Projective representation

In the field of representation theory in mathematics, a projective representation of a group "G" on a vector space "V" over a field "F" is a group homomorphism from "G" to the projective linear group
formula_1
where GL("V") is the general linear group of invertible linear transformations of "V" over "F", and "F"∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation).
In more concrete terms, a projective representation of formula_2 is a collection of operators formula_3 satisfying the homomorphism property up to a constant:
for some constant formula_5. Equivalently, a projective representation of formula_2 is a collection of operators formula_7, such that formula_8. Note that, in this notation, formula_9 is a "set" of linear operators related by multiplication with some nonzero scalar.
If it is possible to choose a particular representative formula_10 in each family of operators in such a way that the homomorphism property is satisfied "on the nose", rather than just up to a constant, then we say that formula_11 can be "de-projectivized", or that formula_11 can be "lifted to an ordinary representation". More concretely, we thus say that formula_11 can be de-projectivized if there are formula_10 for each formula_15 such that formula_16. 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" lift a projective representation formula_18 of to a linear representation of a different group , which will be a central extension of . The group formula_19 is the subgroup of formula_20 defined as follows:
where formula_22 is the quotient map of formula_23 onto formula_24. Since formula_18 is a homomorphism, it is easy to check that formula_19 is, indeed, a subgroup of formula_20. If the original projective representation formula_18 is faithful, then formula_19 is isomorphic to the preimage in formula_23 of formula_31.
We can define a homomorphism formula_32 by setting formula_33. The kernel of formula_34 is:
which is contained in the center of formula_19. It is clear also that formula_34 is surjective, so that formula_19 is a central extension of formula_2. We can also define an ordinary representation formula_40 of formula_19 by setting formula_42. The "ordinary" representation formula_40 of formula_19 is a lift of the "projective" representation formula_18 of formula_2 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: discrete Fourier transform.
Consider the field formula_51 of integers mod formula_52, where formula_52 is prime, and let formula_54 be the formula_52-dimensional space of functions on formula_51 with values in formula_57. For each formula_58 in formula_51, define two operators, formula_60 and formula_61 on formula_54 as follows:
We write the formula for formula_61 as if formula_58 and formula_66 were integers, but it is easily seen that the result only depends on the value of formula_58 and formula_66 mod formula_52. The operator formula_60 is a translation, while formula_61 is a shift in frequency space (that is, it has the effect of translating the discrete Fourier transform of formula_72).
One may easily verify that for any formula_58 and formula_66 in formula_51, the operators formula_60 and formula_77 commute up to multiplication by a constant:
We may therefore define a projective representation formula_18 of formula_80 as follows:
where formula_82 denotes the image of an operator formula_83 in the quotient group formula_24. Since formula_60 and formula_77 commute up to a constant, formula_18 is easily seen to be a projective representation. On the other hand, since formula_60 and formula_77 do not actually commute—and no nonzero multiples of them will commute—formula_18 cannot be lifted to an ordinary (linear) representation of formula_80.
Since the projective representation formula_18 is faithful, the central extension formula_19 of formula_80 obtained by the construction in the previous section is just the preimage in formula_23 of the image of formula_18. Explicitly, this means that formula_19 is the group of all operators of the form
for formula_99. This group is a discrete version of the Heisenberg group and is isomorphic to the group of matrices of the form
with formula_101.
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. Specifically, suppose formula_102 is a connected cover of a connected Lie group formula_2, so that formula_104 for a discrete central subgroup formula_105 of formula_102. (Note that formula_102 is a special sort of central extension of formula_2.) Suppose also that formula_109 is an irreducible unitary representation of formula_102 (possibly infinite dimensional). Then by Schur's lemma, the central subgroup formula_105 will act by scalar multiples of the identity. Thus, at the projective level, formula_109 will descend to formula_2. That is to say, for each formula_15, we can choose a preimage formula_115 of formula_116 in formula_102, and define a projective representation formula_18 of formula_2 by setting
where formula_82 denotes the image in formula_24 of an operator formula_83. Since formula_105 is contained in the center of formula_102 and the center of formula_102 acts as scalars, the value of formula_127 does not depend on the choice of formula_115.
The preceding construction is an important source of examples of projective representations. Bargmann's theorem (discussed below) gives a criterion under which "every" irreducible projective unitary representation of formula_2 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 SU(2) in each dimension. When the dimension is odd (the "integer spin" case), the representation descends to an ordinary representation of SO(3). When the dimension is even (the "fractional spin" case), the representation does not descend to an ordinary representation of SO(3) but does (by the result discussed above) descend to a projective representation of SO(3). Such projective representations of SO(3) (the ones that do not come from ordinary representations) are referred to as "spinorial representations."
By an argument discussed below, every finite-dimensional, irreducible "projective" representation of SO(3) comes from a finite-dimensional, irreducible "ordinary" representation of SU(2).
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 formula_18 of a Lie group formula_2 on the quantum Hilbert space, that is, a continuous homomorphism
where formula_133 is the quotient of the unitary group formula_134 by the operators of the form formula_135. 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. 
A finite-dimensional projective representation of formula_2 then gives rise to a projective unitary representation formula_137 of the Lie algebra formula_138 of formula_2. In the finite-dimensional case, it is always possible to "de-projectivize" the Lie-algebra representation formula_137 simply by choosing a representative for each formula_141 having trace zero. In light of the homomorphisms theorem, it is then possible to de-projectivize formula_18 itself, but at the expense of passing to the universal cover formula_143 of formula_2. That is to say, every finite-dimensional projective unitary representation of formula_2 arises from an ordinary unitary representation of formula_143 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 formula_2 arises from a "determinant-one" ordinary unitary representation of formula_143 (i.e., one in which each element of formula_143 acts as an operator with determinant one). If formula_138 is semisimple, then every element of formula_138 is a linear combination of commutators, in which case "every" representation of formula_138 is by operators with trace zero. In the semisimple case, then, the associated linear representation of formula_143 is unique.
Conversely, if formula_18 is an "irreducible" unitary representation of the universal cover formula_143 of formula_2, then by Schur's lemma, the center of formula_143 acts as scalar multiples of the identity. Thus, at the projective level, formula_18 descends to a projective representation of the original group formula_2. Thus, there is a natural one-to-one correspondence between the irreducible projective representations of formula_2 and the irreducible, determinant-one ordinary representations of formula_143. (In the semisimple case, the qualifier "determinant-one" may be omitted, because in that case, every representation of formula_143 is automatically determinant one.)
An important example is the case of SO(3), whose universal cover is SU(2). Now, the Lie algebra formula_163 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.
The results of the previous subsection do not hold in the infinite-dimensional case, simply because the trace of formula_141 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 formula_165, acting on the Hilbert space formula_166. These operators are defined as follows:
for all formula_168. These operators are simply continuous versions of the operators formula_60 and formula_61 described in the "First example" section above. As in that section, we can then define a "projective" unitary representation formula_18 of formula_172:
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, which is a one-dimensional central extension of formula_172. (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 two-dimensional Lie algebra cohomology formula_175 of formula_138 is trivial, then every projective unitary representation of formula_2 can be de-projectivized after passing to the universal cover. More precisely, suppose we begin with a projective unitary representation formula_18 of a Lie group formula_2. Then the theorem states that formula_18 can be lifted to an ordinary unitary representation formula_181 of the universal cover formula_102 of formula_2. This means that formula_181 maps each element of the kernel of the covering map to a scalar multiple of the identity—so that at the projective level, formula_181 descends to formula_2—and that the associated projective representation of formula_2 is equal to formula_18.
The theorem does not apply to the group formula_172—as the previous example shows—because the two-dimensional cohomology 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 formula_19 of formula_2, constructed similarly to the section above on linear representations and projective representations, as a subgroup of the direct product group formula_192, where formula_193 is the Hilbert space on which formula_18 acts and formula_195 is the group of unitary operators on formula_193. The group formula_19 is defined as
As in the earlier section, the map formula_199 given by formula_200 is a surjective homomorphism whose kernel is formula_201 so that formula_19 is a central extension of formula_2. Again as in the earlier section, we can then define a linear representation formula_40 of formula_19 by setting formula_206. Then formula_40 is a lift of formula_18 in the sense that formula_209, where formula_22 is the quotient map from formula_195 to formula_212. 
A key technical point is to show that formula_19 is a "Lie" group. (This claim is not so obvious, because if formula_193 is infinite dimensional, the group formula_192 is an infinite-dimensional topological group.) Once this result is established, we see that formula_19 is a one-dimensional Lie group central extension of formula_2, so that the Lie algebra formula_218 of formula_19 is also a one-dimensional central extension of formula_138 (note here that the adjective "one-dimensional" does not refer to formula_19 and formula_222, but rather to the kernel of the projection map from those objects onto formula_2 and formula_224 respectively). But the cohomology group formula_175 may be identified with the space of one-dimensional (again, in the aforementioned sense) central extensions of formula_138; if formula_175 is trivial then every one-dimensional central extension of formula_138 is trivial. In that case, formula_218 is just the direct sum of formula_138 with a copy of the real line. It follows that the universal cover formula_231 of formula_19 must be just a direct product of the universal cover of formula_2 with a copy of the real line. We can then lift formula_40 from formula_19 to formula_231 (by composing with the covering map) and finally restrict this lift to the universal cover formula_143 of formula_2.