In the field of representation theory in mathematics, a projective representation of a group on a vector space over a field is a group homomorphism from to the projective linear group where GL(V) is the general linear group of invertible linear transformations of over , and is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation).

Just as linear representations study the possible actions of the group on vector spaces via linear transformation, the projective representations study the actions on lines in these vector spaces (namely (V \{0}) / F*) via linear transformations.

satisfying the homomorphism property up to a constant:

is a set of linear operators related by multiplication with some nonzero scalar.

in each family of operators in such a way that the homomorphism property is satisfied exactly. This possibility is discussed further below.

Linear representations and projective representations

One way in which a projective representation can arise is by taking a linear group representation of on and applying the quotient map

which is the quotient by the subgroup of scalar transformations (diagonal matrices with all diagonal entries equal). The interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to an ordinary linear representation. A general projective representation cannot be lifted to a linear representation , and the obstruction to this lifting can be understood via group cohomology, as described below.

However, one can of to a linear representation of a different group , which will be a central extension of defined as follows:

1. ,

.

is:

1. ,

. The ordinary is a lift of the projective in the sense that:

1. .

If is a perfect group there is a single universal perfect central extension of that can be used.

Group cohomology

The analysis of the lifting question involves group cohomology. Indeed, if one fixes for each in a lifted element in lifting from back to , the lifts then satisfy

for some scalar in . It follows that the 2-cocycle or Schur multiplier satisfies the cocycle equation

for all in . This depends on the choice of the lift ; a different choice of lift will result in a different cocycle

cohomologous to . Thus defines a unique class in . This class might not be trivial. For example, in the case of the symmetric group and alternating group, Schur established that there is exactly one non-trivial class of Schur multiplier, and completely determined all the corresponding irreducible representations.

In general, a nontrivial class leads to an extension problem for . If is correctly extended we obtain a linear representation of the extended group, which induces the original projective representation when pushed back down to . The solution is always a central extension. From Schur's lemma, it follows that the irreducible representations of central extensions of , and the irreducible projective representations of , are essentially the same objects.

First example: Finite abelian groups

The group with 2 elements

. This group has two (irreducible) linear representations, both 1-dimensional:

• The trivial representation

will produce the same 1-dimensional projective representation. Namely, there is a unique 1-dimensional projective representation, and it is the trivial representation (and this holds for any group).

rotation:

are the same up to a sign:

instead of the real numbers, this projective representation is the same as

.

General cyclic group

be a choice of representatives for a projective representation:

• The trivial element without changing the representation.

• The nontrivial elements.

such that

Over the complex numbers (or generally over algebraically closed field will produce a standard linear representation, or in other words all the projective representations arise from ordinary linear representations

Product of cyclic groups

, a similar process can be done by changing the choice of representatives to satisfy:

This reduces to three conditions:

1. , and

2. .

This new scalar must satisfy:

• .

• .

which satisfy these 3 conditions will define a projective representation.

.

be such a root of unity, define:

.

.

distinct roots of unity are distinct representations.

The projective representations for general product of cyclic groups is done in a similar manner.

Discrete Fourier transform

as mentioned above, are many times viewed through the lens of Fourier transform.

as follows:

is a shift in frequency space (that is, it has the effect of translating the discrete Fourier transform).

commute up to multiplication by a constant:

1. .

as follows:

1. ,

in the quotient group.

is the group of all operators of the form

. This group is a discrete version of the Heisenberg group and is isomorphic to the group of matrices of the form

.

Projective representations of Lie groups

Studying projective representations of Lie groups leads one to consider true representations of their central extensions (see ). In many cases of interest it suffices to consider representations of covering groups is an irreducible unitary representation (possibly infinite dimensional). Then by Schur's lemma by setting

1. ,

acts as scalars.

The preceding construction is an important source of examples of projective representations. Bargmann's theorem (discussed below) gives a criterion under which every arises in this way.

Projective representations of SO(3)

A physically important example of the above construction comes from the case of the rotation group SO(3), whose universal cover is SU(2). According to the representation theory of SU(2), there is exactly one irreducible representation of in each dimension. When the dimension is odd (the "integer spin" case), the representation descends to an ordinary representation of . When the dimension is even (the "fractional spin" case), the representation does not descend to an ordinary representation of but does (by the result discussed above) descend to a projective representation of . Such projective representations of (the ones that do not come from ordinary representations) are referred to as "spinorial representations", whose elements (vectors) are called spinors.

By an argument discussed below, every finite-dimensional, irreducible projective representation of comes from a finite-dimensional, irreducible ordinary representation of .

Examples of covers, leading to projective representations

Notable cases of covering groups giving interesting projective representations:

• The special orthogonal group is doubly covered by the Spin group .

• In particular, the group SO(3) (the rotation group in 3 dimensions) is doubly covered by SU(2). This has important applications in quantum mechanics, as the study of representations of SU(2) leads to a nonrelativistic (low-velocity) theory of spin.

• The group SO+(3;1), isomorphic to the Möbius group, is likewise doubly covered by SL2(C). Both are supergroups of aforementioned and respectively and form a relativistic spin theory.

• The universal cover of the Poincaré group is a double cover (the semidirect product of with ). The irreducible unitary representations of this cover give rise to projective representations of the Poincaré group, as in Wigner's classification. Passing to the cover is essential, in order to include the fractional spin case.

• The orthogonal group is double covered by the Pin group .

• The symplectic group (not to be confused with the compact real form of the symplectic group, sometimes also denoted by ) is double covered by the metaplectic group . An important projective representation of comes from the metaplectic representation of .

Finite-dimensional projective unitary representations

In quantum physics, symmetry of a physical system on the quantum Hilbert space, that is, a continuous homomorphism

. The reason for taking the quotient is that physically, two vectors in the Hilbert space that are proportional represent the same physical state. [That is to say, the space of (pure) states is the set of equivalence classes of unit vectors, where two unit vectors are considered equivalent if they are proportional.] Thus, a unitary operator that is a multiple of the identity actually acts as the identity on the level of physical states.

having trace zero. In light of the homomorphisms theorem. by the procedure mentioned at the beginning of this section.

arises from a determinant-one is a linear combination of commutators, in which case every is unique.

is an irreducible, then by Schur's lemma is automatically determinant one.)

An important example is the case of SO(3), whose universal cover is SU(2) is semisimple. Furthermore, since SU(2) is a compact group, every finite-dimensional representation of it admits an inner product with respect to which the representation is unitary. Thus, the irreducible projective representations of SO(3) are in one-to-one correspondence with the irreducible ordinary representations of SU(2).

Infinite-dimensional projective unitary representations: the Heisenberg case

. These operators are defined as follows:

described in the "First example" section above. As in that section, we can then define a projective:

because the operators commute up to a phase factor. But no choice of the phase factors will lead to an ordinary unitary representation, since translations in position do not commute with translations in momentum (and multiplying by a nonzero constant will not change this). These operators do, however, come from an ordinary unitary representation of the Heisenberg group. (See also the Stone–von Neumann theorem.)

Infinite-dimensional projective unitary representations: Bargmann's theorem

On the other hand, Bargmann's theorem states that if the second Lie algebra cohomology can be de-projectivized after passing to the universal cover..

—as the previous example shows—because the second cohomology group of the associated commutative Lie algebra is nontrivial. Examples where the result does apply include semisimple groups (e.g., SL(2,R)) and the Poincaré group. This last result is important for Wigner's classification of the projective unitary representations of the Poincaré group.

The proof of Bargmann's theorem goes by considering a central extension is defined as

.

is a Lie is an infinite-dimensional topological group may be identified.

See also

• Affine representation

• Group action

• Central extension

• Particle physics and representation theory

• Spin-1/2

• Spinor

• Symmetry in quantum mechanics

• Heisenberg group

Notes

References