An ordinary representation preserves the group law exactly. A projective representation preserves it only up to a nonzero scalar. That small-looking relaxation creates a new layer of mathematics: cocycles, central extensions, Schur multipliers and the representation theory behind quantum-mechanical phase and spin.
Projective representations appear whenever the physically or geometrically meaningful state is a line rather than a chosen vector, or whenever symmetry operators are determined only up to scalar multiples. The resulting ambiguity is not uncontrolled. Associativity forces the scalars to satisfy a precise 2-cocycle equation, and changing the phases of the chosen operators changes the cocycle only by a coboundary.
This guide extends the BTT routes through group representations, tensor constructions and Lie groups and Lie algebras. It also prepares the spin-group discussion later in this batch.
Definition · Cocycles · Rephasing · Pauli example · Central extensions · SO(3) and SU(2) · Practice
Why a scalar multiple can represent the same physical state
In ordinary vector-space mathematics, v and λv are different vectors whenever λ≠1. In projective space, however, all nonzero scalar multiples of v determine the same line.
Quantum mechanics supplies the most familiar example. A pure state is represented by a ray in Hilbert space. Multiplying a state vector by a nonzero complex phase does not change the ray. Therefore a symmetry acting on rays need not select one uniquely normalised operator on vectors.
This means a symmetry group can act perfectly well on projective space even when no choice of linear operators satisfies the group law exactly. The obstruction is measured by a cocycle.
The projective representation definition
Let G be a group and V a vector space over a field F. A projective representation can be described as a group homomorphism
G → PGL(V),
where PGL(V)=GL(V)/F× identifies invertible linear maps that differ by a nonzero scalar.
Choose one representative operator Ug∈GL(V) for each projective class. Because multiplication is correct only after scalar multiples have been identified, there exist nonzero scalars α(g,h) such that
UgUh=α(g,h)Ugh.
The function α:G×G→F× is called a factor set or 2-cocycle once the associativity condition below is imposed.
Associativity forces the 2-cocycle equation
Compute UgUhUk in two ways.
First group the first two operators:
(UgUh)Uk=α(g,h)α(gh,k)Ughk.
Now group the last two:
Ug(UhUk)=α(h,k)α(g,hk)Ughk.
Ordinary operator multiplication is associative, so the coefficients must agree:
α(g,h)α(gh,k)=α(h,k)α(g,hk).
This is the 2-cocycle equation. Projective ambiguity is therefore constrained by associativity; arbitrary scalar errors are not permitted.
When convenient, one can choose Ue=I and normalise the cocycle so α(e,g)=α(g,e)=1. This does not change the underlying projective representation.
Rephasing changes the cocycle without changing the projective action
The representatives Ug are not unique. Choose a nonzero scalar β(g) for every group element and define
U′g=β(g)Ug.
Then
U′gU′h=α′(g,h)U′gh
with
α′(g,h)=β(g)β(h)β(gh)−1α(g,h).
The multiplier β(g)β(h)/β(gh) is a 2-coboundary. Thus cocycles related by coboundaries describe equivalent phase choices for the same projective representation.
The appropriate equivalence classes form the second cohomology group H²(G,F×) for the trivial G-action on the coefficient scalars in this elementary setting. Over C× for finite groups, this is closely related to the Schur multiplier.
When can the projective representation be made ordinary?
If α is a coboundary, choose β so that α′(g,h)=1 everywhere. Then the rephased operators satisfy
U′gU′h=U′gh.
The projective representation has been lifted to an ordinary linear representation of G.
If the cocycle class is nontrivial, no global rephasing removes every factor simultaneously. The obstruction is structural, not a failure to choose phases cleverly enough.
trivial cocycle class → ordinary representation after rephasing; nontrivial class → genuine projective representation.
A first cyclic example where the cocycle can be removed
Suppose C₂={e,g} acts projectively on a complex vector space and a chosen lift satisfies Ug²=−I. Then α(g,g)=−1.
Over C we can rephase Ug by i. Let U′g=iUg. Then
(U′g)²=i²Ug²=(−1)(−I)=I.
So this particular multiplier is removable. The example is useful because it shows that a visible phase factor in one set of representatives is not automatically a nontrivial cohomology class.
Worked example: a genuinely projective representation of C2 × C2
Let G=C₂×C₂, written additively with elements (a,b), where a,b∈{0,1}. On C² use the Pauli matrices
X = [0 1] Z = [1 0]
[1 0] [0 -1]
They satisfy X²=Z²=I but
ZX=−XZ.
Define U(a,b)=XaZb. Then
U(a,b)U(c,d)=(−1)bcU(a+c,b+d),
with exponents reduced modulo 2. The cocycle is α((a,b),(c,d))=(−1)bc.
The underlying group is abelian, so its elements commute. The lifted matrices need not commute; their commutator can be a scalar. Indeed XZ=−ZX, and the minus sign disappears after passing to PGL₂(C). Thus the projective classes commute exactly as C₂×C₂ requires.
This is a decisive example: projective representation theory can encode noncommuting operator lifts of an abelian symmetry.
Check the cocycle equation explicitly
For α((a,b),(c,d))=(−1)bc, take a third element (e,f). The left side of the cocycle equation has exponent
bc+(b+d)e.
The right side has exponent
de+b(c+e).
Modulo 2 these are both bc+be+de. The cocycle equation holds.
This calculation is small but important. It verifies that the phase factor came from an associative operator multiplication rather than from an arbitrary sign assignment.
The commutator phase is gauge-invariant in the abelian case
For an abelian group define, when appropriate,
c(g,h)=α(g,h)α(h,g)−1.
This is the scalar by which UgUh differs from UhUg. Rephasing Ug and Uh does not change this commutator scalar.
For the Pauli example c((1,0),(0,1))=−1. Therefore no rephasing can make X and Z commute. This immediately proves the projective class is nontrivial.
Central extensions turn projective representations into ordinary ones
A central extension of G by an abelian group A is an exact sequence
1→A→G̃→G→1
in which the image of A lies in the centre of G̃.
Given a cocycle α:G×G→A, one can construct an extension on the set A×G with multiplication
(a,g)(b,h)=(abα(g,h),gh).
The cocycle equation is exactly what makes this multiplication associative.
A projective representation of G with factor set α often becomes an ordinary representation of the associated central extension G̃, with the central subgroup acting by scalars.
This is the main repair mechanism:
projective action of G → ordinary linear action of a central extension of G.
The Pauli matrices reveal a central extension
The matrices generated by X and Z, together with scalar signs, form a nonabelian group lying above C₂×C₂. Modding out by the scalar centre recovers the abelian projective symmetry.
Depending on the exact phase set included, one obtains versions of the Pauli or extraspecial 2-group structure. The key point is stable: the noncommutativity lives in the central scalar layer, and the quotient by that layer is the intended abelian group.
This pattern recurs throughout mathematics and physics. A projective symmetry often signals that the “true” linear symmetry is a covering or central extension.
Schur covers and the Schur multiplier
For finite groups, projective representations over C can often be studied by passing to a suitable central extension called a representation group or Schur cover.
The Schur multiplier is classically related to H²(G,C×), or equivalently to H₂(G,Z) through standard finite-group identifications. It measures possible projective twisting in a group-theoretic way.
The terminology should be used carefully: a Schur cover is not uniquely determined as a concrete group in every naive sense, and cohomology classes classify extension data only with the correct notion of equivalence and coefficient action.
Twisted group algebras package the cocycle into multiplication
Given a cocycle α, form a vector space with basis symbols eg for g∈G and multiply by
egeh=α(g,h)egh.
The cocycle equation ensures associativity. This algebra is a twisted group algebra.
Projective representations with multiplier α correspond to ordinary modules over this twisted algebra. We have moved the projective defect from the representation law into the multiplication law of the algebra.
This is another common mathematical strategy: relocate the complication into the ambient structure so that the objects inside it become ordinary.
Rotations, SU(2) and the projective representation of SO(3)
The rotation group SO(3) has a double cover
SU(2)→SO(3)
with kernel {±I}. Each rotation in SO(3) corresponds to two elements ±U in SU(2).
The two-dimensional defining representation of SU(2) does not descend to an ordinary representation of SO(3), because −I acts as −I rather than as the identity. But ±U induce the same projective transformation on rays. Therefore the spin-1/2 action gives a projective representation of SO(3).
This is why a 360° rotation can act by −I on a spinor while a 720° rotation returns the vector to itself. On projective rays, the intermediate minus sign is invisible.
The Clifford-algebra article later in this batch constructs the same double-cover phenomenon geometrically through spin groups and rotors.
Integer spin descends; half-integer spin does not
Irreducible finite-dimensional SU(2) representations are indexed by highest weight n≥0 and have dimension n+1. The central element −I acts by (−1)n.
If n is even, −I acts trivially, so the representation descends to SO(3). These correspond to integer spin j=n/2.
If n is odd, −I acts by −1, so the representation does not descend linearly to SO(3). It gives a genuine projective SO(3) representation, corresponding to half-integer spin.
This is a clean example of a global topological obstruction visible through representation theory.
Projective unitary representations in quantum symmetry
Wigner’s theorem says, under its usual assumptions, that transformations preserving transition probabilities on quantum rays are implemented by unitary or antiunitary operators on Hilbert space, determined up to phase.
For continuously connected symmetry groups, unitary implementations are central to the usual representation theory of quantum systems. Composition can produce phase multipliers, giving projective unitary representations.
Passing to a suitable central extension can restore ordinary unitary representations. The Heisenberg group and the Galilean group provide further classical settings where central extensions carry physically meaningful parameters.
The physical interpretation requires the broader quantum theory. Mathematically, the phase ambiguity is controlled by the same cocycle machinery developed above.
Lie algebra central extensions
For Lie groups, projective representations can also be studied infinitesimally. A central extension of a Lie algebra adds a central generator Z and modifies brackets by a Lie-algebra 2-cocycle.
At the group level, topological and integrality conditions determine whether a Lie algebra extension integrates to a global Lie group extension.
This again separates local from global information. A projective representation can have an infinitesimal cocycle that is easy to write while the corresponding global lift depends on the topology of the group.
A practical diagnostic workflow
- Write the intended projective group law.
- Choose explicit operator representatives Ug.
- Compute α(g,h) from UgUh=α(g,h)Ugh.
- Verify the cocycle equation on generators or all elements in a small group.
- Test whether rephasing removes α.
- For abelian groups, inspect commutator phases as a fast obstruction.
- Construct or identify the relevant central extension.
- Check whether the representation becomes ordinary on the extension.
Common mistakes
- Calling any almost-homomorphism projective: the scalar defect must satisfy the cocycle equation.
- Treating a chosen cocycle as unique: rephasing changes it by a coboundary.
- Assuming every visible phase is nontrivial: some can be removed globally.
- Confusing projective equivalence with ordinary matrix equality: scalar multiples define the same PGL element.
- Forgetting topology for Lie groups: infinitesimal and global lifting problems can differ.
- Calling spin-1/2 an ordinary SO(3) representation: it is linear on SU(2) and projective on SO(3).
Practice with worked answers
1. Ordinary versus projective
If UgUh=Ugh for every g,h, what is α? Answer: α(g,h)=1 everywhere, so the projective representation is already ordinary.
2. Cocycle check
What equation must α satisfy? Answer: α(g,h)α(gh,k)=α(h,k)α(g,hk).
3. Rephasing
If U′g=β(g)Ug, how does α change? Answer: α′(g,h)=β(g)β(h)β(gh)−1α(g,h).
4. Pauli commutator
Given XZ=−ZX, do the projective classes [X] and [Z] commute? Answer: yes. The minus sign is a central scalar, so [XZ]=[ZX] in PGL₂(C).
5. Central extension
Why does the cocycle equation make (a,g)(b,h)=(abα(g,h),gh) associative? Answer: the two ways of multiplying three elements differ exactly by the two sides of the cocycle equation.
6. Spin descent
An SU(2) representation sends −I to −I. Can it descend to SO(3)? Answer: not as an ordinary representation, because the kernel element of SU(2)→SO(3) would need to act trivially.
7. Integer spin
For highest weight n=4, how does −I act and does the representation descend? Answer: by (−1)⁴=1, so it descends to SO(3).
8. Half-integer spin
For n=3, what is j and does it descend? Answer: j=3/2; −I acts by −1, so it is projective rather than ordinary on SO(3).
The structural lesson
Projective representations show that failure of exact multiplication can itself be structured information. The scalar defect is not noise; it can encode topology, central charge, spin or a nontrivial cohomology class.
group law on rays → phase cocycle → cohomology class → central extension → ordinary linear representation upstairs.
This is representation theory doing what it does best: preserving the meaningful structure while moving the problem into a setting where the obstruction can be named and calculated.
References for further study
For standard treatments, see Jean-Pierre Serre, Linear Representations of Finite Groups; Gregory Karpilovsky, Projective Representations of Finite Groups; and Brian C. Hall, Lie Groups, Lie Algebras, and Representations. For the quantum-symmetry interpretation, Wigner’s theorem and the theory of central extensions provide the wider context.
Continue Representation Mathematics — Batch 05
Move from scalar twisting to algebraic tensor structure in Hopf Algebras, Coproducts and Antipodes. Deform that Hopf structure in Quantum Groups and Uq(sl₂). Build the rotation double cover geometrically in Clifford Algebras, Spin Groups and Spinors. Return to the BTT Mathematics Learning Hub.