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 of 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 ) 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 (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 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 (irreducible) linear representations, both 1-dimensional:
The trivial representation: The nontrivial representation:
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 (and this holds for any group).
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 (or generally over algebraically closed field), 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:
, , and .
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 (that is, it has the effect of translating the discrete Fourier transform of ).
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 (linear) 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 (see ). 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 . (Note that is a special sort of central extension of .) Suppose also that is an irreducible unitary representation of (possibly infinite dimensional). 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 (discussed below) 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 , 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 . Section 4.7 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 (the rotation group in 3 dimensions) is doubly covered by . This has important applications in quantum mechanics, as the study of representations of leads to a nonrelativistic (low-velocity) theory of spin. The group , isomorphic to the Möbius group, is likewise doubly covered by (C). Both are supergroups of aforementioned and respectively and form a relativistic spin theory.