Abstract Nonsense

Crushing one theorem at a time

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.

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March 30, 2011 - Posted by | Algebra, Representation Theory | , , , ,

2 Comments »

  1. […] our last post we have seen that -representations satisfying the real condition are important since they […]

    Pingback by Representation Theory: A Way of Creating C-representations Satisfying the Real Condition With No (rho,J)-invariant Subspaces « Abstract Nonsense | April 2, 2011 | Reply

  2. […] about the latter for a specific group what could we say about the former. We began this search by showing that there was a bijection between equivalency classes of -representations satisfying the real […]

    Pingback by Representation Theory: A Classification of C-representations With No (rho,J)-invariant Subspaces « Abstract Nonsense | April 4, 2011 | Reply


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