Projective representations and normalized factor sets

Let $Q$ be a finite group and let $V$ be a nonzero finite-dimensional complex vector space. A normalized projective representation of $Q$ on $V$ is a map $P:Q⟶\mathrm{GL}(V)$ with $P(1)={\mathrm{id}}_V$ for which there are scalars $\alpha (q,r)\in {\mathbb{C}}^{\times }$ satisfying $P(q)P(r)=\alpha (q,r)P(qr)(q,r\in Q)\mathrm{.}$ The function $\alpha$ is the factor set of $P$; in this convention the scalar multiplies $P(qr)$, so that the relation with $\alpha \equiv 1$ is the multiplicativity of an ordinary representation. The degree of $P$ is ${\dim }_{\mathbb{C}}V$.

The factor set is determined by $P$. If $P(q)P(r)=\alpha (q,r)P(qr)$ and $P(q)P(r)=\beta (q,r)P(qr)$ for scalars, then $(\alpha (q,r)-\beta (q,r))P(qr)=0$ and $P(qr)$ is invertible, so $\alpha (q,r)=\beta (q,r)$. Thus a map $P$ either has no factor set or has exactly one, and the equation is a genuine condition on $P$: it says that the composite of $P$ with the quotient map $\mathrm{GL}(V)\to \mathrm{PGL}(V)$ is a group homomorphism, since $P(q)P(r)P(qr{)}^{-1}$ is a scalar exactly when the displayed relation holds. The scalars are automatically nonzero, because $P(q)P(r)$ is invertible and $P(q)P(r)=\alpha (q,r)P(qr)$. The normalization $P(1)={\mathrm{id}}_V$ fixes the representative of a projective representation up to scalars; with it the relation at $q=r=1$ forces $\alpha (1,1)=1$, and the general normalization identities $\alpha (1,q)=\alpha (q,1)=1$ are proved in The factor set satisfies the two-cocycle equation.

Similarity. Projective representations $P$ of $Q$ on $V$ and $P^{\mathrm{\prime }}$ of $Q$ on $V^{\mathrm{\prime }}$ are similar when there is a $\mathbb{C}$-linear isomorphism $M:V\to V^{\mathrm{\prime }}$ with $P^{\mathrm{\prime }}(q)=MP(q)M^{-1}(q\in Q)\mathrm{.}$ Similar projective representations have the same factor set: multiplying $P^{\mathrm{\prime }}(q)P^{\mathrm{\prime }}(r)=MP(q)P(r)M^{-1}=\alpha (q,r)MP(qr)M^{-1}=\alpha (q,r)P^{\mathrm{\prime }}(qr)$ identifies the factor set of $P^{\mathrm{\prime }}$ with that of $P$. Similarity is an equivalence relation on the projective representations of $Q$ with a fixed factor set, and it preserves the degree.

Irreducibility. A normalized projective representation $P$ of $Q$ on $V$ is reducible when there is a subspace $W\le V$ with $0\mathrm{\ne }W\mathrm{\ne }V$ and $P(q)W\subseteq W$ for every $q\in Q$, and irreducible otherwise. Because every $P(q)$ is invertible, $P(q)W\subseteq W$ for all $q$ is equivalent to $P(q)W=W$ for all $q$. A similarity $M:P\to P^{\mathrm{\prime }}$ carries $P$-invariant subspaces bijectively onto $P^{\mathrm{\prime }}$-invariant subspaces, so reducibility and irreducibility are similarity invariants; this is the notion of irreducibility used in The projective Clifford correspondence for an invariant irreducible representation.

• Why $V\mathrm{\ne }0$. For the zero space the group $\mathrm{GL}(0)$ is trivial, so the relation $P(q)P(r)=\alpha (q,r)P(qr)$ holds for every scalar function $\alpha$ and no factor set is determined. The zero module reappears in the module dictionary of Projective representations and twisted algebra modules as the common zero object of all twisted module categories, which is why the correspondence is stated for nonzero modules.

• Conventions. The factor set here is the map $\alpha$ of Späth, Definition 1.4, normalized by $P(1)=\mathrm{id}$; its cocycle equation and its behaviour under rephasing are recorded in The factor set satisfies the two-cocycle equation and Rephasing changes factor sets by coboundaries. A projective representation in this sense is not a representation: the scalar $\alpha (q,r)$ need not be $1$, and the maps $P(q)$ need not multiply.