Projective unitary group

Definition 10.6 Suppose V is a complex scalar product space. projective unitary group of V is... 

We start by defining the projective unitary representations. Recall the unitary group ZT (V) of a complex scalar product space V from Definition 4.2. The following definition is an analog of Definition 4.11. 

Definition 10.7 Suppose G is a group and V is a complex scalar product space. Then the triple G, V, p) is called a projective unitary representation if and only if p is a group homomorphism from G to PIT (V). 

Sometimes, to stress the distinction between unitary group representations as defined in Chapter 4 and projective unitary representations, we will call the former linear unitary representations. Any (linear) unitary representation descends to a projective unitary representation. More specifically, suppose G is a group, suppose V is complex scalar product space and suppose p G U (U) is a (linear) unitary representation. Then we can define a projective unitary representation p G P(V) by... 

So pi is a projective unitary representation of SO(3). In fact, pi is a bona fide projective Lie group representation, i.e,. it is a differentiable ftinction, as we will show in Proposition 10.5. However, pi does not descend from any linear unitary representation of St/(2) (Exercise 10.20). 

Definition 10.8 Suppose G is a group, V is a complex scalar product space and p. G PU (V) is a projective unitary representation. We say that p is irreducible if the only subspace W of V such that [VT] is invariant under p is V itself. 

The results of this section are another confirmation of the philosophy spelled out in Section 6.2. We expect that the irreducible representations of the symmetry group determined by equivalent observers should correspond to the elementary systems. In fact, the experimentally observed spin properties of elementary particles correspond to irreducible projective unitary representations of the Lie group SO(3). Once again, we see that representation theory makes a testable physical prediction. 

In this appendix we prove Proposition 10.6 from Section 10.4, which states that the irreducible projective unitary Lie group representations of SO(3) are in one-to-one correspondence with the irreducible (linear) unitary Lie group representations of St/ (2). The proof requires some techniques from topology and differential geometry. 

Proof, (of Proposition 10.6) First we suppose that (S(/(2), V, p) is a linear irreducible unitary Lie group representation. By Proposition 6.14 we know that p is isomorphic to the representation R for some n. By Proposition 10.5 we know that R can be pushed forward to an irreducible projective representation of SO(3). Hence p can be pushed forward to an irreducible projective Lie group representation of SO(3). 

We begin by reiterating the definition of a PR and listing some conventions regarding PFs. A projective unitary representation of a group G = g, of dimension g is a set of matrices that satisfy the relations... 

Conversely, suppose that (SO(3), P( V), cr) is a finite-dimensional projective rmitary representation. We want to show that cr is the pushforward of the projectivization of a linear unitary representation p of SO (2). In other words, we must show that there is a Lie group representation p that makes the diagram in Figure B.2 commutative, and that this p is a Lie group representation. 

Equation (26) shows that (T(q, R) forms a unitary projective (or multiplier) representation of R j - P(q). Only for non-symmorphic groups with b different from zero (that is, when q lies on the surface of the BZ) are the projective factors exp [ib,.w,] in eq. (26) different from unity. 

The representations provided in the basis of degenerate eigenfunctions are usually irreducible and can be chosen to be unitary matrices (in fact usually orthogonal matrices if the functions are real functions). In practice, what one is usually faced with is a collection of functions which have arbitrary (but known) transformation properties and what one actually wants to do is to adapt these functions so that they actually transform like the true eigenfunctions of the problem. This can be done by means of the group theoretical projection operator. 

If the unitary matrix irreducible representation of the point group is known and denoted by (R), where labels the irreducible representation, then the projection operator is... 

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