# Abstract Nonsense

## 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.

Motivation

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 $(x,y)\in\mathscr{V}\oplus\mathscr{V}$

$\widetilde{J}^2(x,y)=\widetilde{J}\left(J(y),J(x)\right)=\left(J^2(x),J^2(y)\right)=(x,y)$

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}

and

\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

$\left(\rho_g\oplus\text{Conj}_\rho^J(g)\right)\left(v,JU^{-1}JU(v)\right)=\left(\rho_g(v),J\rho_g(v)U^{-1}JU(v)\right)$

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.

References:

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