post: In this post we discuss a way of taking a \mathbb{C}-representation and using it to create another representation which satisfies the real condition. Moreover, when the original \mathbb{C}-representation is an irrep which is quaternionic or complex then the resulting representation will have no non-trivial (\rho,J)-invariant subspaces.


In our last post we have seen that \mathbb{C}-representations satisfying the real condition are important since they correspond to real representations in a natural way. Moreover, we have seen that \mathbb{C}-representations satisfying the real condition with realizer J with no non-trivial proper \left(\rho,J\right)-invariant subspaces are even more interesting since they correspond naturally to irreducible real representations. We shall now show a method which takes a general \mathbb{C}-representations and produces a \mathbb{C}-representation which satisfies the real condition. Moreover, we shall see that if the original irrep happens to be complex or quaternionic irrep then the corresponding representation won’t have any non-trivial proper (\rho,J)-invariant subspaces.


The Method

We now describe the aforementioned method and show that it does what it claims. Namely:

Theorem: Let \rho:G\to\mathcal{U}\left(\mathscr{V}\right) be a \mathbb{C}-representation and J a complex conjugate on \mathscr{V}. Then, \widetilde{J}:\mathscr{V}\oplus\mathscr{V}\to\mathscr{V}\oplus\mathscr{V} by \widetilde{J}(x,y)=\left(J(y),J(x)\right) is a complex conjugate on \mathscr{V}\oplus\mathscr{V} (the direct sum) and \rho\oplus\text{Conj}_\rho^J is a \mathbb{C}-representation (where, as usual, this is the direct sum representation which satisfies the real condition with realizer \widetilde{J}. Moreover, \mathscr{V}\oplus\mathscr{V} will have no non-trivial proper \left(\rho\oplus\text{Conj}_\rho^j,\widetilde{J}\right)-invariant if and only if \rho is a complex or quaternionic irrep.

Proof: To see that \widetilde{J} is a complex conjugate we merely note that for every (x,y)\in\mathscr{V}\oplus\mathscr{V}




so that by the arbitrariness of (x,y) we may conclude that \widetilde{J}^2=\mathbf{1}. Next,


\begin{aligned}\widetilde{J}\left(\alpha(u,v)+\beta(x,y)\right) &=\widetilde{J}\left(\alpha u+\beta x,\alpha v+\beta y\right)\\ &=\left(J\left(\alpha v+\beta y\right),J\left(\alpha u+\beta x\right)\right)\\ &=\left(\overline{\alpha}J(v)+\overline{\beta}J(y),\overline{\alpha}J(u)+\overline{\beta}J(x)\right)\\ &=\overline{\alpha}\left(J(v),J(u)\right)+\overline{\beta}\left(J(y),J(x)\right)\\ &= \overline{\alpha}\widetilde{J}(u,v)+\overline{\beta}\widetilde{J}(x,y)\end{aligned}



\begin{aligned}\left\langle \widetilde{J}(u,v),\widetilde{J}(x,y)\right\rangle &= \left\langle (J(v),J(u)),(J(y),J(x))\right\rangle\\ &= \left\langle J(v),J(y)\right\rangle+\left\langle J(u),J(x)\right\rangle\\ &= \left\langle y,v\right\rangle+\left\langle x,u\right\rangle\\ &= \left\langle (x,y),(u,v)\right\rangle\end{aligned}


from where the fact that J\oplus J is complex conjugate on \mathscr{V}\oplus\mathscr{V} follows. Now, to see that J is a realizer for \rho\oplus\text{Conj}_\rho^J note that for any g\in G and (u,v)\in\mathscr{V}\oplus\mathscr{V} one has

\begin{aligned}\widetilde{J}\left(\rho\oplus\text{Conj}_\rho^J\right)\widetilde{J}(u,v) &=\widetilde{J}\left(\rho\oplus \text{Conj}_\rho^j\right)\left(J(u),J(v)\right)\\ &=\widetilde{J}\left(\rho_gJ(v),J\rho_g(u)\right) \\ &=\left(\rho_g(u),\rho_g(v)\right)\\ &=\left(\rho\oplus\text{Conj}_\rho^J\right)(u,v)\end{aligned}


so that in fact \rho\oplus\text{Conj}_\rho^J does satisfy the real condition with realizer J.


Now, to verify our second claim we show firstly that if \{\bold{0}\}<\mathscr{W}<\mathscr{V} is \rho-invariant then \rho\oplus\text{Conj}_\rho^J can’t possibly hope to have no non-trivial proper \left(\rho\oplus\text{Conj}_\rho^j,\widetilde{J}\right)-invariant subspaces. Indeed, suppose that \mathscr{W} is such a subspace and define \mathscr{S}=\left\{(w,J(w)):w\in\mathscr{W}\right\}, clearly \{\bold{0}\}<\mathscr{S}<\mathscr{V}\oplus\mathscr{V}. Moreover, it’s clear that \widetilde{J}\left(\mathscr{S}\right)=\mathscr{S} since evidently \widetilde{J} is the identity map on \mathscr{S}! Lastly, it’s clear that it is \rho\oplus\text{Conj}_\rho^J-invariant since for every g\in G and (w,J(w))\in\mathscr{S} one has \left(\rho\oplus\text{Conj}_\rho^j\right)(w,J(w))=\left(\rho_g(w),J\rho_gJJ(w)\right)=\left(\rho_g(w),J\rho_g(w)\right)\in\mathscr{S}. Thus, to find when \mathscr{V}\oplus\mathscr{V} has non non-trivial proper \left(\rho\oplus\text{Conj}_\rho^J,\widetilde{J}\right)-invariant we may restrict our attention to irreps. That said, suppose that \rho is a real irrep so that there exists some unitary U\in\mathscr{U}\left(\mathscr{V}\right) such that JU\rho_g U^{-1}J=U\rho_g U^{-1} for every g\in G. We claim then that \mathscr{X}=\left\{\left(v,JU^{-1}JU(v)\right):v\in\mathscr{V}\right\} is \left(\rho\oplus\text{Conj}_\rho^J,\widetilde{J}\right)-invariant. Indeed, we note that for every g\in G and  \left(v,JU^{-1}JU(v)\right)\in\mathscr{X} one has that



but, by definition we have that U^{-1}JU commutes with \rho_g for each g\in G so that the above can be written \left(\rho_g(v),JU^{-1}JU\rho_g(v)\right) which is doubtlessly in \mathscr{X} so that by the arbitrariness of the above calculation we see that \mathscr{X} is \rho\oplus\text{Conj}_\rho^J-invariant. Similarly, for every \left(v,JU^{-1}JU(v)\right)\in\mathscr{X} we have that \widetilde{J}\left(v,JU^{-1}JU(v)r\right)=\left(U^{-1}JU(v),J(v)\right) but evidently JU^{-1}JUU^{1}JU(v)=J(v) so that \widetilde{J}\left(v,JU^{-1}UJ(v)\right)\in\mathscr{X}. Conversely, if x\in\mathscr{X} then \widetilde{J}(x)\in\mathscr{X} and so x=J^2(x)\in J\left(\mathscr{X}\right) so that \mathscr{X}=J\left(\mathscr{X}\right) as desired. Clearly then since \{\bold{0}\}<\mathscr{X}<\mathscr{V}\oplus\mathscr{V} we may conclude that real irreps cannot produce \mathbb{C}-representations with no \left(\rho\oplus \text{Conj}_\rho^J,\widetilde{J}\right)-invariant subspaces we may conclude that the ‘only if’ part of the theorem is true.




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


April 2, 2011 - Posted by | Algebra, Representation Theory | , , , ,


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