Projective Representations — Mathematics Knowledge System

Overview

A projective representation of a group G is a group homomorphism ρ: G → PGL(V) = GL(V)/k×, i.e., a representation where the group law holds only up to scalars. Equivalently, it is a map ρ: G → GL(V) satisfying ρ(g)ρ(h) = c(g,h)ρ(gh) for a 2-cocycle c: G×G → k×. Projective representations arise naturally in quantum mechanics (where states are rays in Hilbert space), in the study of spin (covering groups), and in Clifford theory (when an irrep of a normal subgroup does not extend to the whole group).

Intuition

A projective representation is like an ordinary representation, but the group law is satisfied only up to non-zero scalar factors. The scalars form a 2-cocycle, and the collection of all projective representations is controlled by the second cohomology group H²(G, k×). The Schur multiplier M(G) = H²(G, ℂ×) measures the 'obstruction' to lifting projective representations to ordinary ones.

Formal Definition

Definition

Let G be a group and k an algebraically closed field. A projective representation of G is a group homomorphism ρ: G → PGL(V), or equivalently a map ρ: G → GL(V) with

\rho(g)\rho(h) = c(g,h)\,\rho(gh), \quad c(g,h) \in k^\times \quad (\text{for all } g,h \in G)

Projective representation: group law holds up to scalar c

c: G \times G \to k^\times \text{ is a 2-cocycle: } c(g,h)c(gh,k) = c(g,hk)c(h,k)

Cocycle condition on the factor set c

M(G) = H^2(G, \mathbb{C}^\times) \quad (\text{Schur multiplier})

Schur multiplier: classifies projective representations up to equivalence

1 \to \mathbb{C}^\times \to \tilde{G} \to G \to 1 \quad (\text{Schur cover / representation group})

Schur cover: lifting projective reps to ordinary ones

Notation

Notation	Meaning
$\mathrm{PGL}(V)$	Projective general linear group GL(V)/k×
$c(g,h)$	Factor set (2-cocycle) of the projective representation
$M(G)$	Schur multiplier H²(G, ℂ×)
$\tilde{G}$	Schur cover (representation group) of G

Properties

Coboundary equivalence

Two factor sets c and c' are equivalent if c'(g,h) = c(g,h)·f(g)f(h)/f(gh) for some f: G → k×. Equivalence classes form H²(G, k×).

Ordinary reps as projective reps

Ordinary representations are the special case where c ≡ 1 (the trivial cocycle). They correspond to the identity class in H²(G, k×).

Theorems

Theorem 1: Schur's Theorem on Projective Representations

Every projective representation of G over ℂ lifts to an ordinary representation of the Schur cover G̃, which is a central extension 1 → M(G) → G̃ → G → 1. In particular, projective representations of G correspond to ordinary representations of G̃.

Theorem 2: Classification of Projective Reps

Equivalence classes of projective representations of G with cocycle class [c] ∈ H²(G, ℂ×) are in bijection with ordinary representations of the corresponding central extension of G by ℂ×.

Theorem 3: Schur Multiplier of Symmetric Groups

M(Sₙ) ≅ ℤ/2ℤ for n ≥ 4. The corresponding non-trivial projective representations of Sₙ are the 'spin representations', arising from the double cover Sₙ̃ (the Schur cover).

Worked Examples

1
Check: ρ(1)² = [[0,1],[−1,0]]² = [[−1,0],[0,−1]] = −I. If ρ were an ordinary representation, ρ(1)² = ρ(0) = I. Instead ρ(1)² = −I = c(1,1)·ρ(0) with c(1,1) = −1.
$\rho(1)^2 = -I = (-1) \cdot I = c(1,1) \cdot \rho(0)$
2
So ρ is a projective representation of ℤ/2ℤ with cocycle c(1,1) = −1 ∈ ℂ×.
$c: \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \to \mathbb{C}^\times, \quad c(1,1) = -1$
3
This projective representation does not lift to an ordinary representation of ℤ/2ℤ, but does lift to an ordinary representation of ℤ/4ℤ (the Schur cover), where the generator maps to ρ(1).
$\tilde{G} = \mathbb{Z}/4\mathbb{Z}: \quad g \mapsto \rho(1) \text{ (order 4 matrix)}$

✓ Answer

ρ is a projective representation of ℤ/2ℤ with non-trivial cocycle c(1,1)=−1. It lifts to an ordinary rep of ℤ/4ℤ.

Practice Problems

Mediumfree response

Verify the 2-cocycle condition c(g,h)c(gh,k) = c(g,hk)c(h,k) for the cocycle of a projective representation, by computing both sides from the associativity of ρ.

Hardfree response

What is the Schur multiplier of A₄? Use it to determine whether A₄ has non-trivial projective representations.

Quiz

Summary

A projective representation satisfies ρ(g)ρ(h) = c(g,h)ρ(gh) where c is a 2-cocycle G×G → k×.
Equivalence classes of factor sets are classified by the Schur multiplier M(G) = H²(G, ℂ×).
Every projective rep of G lifts to an ordinary rep of the Schur cover G̃.
Physical example: spin-1/2 is a projective rep of SO(3), lifting to an ordinary rep of SU(2).
M(Sₙ) ≅ ℤ/2ℤ for n ≥ 4; spin representations of Sₙ arise from this.

References

BookKarpilovsky, G. — The Schur Multiplier (1987)
BookIsaacs, I.M. — Character Theory of Finite Groups (1976), Chapter 11