# Abstract Nonsense

## 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.