Irreducible sectors of the finite-group double

In any irreducible $D(G)$-module $V$, put $p_h=\rho (d_{h,e})$ and $u_g=\rho ({\sum }_{r\in G}{d}_{r,g})$. The multiplication in [F1] gives $p_h{p}_k={\mathbf{1}}_{h=k}{p}_h$, ${\sum }_h{p}_h=I$, $u_g{u}_l=u_{gl}$, $u_g{p}_h{u}_g^{-1}=p_{gh{g}^{-1}}$, and $\rho (d_{h,g})=p_h{u}_g$. Hence $V={⨁}_{h\in G}{V}_h$ with $V_h=p_{h}V$, and $u_g$ maps $V_h$ isomorphically onto $V_{gh{g}^{-1}}$.

Conversely, fix a conjugacy class $C$, representative $c$, and irreducible representation $\pi :Z_G(c)\to \mathrm{GL}(W)$ as in [F3]–[F4]. For each $h\in C$ choose $t_h$ with $t_{h}c{t}_h^{-1}=h$ and take $t_c=e$. On $M_{C,\pi }:={⨁}_{h\in C}{W}_h$ define $p_a$ as projection onto $W_a$ (zero if $a\mathrm{\notin }C$), and define $u_g(h,w)=(gh{g}^{-1},\pi (t_{gh{g}^{-1}}^{-1}g{t}_h)w)$. With $h^{\mathrm{\prime }}=gh{g}^{-1}$, the factor $z=t_{h^{\mathrm{\prime }}}^{-1}g{t}_h$ centralizes $c$, since $zc{z}^{-1}=t_{h^{\mathrm{\prime }}}^{-1}gh{g}^{-1}{t}_{h^{\mathrm{\prime }}}=c$. For $k\in G$, the product of the two centralizer factors in $u_g{u}_k$ is $(t_{gkh{k}^{-1}{g}^{-1}}^{-1}g{t}_{kh{k}^{-1}})(t_{kh{k}^{-1}}^{-1}k{t}_h)=t_{gkh{k}^{-1}{g}^{-1}}^{-1}gk{t}_h$, so $u_g{u}_k=u_{gk}$. The fiber definition also gives $u_g{p}_a{u}_g^{-1}=p_{ga{g}^{-1}}$.

The set of $h$ for which $V_h\mathrm{\ne }0$ is a union of conjugacy classes. For any such class $C$, the subspace ${⨁}_{h\in C}{V}_h$ is invariant under every $p_a$ and $u_g$, hence under every $d_{a,g}$. Irreducibility therefore forces the support to be one conjugacy class $C$. Fix $c\in C$.

Define $\rho (d_{a,g})=p_a{u}_g$. The identities of step 1.2 imply $\rho (d_{a,g})\rho (d_{b,l})={\mathbf{1}}_{a=gb{g}^{-1}}\rho (d_{a,gl})$, exactly the multiplication in [F1], and $\rho (1_D)={\sum }_a{p}_a=I$. Thus $M_{C,\pi }$ is a unital $D(G)$-module. If $N$ is a nonzero invariant subspace, the projections $p_h$ decompose it into fibers because they are pairwise orthogonal and sum to $I$; $N_c$ is $\pi$-invariant, so by [F4] it is either zero or all of $W_c$. In the first case, for each $h$ the isomorphism $u_{t_h}$ carries $N_c$ onto $N_h$, so every fiber vanishes. In the second case, those images fill every fiber. Thus $N=0$ or $N=M_{C,\pi }$, proving irreducibility.

By [F3], the restrictions $u_z{\mathrm{\mid }}_{V_c}$ for $z\in Z_G(c)$ form a finite-dimensional group representation. If it had a nonzero proper invariant subspace $W$, choose for each $h\in C$ an element $t_h$ with $t_{h}c{t}_h^{-1}=h$. For $h^{\mathrm{\prime }}=gh{g}^{-1}$ the element $z=t_{h^{\mathrm{\prime }}}^{-1}g{t}_h$ satisfies $zc{z}^{-1}=t_{h^{\mathrm{\prime }}}^{-1}gh{g}^{-1}{t}_{h^{\mathrm{\prime }}}=c$, so $z\in Z_G(c)$. Therefore $u_g{u}_{t_h}W=u_{t_{h^{\mathrm{\prime }}}}{u}_{z}W\subseteq u_{t_{h^{\mathrm{\prime }}}}W$. The direct sum ${⨁}_{h\in C}{u}_{t_h}W$ is invariant under every $u_g$ and $p_a$; it is nonzero, and it is proper because each $u_{t_h}$ is an isomorphism from $V_c$ to $V_h$ while $W⊊V_c$. It is thus a nonzero proper $D(G)$-submodule, contradicting irreducibility. Hence $V_c$ is irreducible as a $Z_G(c)$-representation.

The constructed module has one fiber $W_h\cong W$ for each $h\in C$, so its dimension is $\mathrm{\mid }C\mathrm{\mid }\dim \pi$. The map $G\mathrm{/}{Z}_G(c)\to C$, $g{Z}_G(c)\mapsto gc{g}^{-1}$, is well-defined and onto; equality of two conjugates implies the two representatives differ by an element of $Z_G(c)$, so it is injective. Therefore $\mathrm{\mid }C\mathrm{\mid }\mathrm{\mid }{Z}_G(c)\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }$.

For an arbitrary irreducible $V$ with support $C$, set $W=V_c$ and let $\pi$ be its centralizer action from step 3.1. The maps $u_{t_h}$ then identify each fiber $V_h$ with the constructed copy $W_h$, and the relation $u_g{u}_{t_h}=u_{t_{gh{g}^{-1}}}{u}_{t_{gh{g}^{-1}}^{-1}g{t}_h}$ shows that this fiberwise identification intertwines the full $D(G)$-action. Distinct classes have distinct supports; for a fixed $C$, an isomorphism restricts on the $p_c$ fiber to an isomorphism of centralizer representations. Hence the pairs $(C,[\pi ])$ classify the irreducibles without duplication.

If $c^{\mathrm{\prime }}=rc{r}^{-1}$ is another representative of $C$, then $Z_G(c^{\mathrm{\prime }})=r{Z}_G(c)r^{-1}$. Sending $\pi$ to ${\pi }^{\mathrm{\prime }}(z^{\mathrm{\prime }})=\pi (r^{-1}{z}^{\mathrm{\prime }}r)$ gives a bijection of the irreducible representation classes of the two centralizers. The fiber map $u_r:V_c\to V_{c^{\mathrm{\prime }}}$ implements this relabeling because $u_{z^{\mathrm{\prime }}}{u}_r=u_r{u}_{r^{-1}{z}^{\mathrm{\prime }}r}$, by $u_g{u}_l=u_{gl}$ from step 1.1. Thus the parametrization is independent of the representative up to the stated conjugation.

The finite centralizer $Z_G(c)$ satisfies the finite-group and characteristic hypotheses of [F5], and [F6] supplies the resulting squared-dimension sum ${\sum }_{[\pi ]\in {\mathrm{Irr}}_{\mathbb{C}}(Z_G(c))}(\dim \pi {)}^2=\mathrm{\mid }{Z}_G(c)\mathrm{\mid }$. Summing the squared dimensions from step 3.2 over conjugacy classes gives ${\sum }_C\mathrm{\mid }C{\mathrm{\mid }}^2\mathrm{\mid }{Z}_G(c)\mathrm{\mid }={\sum }_C\mathrm{\mid }C\mathrm{\mid }\mathrm{\mid }G\mathrm{\mid }=\mathrm{\mid }G{\mathrm{\mid }}^2$, since the conjugacy classes partition $G$. Finally, [F1] gives $\dim D(G)=\mathrm{\mid }G{\mathrm{\mid }}^2$.