Unitary irreducible representations

We chose a unitary irreducible representation R of the group G, as well as a normalized state the, so-called, reference state R). The choice of the reference state is in principle arbitrary but not unessential. Usually it is an extremal state (the highest-weight state), the state anihilated by Ea, namely, Erl R) = 0. 

Note that it suffices to define the scalar product on basis elements. This proposition plays a crucial role in the proof of Proposition 6.14, the classification of the unitary irreducible representations of the group 51/(2). 

For unitary representations we have a converse to Proposition 6.3. Unitary irreducible representations are sometimes called unirreps for short. 

The next proposition shows that characters of unitary irreducible representations form a Hermitian orthonormal subset of the vector space of complexvalued functions on the group. This theoretical result will help us to ascertain that we have found all irreducible representations when the characters of the irreducible representations we know span the set of functions invariant under conjugation. 

In other words, the representations (SU(2), P , Rfi), for nonnegative integers n, form a complete list of the finite-dimensional unitary irreducible representations of SU(2), without repeats. Complete lists without repeats are called classifications. 

Proposition 6.16 Every finite-dimensional, unitary, irreducible representation of SO(3) is isomorphic to Qn for some even n. In addition, Qn is isomorphic to Qn if and only ifn = n. ... 

Now W is a finite-dimensional, unitary, irreducible representation, so by Proposition 6.16 there must be a nonnegative even integer h and an isomorphism T P W of representations. Because T is an isomorphism, the fist of weights for P must be the same as the fist of weights for W. Hence... 

For a given unitary irreducible representation r there will be nj matrix elements corresponding to each R. If the operations are R Rv Rtf. .. Rr, then the g matrix elements of a chosen i and j value,... 

Fig. 1. Regions in the (k, q) plane where the -eigenvalue spectrum corresponds to unitary irreducible representations of so(3) (horizontal shading) and so(2, 1) (vertical shading). See text for details. 

The Casimir operator for so(3) commutes with all generators and is given by J2 = J + J + J3. The unitary irreducible representations are characterized by a single quantum number j = 0,1, ,..., which also includes the spin representations of su(2) corresponding to the special unitary group SU(2) when j = j,, and are given by... 

We are primarily interested in the unitary irreducible representations (cf. Section III). In this case the generators Jt and Vt must be Hermitian (Jj = Jt, V = Vj). As in the so(2,1) case this imposes restrictions on the types of irreps we have obtained in the general case. Since V3 is Hermitian then... 

Fig. 4. The collection of so(4) subtowers for n = 1,2, 3,... forms an so(4, 2) tower of scaled hydrogenic eigenfunctions such that all eigenfunctions belong to a single unitary irreducible representation of so(4, 2). The top state in any so(4) subtower can be reached by successive application of the so(2, 1) operator T+ to the ground state. See text for details.

The CS construction is based on the Hilbert space of states H that carries unitary irreducible representations TZ G) of a compact group G on H and a one-dimensional representation of a subgroup K groups associated with A-electron state space and one-electron state space, respectively. In the finite-dimensional case these groups are the classical matrix groups and U H ) = and... 

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