projective representation



A projective representation of a group GG is a representation up to a central term: a group homomorphism GPGL(V)G\longrightarrow PGL(V), to the projective general linear group of some 𝕂\mathbb{K}-vector space VV.


The group extension and its cocycle

By construction, there is a short exact sequence

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

which exhibits GL(V)GL(V) as a group extension of PGl(V)PGl(V) by the group of units 𝕂 ×\mathbb{K}^\times of the ground field. This is classified by a 2-cocycle cc in group cohomology H Grp 2(PGL(V),𝕂 ×)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 ϕ:GH\phi \colon G\longrightarrow H are equivalently maps Bϕ:BGBH\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

B𝕂 * BGL(V) * * BPGL(V) c B 2𝕂 ×. \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)PGL(V)=GL(V)/𝕂 ×GL(V)\to PGL(V)=GL(V)/\mathbb{K}^\times, every linear representation of GG 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 Bρ:BGBPGL(V)\mathbf{B}\rho \colon \mathbf{B}G\longrightarrow \mathbf{B} PGL(V) to a linear representation ρ^\hat \rho

BGL(V) * Bρ^ BG Bρ BPGL(V) c B 2𝕂 × \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(ρ)cBρc(\rho)\coloneqq c \circ \mathbf{B}\rho in the group cohomology class H Grp 2(G,𝕂 ×)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 Bc:BPGL(V)B 2𝕂 ×\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

B(𝕂 ×GL(V)) B(𝕂 ×1) BPGL(V). \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 B(𝕂 ×GL(V))\mathbf{B}(\mathbb{K}^\times \to GL(V)) looks schematically like

{ * p 1 k p 2 * kp 1p 2 *} \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 BGBPGL(V)\mathbf{B}G\to \mathbf{B}PGL(V) may equivalently be represented by two functions (not group homomorphisms in general!)

  1. ρ:GGL(V)\rho \colon G\to GL(V)

  2. λ:G×G𝕂 ×\lambda \colon G\times G\to \mathbb{K}^\times

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

  1. ρ(g)ρ(h)=λ(g,h)ρ(gh)\rho(g)\rho(h)=\lambda(g,h)\rho(g h)

  2. λ\lambda is a group 2-cocycle on GG with values in 𝕂 *\mathbb{K}^* representing the above cohomology class of c ρc_\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

BGL(V) * BG Bρ BPGL(V) c B 2𝕂 × \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 Bρ\mathbf{B}\rho. By the pasting law this is (the delooping of) the group extension G^\hat G of GG which is classified by c(ρ)c(\rho):

BG^ Bρ˜ BGL(V) * BG Bρ BPGL(V) c B 2𝕂 × \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 ρ\rho of GG induces a genuine linear representation ρ˜\tilde \rho of G^\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 GG and genuine representations of G^\hat G.

Revised on January 2, 2015 15:34:32 by Urs Schreiber (