The Clifford obstruction class of an invariant irreducible representation

Let $N⊴G$ be finite groups, let $\rho :N\to \mathrm{GL}(S)$ be an irreducible representation on a nonzero finite-dimensional complex space $S$, let $\theta$ be its character, let $I=I_G(\theta )$ be the inertia group (Inertia group and characters lying above a normal type), and put $Q:=I\mathrm{/}N$. By An invariant irreducible normal representation yields projective inertia operators there are operators $P(i)\in \mathrm{GL}(S)$ with $P(n)=\rho (n)$, $P(ni)=\rho (n)P(i)$, $P(in)=P(i)\rho (n)$ and $P(i)P(j)=\alpha (iN,jN)P(ij)$, where $\alpha$ is a normalized two-cocycle on $Q$ with values in the abelian group ${\mathbb{C}}^{\times }$ (trivial $Q$-action, written multiplicatively). The Clifford obstruction of $\rho$ (equivalently, of $\theta$) is the cohomology class $[\alpha ]\in H^2(Q,{\mathbb{C}}^{\times })$ in the factor-set model of Second cohomology by factor sets. The class vanishes precisely when the projective operators can be made multiplicative, which by An invariant irreducible representation extends to its inertia group exactly when the Clifford obstruction vanishes is exactly the extendibility of $\rho$ to $I$; in particular the class is a complete obstruction to extension, not merely a necessary condition.

The class does not depend on the operators. If $P^{\mathrm{\prime }}$ is a second family with the same normalization and the same three identities, then by An invariant irreducible normal representation yields projective inertia operators there is a function $c:Q\to {\mathbb{C}}^{\times }$ with $c(N)=1$ and $P^{\mathrm{\prime }}(i)=c(iN)P(i)$ for all $i\in I$, and the factor set of $P^{\mathrm{\prime }}$ is ${\alpha }^{\mathrm{\prime }}(q,r)=c(q)c(r)c(qr{)}^{-1}\alpha (q,r)$. Read as a one-cochain on $Q$ with the trivial action, $c$ has coboundary $\delta c(q,r)=c(q)c(r)c(qr{)}^{-1}$ and ${\alpha }^{\mathrm{\prime }}=(\delta c)\cdot \alpha$, so ${\alpha }^{\mathrm{\prime }}$ and $\alpha$ have the same class in $H^2(Q,{\mathbb{C}}^{\times })$; this is the rephasing computation of Rephasing changes factor sets by coboundaries, applied to the projective representation of $I$ afforded by $P$, and it is the step that makes the class depend only on the character triple.

Equivalent representations give the same class. If ${\rho }^{\mathrm{\prime }}=\iota \rho {\iota }^{-1}$ for a linear isomorphism $\iota :S\to S^{\mathrm{\prime }}$, then the operators $P^{\mathrm{\prime }}(i):=\iota P(i){\iota }^{-1}$ satisfy the same identities with the same function $\alpha$, because $P^{\mathrm{\prime }}(n)={\rho }^{\mathrm{\prime }}(n)$ and all scalar identities are unchanged by conjugation. So replacing $\rho$ by an equivalent representation, possibly on another space, leaves the class $[\alpha ]$ literally unchanged; note that $I$ and $\theta$ are unchanged as well.

Degenerate case $I=N$. Here $Q$ is the trivial group, whose only normalized two-cocycle is the constant function $1$, so $H^2(Q,{\mathbb{C}}^{\times })$ is the trivial group and the Clifford obstruction is automatically zero. This is the trivial case of the extension criterion: $\rho$ is already a representation of $I=N$, and indeed the identity map $\rho$ is an extension. The first interesting case is therefore a proper invariant type, and the class measures exactly how far the associated projective operators are from an extension.