nLab projective representation

Definition

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 $V$.

Properties

The group extension and its cocycle

By construction, there is a short exact sequence

$$1 \to \mathbb{K}^\times \longrightarrow GL(V) \longrightarrow PGL(V)\to 1$$

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

$$\array{ \mathbf{B}\mathbb{K}^*& \longrightarrow & \mathbf{B}GL(V) &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow \\ \ast &\longrightarrow&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times } \,.$$

Relation to genuine representations

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 $\hat \rho$

$$\array{ & & \mathbf{B}GL(V) &\longrightarrow& \ast \\ & {}^{\mathllap{\mathbf{B}\hat \rho}}\nearrow & \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times }$$

is the class of the 2-cocycle $c(\rho)\coloneqq c \circ \mathbf{B}\rho$ in the group cohomology class $H^2_{Grp}(G,\mathbb{K}^\times)$.

As twisted linear representations

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

$$\array{ \mathbf{B}(\mathbb{K}^\times \to GL(V)) &\longrightarrow& \mathbf{B}(\mathbb{K}^\times \to 1) \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}PGL(V) } \,.$$

Here the 2-groupoid $\mathbf{B}(\mathbb{K}^\times \to GL(V))$ looks schematically like

$$\left\{ \array{ && \ast \\ & {}^{\mathllap{p_1}}\nearrow &\Downarrow^{\mathrlap{k}}& \searrow^{\mathrlap{p_2}} \\ \ast && \underset{k p_1 p_2}{\longrightarrow}&& \ast } \right\}$$

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!)

1. $\rho \colon G\to GL(V)$

2. $\lambda \colon G\times G\to \mathbb{K}^\times$

such that for all $g,h \in G$

1. $\rho(g)\rho(h)=\lambda(g,h)\rho(g h)$

2. $\lambda$ 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.

As genuine representations after extensions

Alternatively, one may consider the above diagram

$$\array{ & & \mathbf{B}GL(V) &\longrightarrow& \ast \\ & & \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times }$$

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 $c(\rho)$:

$$\array{ \mathbf{B}\hat G & \stackrel{\mathbf{B}\tilde \rho}{\longrightarrow} & \mathbf{B}GL(V) &\longrightarrow& \ast \\ \downarrow & & \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{B}\rho}{\longrightarrow}&\mathbf{B}PGL(V) &\stackrel{c}{\longrightarrow}& \mathbf{B}^2 \mathbb{K}^\times }$$

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 $\hat G$.

Related concepts

• projectively flat connection

References

(…)

Relation to twisted equivariant K-theory:

• José Manuel Gómez, Johana Ramírez, A Decomposition of twisted equivariant K-theory (arXiv:2001.02164)