Projective representation


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 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.
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 via linear transformations.
In more concrete terms, a projective representation of can be represented as a collection of operators satisfying the homomorphism property up to a constant:
for some constant. Two such choices of operators define the same projective representation if for any the choices are the same up to a scalar. Equivalently, a projective representation of is a collection of operators, such that. Note that, in this notation, is a set of linear operators related by multiplication with some nonzero scalar.
If it is possible to choose a particular representative in each family of operators in such a way that the homomorphism property is satisfied exactly, rather than just up to a constant, then we say that can be "de-projectivized", or that can be "lifted to an ordinary representation". More concretely, we thus say that can be de-projectivized if there are for each such that. 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. 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 lift a projective representation of to a linear representation of a different group, which will be a central extension of. The group is the subgroup of defined as follows:
where is the quotient map of onto. Since is a homomorphism, it is easy to check that is, indeed, a subgroup of. If the original projective representation is faithful, then is isomorphic to the preimage in of.
We can define a homomorphism by setting. The kernel of is:
which is contained in the center of. It is clear also that is surjective, so that is a central extension of. We can also define an ordinary representation of by setting. The ordinary representation of is a lift of the projective representation of in the sense that:
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

Consider first the group with two elements denoted by, where is the trivial element and the group is defined by the identity. In every projective representation is sent to the identity map, so the representation is determined completely by the image of. This group has two linear representations, both 1-dimensional:
While these are distinct as linear representations, namely their actions on the points are different, they are the same as projective representations. In a 1-dimensional space, there is a single line, and both send that line to itself. More generally, any choice of the image will produce the same 1-dimensional projective representation. Namely, there is a unique 1-dimensional projective representation, and it is the trivial representation.
Over the 2-dimensional plane, we can send to the rotation:
Two rotations, namely rotation, returns the lines in the plane to themselves, hence it defines a projective representation. More formally, are the same up to a sign:
Unlike the 1-dimensional case, this projective representation doesn't arise from a standard linear representation. However, working over the complex numbers instead of the real numbers, this projective representation is the same as
in which case it is a linear representation, since.

General cyclic group

Now let which we write multiplicatively as, and let be a choice of representatives for a projective representation:The trivial element: Since, and is invertible, it must be the scalar. Hence, we can change the choice to be without changing the representation.The nontrivial elements: Since are the same up to a scalar, we can similarly choose the representatives so that for.
After these reductions, we are left with a choice for and a scalar such that
On the other hand, any choice of a root for some scalar can define a projective representation by setting
Over the complex numbers, changing the choice of into will produce a standard linear representation, or in other words all the projective representations arise from ordinary linear representations

Product of cyclic groups

In a product of two 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:
  • , so that.
  • Similarly,.
Both together imply that. Any such choices of which satisfy these 3 conditions will define a projective representation.
Over the complex numbers, as previously we can assume that, so the only parameter is.
For example, taking we have 3 conditions and where. These equations have solutions in for any and any choice of root of unity. More specifically, letting be such a root of unity, define:
Both matrices define an "-rotation": rotates each coordinate separately as complex numbers, while rotates the coordinates of the vector, and in particular. In addition we have that.
Note that is independent of the choices of representatives. It follows that the projective representations defined above for the 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

The projective representations of as mentioned above, are many times viewed through the lens of Fourier transform.
Consider the field of integers mod, where is prime, and let be the -dimensional space of functions on with values in. For each in, define two operators, and on as follows:
We write the formula for as if and were integers, but it is easily seen that the result only depends on the value of and mod. The operator is a translation, while is a shift in frequency space.
One may easily verify that for any and in, the operators and commute up to multiplication by a constant:
We may therefore define a projective representation of as follows:
where denotes the image of an operator in the quotient group. Since and commute up to a constant, is easily seen to be a projective representation. On the other hand, since and do not actually commute—and no nonzero multiples of them will commute— cannot be lifted to an ordinary representation of.
Since the projective representation is faithful, the central extension of obtained by the construction in the previous section is just the preimage in of the image of. Explicitly, this means that is the group of all operators of the form
for. This group is a discrete version of the Heisenberg group and is isomorphic to the group of matrices of the form
with.

Projective representations of Lie groups

Studying projective representations of Lie groups leads one to consider true representations of their central extensions. In many cases of interest it suffices to consider representations of covering groups. Specifically, suppose is a connected cover of a connected Lie group, so that for a discrete central subgroup of. Suppose also that is an irreducible unitary representation of . Then by Schur's lemma, the central subgroup will act by scalar multiples of the identity. Thus, at the projective level, will descend to. That is to say, for each, we can choose a preimage of in, and define a projective representation of by setting
where denotes the image in of an operator. Since is contained in the center of and the center of acts as scalars, the value of does not depend on the choice of.
The preceding construction is an important source of examples of projective representations. Bargmann's theorem gives a criterion under which every irreducible projective unitary representation of arises in this way.

Projective representations of

A physically important example of the above construction comes from the case of the rotation group , whose universal cover is . According to the representation [theory of SU(2)|representation theory of ], there is exactly one irreducible representation of in each dimension. When the dimension is odd, the representation descends to an ordinary representation of. When the dimension is even, the representation does not descend to an ordinary representation of but does descend to a projective representation of. Such projective representations of are referred to as "spinorial representations", whose elements 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:

Finite-dimensional projective unitary representations

In quantum physics, symmetry of a physical system is typically implemented by means of a projective unitary representation of a Lie group on the quantum Hilbert space, that is, a continuous homomorphism
where is the quotient of the unitary group by the operators of the form. The reason for taking the quotient is that physically, two vectors in the Hilbert space that are proportional represent the same physical state. Thus, a unitary operator that is a multiple of the identity actually acts as the identity on the level of physical states.
A finite-dimensional projective representation of then gives rise to a projective unitary representation of the Lie algebra of. In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation simply by choosing a representative for each having trace zero. In light of the homomorphisms theorem, it is then possible to de-projectivize itself, but at the expense of passing to the universal cover of. That is to say, every finite-dimensional projective unitary representation of arises from an ordinary unitary representation of by the procedure mentioned at the beginning of this section.
Specifically, since the Lie-algebra representation was de-projectivized by choosing a trace-zero representative, every finite-dimensional projective unitary representation of arises from a determinant-one ordinary unitary representation of . If is semisimple, then every element of is a linear combination of commutators, in which case every representation of is by operators with trace zero. In the semisimple case, then, the associated linear representation of is unique.
Conversely, if is an irreducible unitary representation of the universal cover of, then by Schur's lemma, the center of acts as scalar multiples of the identity. Thus, at the projective level, descends to a projective representation of the original group. Thus, there is a natural one-to-one correspondence between the irreducible projective representations of and the irreducible, determinant-one ordinary representations of.
An important example is the case of SO(3), whose universal cover is SU(2). Now, the Lie algebra is semisimple. Furthermore, since SU 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 are in one-to-one correspondence with the irreducible ordinary representations of SU.

Infinite-dimensional projective unitary representations: the Heisenberg case

The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of is typically not well defined. Indeed, the result fails: Consider, for example, the translations in position space and in momentum space for a quantum particle moving in, acting on the Hilbert space. These operators are defined as follows:
for all. These operators are simply continuous versions of the operators and described in the "First example" section above. As in that section, we can then define a projective unitary representation of :
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. These operators do, however, come from an ordinary unitary representation of the Heisenberg group, which is a one-dimensional central extension of.

Infinite-dimensional projective unitary representations: Bargmann's theorem

On the other hand, Bargmann's theorem states that if the second Lie algebra cohomology group of is trivial, then every projective unitary representation of can be de-projectivized after passing to the universal cover. More precisely, suppose we begin with a projective unitary representation of a Lie group. Then the theorem states that can be lifted to an ordinary unitary representation of the universal cover of. This means that maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level, descends to —and that the associated projective representation of is equal to.
The theorem does not apply to the group —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 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 of, constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group, where is the Hilbert space on which acts and is the group of unitary operators on. The group is defined as
As in the earlier section, the map given by is a surjective homomorphism whose kernel is so that is a central extension of. Again as in the earlier section, we can then define a linear representation of by setting. Then is a lift of in the sense that, where is the quotient map from to.
A key technical point is to show that is a Lie group. Once this result is established, we see that is a one-dimensional Lie group central extension of, so that the Lie algebra of is also a one-dimensional central extension of . But the cohomology group may be identified with the space of one-dimensional central extensions of ; if is trivial then every one-dimensional central extension of is trivial. In that case, is just the direct sum of with a copy of the real line. It follows that the universal cover of must be just a direct product of the universal cover of with a copy of the real line. We can then lift from to and finally restrict this lift to the universal cover of.