geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
A projective representation of a group $G$ is a representation up to a central term: a group homomorphism $G\longrightarrow PGL(V)$, to the projective general linear group of some $\mathbb{K}$-vector space $V$.
By construction, there is a short exact sequence
which exhibits $GL(V)$ as a group extension of $PGl(V)$ by the group of units $\mathbb{K}^\times$ of the ground field. This is classified by a 2-cocycle $c$ in group cohomology $H^2_{Grp}(PGL(V),\mathbb{K}^\times)$.
It is useful to re-express this equivalently in terms of homotopy theory via the discussion at looping and delooping, by which group homomorphisms $\phi \colon G\longrightarrow H$ are equivalently maps $\mathbf{B}\phi \colon \mathbf{B}G\longrightarrow \mathbf{B}H$ between their delooping groupoids.
In terms of this the above group extension and its classifying cocycle is exhibited by a homotopy fiber sequence of deloopings of the form
Via the projection $GL(V)\to PGL(V)=GL(V)/\mathbb{K}^\times$, every linear representation of $G$ induces a projective representation.
By the universal property of the homotopy pullback, the discussion above means that the obstruction to lift a given projective representation $\mathbf{B}\rho \colon \mathbf{B}G\longrightarrow \mathbf{B} PGL(V)$ to a linear representation $\hat \rho$
is the class of the 2-cocycle $c(\rho)\coloneqq c \circ \mathbf{B}\rho$ in the group cohomology class $H^2_{Grp}(G,\mathbb{K}^\times)$.
By the above and the discussion at group extension – Central group extensions – Formulation in homotopy theory the cocycle map $\mathbf{B}c \colon \mathbf{B}PGL(V)\to \mathbf{B}^2 \mathbb{K}^\times$ of homotopy types may be represented by a zigzag (“2-anafunctor”) of crossed complexes as
Here the 2-groupoid $\mathbf{B}(\mathbb{K}^\times \to GL(V))$ looks schematically like
This shows that a map $\mathbf{B}G\to \mathbf{B}PGL(V)$ may equivalently be represented by two functions (not group homomorphisms in general!)
$\rho \colon G\to GL(V)$
$\lambda \colon G\times G\to \mathbb{K}^\times$
such that for all $g,h \in G$
$\rho(g)\rho(h)=\lambda(g,h)\rho(g h)$
$\lambda$ is a group 2-cocycle on $G$ with values in $\mathbb{K}^*$ representing the above cohomology class of $c_\rho$.
This is the form in which projective representations are often discussed in the literature.
Alternatively, one may consider the above diagram
and form the further pullback along $\mathbf{B}\rho$. By the pasting law this is (the delooping of) the group extension $\hat G$ of $G$ which is classified by $c(\rho)$:
This way the projective representation $\rho$ of $G$ induces a genuine linear representation $\tilde \rho$ of $\hat G$. One finds (this is a special case of the general discussion at twisted infinity-bundle) that this constitutes an equivalence between projective representations of $G$ and genuine representations of $\hat G$.