Unitary irreducible representations finite dimensional

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... 

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... 

In Section 6.1 we define irreducible representations. Then we state, prove and illustrate Schur s lemma. Schur s lemma is the statement of the all-or-nothing personality of irreducible representations. In the Section 6.2 we discuss the physical importance of irreducible representations. In Section 6.3 we introduce invariant integration and apply it to show that characters of irreducible representations form an orthonormal set. In the optional Section 6.4 we use the technology we have developed to show that finite-dimensional unitary representations are no more than the sum of their irreducible parts. The remainder of the chapter is devoted to classifying the irreducible representations of 5(7(2) and 50(3). 

Proposition 6.6 Suppose V is a finite-dimensional complex vector space with a complex scalar product. Suppose G, V, p) is a unitary representation. Suppose that every linear operator 7 V V that commutes with p is a scalar multiple of the identity. Then G, V, p is irreducible. 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

Proof. Suppose (SU(2), V, p) is a finite-dimensional unitary irreducible Lie group representation. Let / denote its character. Define the function... 

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