# Abstract Nonsense

## A Bijection Between A Subset of The Complex Reps of a Finite Group and the Real Reps (Pt. III)

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

Relation Between The Two Maps

What we now claim is that the above construction and the construction we previously made for turning $\mathbb{C}$-representations into $\mathbb{R}$-representations are intimately related. Indeed:

Theorem: Let $\rho:G\to\mathcal{U}\left(\mathscr{R}\right)$ be a $\mathbb{R}$-representation. Then, $\left(\rho_\Im\right)_\Re\cong\rho$.

Proof: We have that for the constructed $\rho_\Im$ with the distinguished inner product $\mathbf{1}\oplus-\mathbf{1}$, $\mathscr{R}^\mathbb{C}_\Re=\left\{(v,\bold{0}):v\in\mathscr{V}\right\}$. So, we can clearly define $U:V\to\mathscr{R}^\mathbb{C}_\Re:v\mapsto (v,\bold{0})$ evidently this is invertible and $\rho(g)=U\left(\rho_\Im\right)_\Re(g)U^{-1}$ for every $g\in G$. The conclusion follows. $\blacksquare$

Similarly:

Theorem: Let $\rho:G\to\mathcal{U}\left(\mathscr{V}\right)$ be a $\mathbb{C}$-representation satisfying the real condition with realizer $J$. Then, $\left(\rho_\Re\right)_\Im\cong\rho$.

Proof: We have by definition that $\rho_\Re:G\to\mathcal{U}\left(\mathscr{V}_\Re\right)$ where $\mathscr{V}_\Re=\left\{v\in \mathscr{V}:J(v)=v\right\}$. So, define $T:\mathscr{V}_\Re^\mathbb{C}\to\mathscr{V}$ by $(u,v)\mapsto u+iv$. We claim that this is an isomorphism. Indeed, it’s clearly additive, so let’s now prove that $T(z(u,v))=z T((u,v))$ for each $(u,v)\in\mathscr{V}_\Re^\mathbb{C}$. To see this merely note

\begin{aligned}T\left((a+bi)\left(u,v\right)\right) &=T\left(au-bv,bu+av\right)\\ &=(au-bv)+i(bu+av)\\ &=(a+bi)(u+iv)\\ &=(a+bi)T(u,v)\end{aligned}

Note though that $T$ is surjective. Indeed, for any $\displaystyle v\in\mathscr{V}$ we have that there exists $a_r+ib_r\in\mathbb{C}\; r\in[n]$ such that $\displaystyle \sum_{r=1}^{n}(a_r+i b_r)v_r=v$ where $\{v_1\cdots,v_n\}$ is the basis guaranteed by our characterization of complex conjugates such that $J(v_r)=v_r$. Clearly then $\displaystyle v=\sum_{r=1}^{n}a_n v_n+i\sum_{r=1}^{n}b_r v_r$ and since $\displaystyle \sum_{r=1}^{n}a_r v_r,\sum_{r=1}^{n}b_r v_r\in\mathscr{V}_\Re$ we have that $\displaystyle \left(\sum_{r=1}^{n}a_r v_r,\sum_{r=1}^{n}b_r v_r\right)\in\mathscr{V}_\Re^\mathbb{C}$ and evidently $\displaystyle T\left(\sum_{r=1}^{n}a_r v_r,b_r v_r\right)=v$. Thus, $T$ is an epimorphism but since $\dim_\mathbb{C}\mathscr{V}=\dim_{\mathbb{C}}\mathscr{V}_\Re^\mathbb{C}<\infty$ we may conclude that $T$ is an isomorphism. That said, note that for any $g\in G$ and $(u,v)\in\mathscr{V}_\Re^\mathbb{C}$

\displaystyle \begin{aligned}T^{-1}\rho_gT(u,v) &=T^{-1}\rho_g(u+iv)\\ &=T^{-1}(\rho_g(u)+\rho_g(v)i)\\ &=(\rho_g(u),\rho_g(v))\\ &=\left(\rho_\Re\right)_\Im(u,v)\end{aligned}

where we used the fact that $\rho_g(u),\rho_g(v)\in\mathscr{V}_\Re$. The conclusion follows. $\blacksquare$

As a last matter

So why does this matter? Because since satisfying the real condition is a attribute constant on equivalency classes of irreps we may conclude by our previous work that

$F:\left\{\begin{array}{c}\text{Equivalency classes of}\\ \mathbb{C}-\text{representations satisfying}\\ \text{ real condition}\end{array}\right\}\to\left\{\begin{array}{c}\text{Equivalency classes of }\\ \mathbb{R}-\text{representations}\end{array}\right\}$

given by $F([\rho])=[\rho_\Re]$ is a bijection. Moreover, since having no non-trivial proper subspaces which are $(\rho,J)$-invariant (this just means that the subspace $\mathscr{W}$ is $\rho$-invariant and $J(\mathscr{W})=\mathscr{W}$) is also constant on equivalency classes of irreps we may conclude again that

$F:\left\{\begin{array}{c}\text{Equivalency classes of }\\ \mathbb{C}-\text{representations satisfying}\\ \text{the real condition and}\\ \text{having no non-trivial}\\ \text{proper }\left(\rho,J\right)-\text{invariant subspaces}\end{array}\right\}\to \left\{\begin{array}{c}\text{Equivalency classes of}\\ \text{irreducible }\mathbb{R}-\text{representations}\end{array}\right\}$

given by $F([\rho])=[\rho_\Re]$ is a bijection. Thus, if we can identify all the $[\rho]$ which make up the domain of this second function and calculate $\rho_\Re$, or more likely their characters, we will have a lot of information on the irreducible $\mathbb{R}$-representations of $G$. That shall be our focus for the next few posts.

References:

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