# Role of projective representations in representation theory of semidirect products?

I am interested in the representations of a finite group $G=N\rtimes H$. There is an article by A. Reyes that might be helpful, but I can't find it for free anywhere. This is what I know so far.

If $X\subseteq\mathrm{Irr }(N)$ is a set of $G$-orbit representatives, then finding the irreducible characters of the stabilizers $G^\chi$ lying over $\chi\in\mathrm{Irr }(N)$ and inducing them to $G$ produces all irreducible characters of $G$ exactly once as $\chi$ ranges through $X$ (true for any $G$ and $N\trianglelefteq G$ by Clifford correspondence). So, we need to find the irreducible characters of $G^\chi$ lying over $\chi$.

If $N$ is abelian, then $\widetilde{\chi}=nh\mapsto\chi(n)$ is an extension of $\chi$ to $G^\chi=N\rtimes H^\chi$ and Mackey's "little groups method" applies, i.e. the characters of $G^\chi$ lying over $\chi$ are exactly those of the form $\widetilde{\chi}\otimes\widehat{\varphi}$ where $\varphi$ is any irreducible character of $H^\chi$ and $\widehat{\varphi}$ its inflation to $G^\chi$.

When $N$ is nonabelian, all I have read is that "projective representations" (hence central extensions or $H^2(G,\mathbb{C}^\times)$) need to come into play somehow (projective representations I am new to; group cohomology I am slightly familiar). So, I am looking for an explanation or reference on the role of projective representations here (or if not projective representations, anything else useful).

On my own I can see that if $\rho$ is the representation of $\chi\in\mathrm{Irr }(N)$, and we try to extend it to $G^\chi$ by $\widetilde{\rho}=nh\mapsto\rho(n),$ then $\widetilde{\rho}(nhn'h')=\rho(n)\rho^h(n').$ Since $h\in H^\chi$, $\rho$ and $\rho^h$ are isomorphic but not necessarily equal if $\rho$ is not degree 1, so $\widetilde{\rho}$ might not be a representation if $N$ is nonabelian. Following the freely available first page preview of Reyes, $\rho^h(n)=\phi(h)\rho(n)\phi(h)^{-1}$ for some invertible transformation $\phi(h)$, and by Schur's Lemma $\phi(hh')=\alpha(h,h')\phi(h)\phi(h')$ for some nonzero scalar $\alpha(h,h')$. But I don't know how this is useful (article cuts off at that point).

• If you have access to it, this is done in detail in Isaacs' book. – Tobias Kildetoft Oct 24 '17 at 19:42
• Character Theory of Finite Groups? – anon Oct 24 '17 at 19:44
• Yes, that's the one. It has an entire chapter on projective representations. – Tobias Kildetoft Oct 24 '17 at 19:45
• Great, I will look into it. Thanks. – anon Oct 24 '17 at 19:46