By F. Albert Cotton

Keeps the easy-to-read structure and casual style of the former versions, and contains new fabric at the symmetric houses of prolonged arrays (crystals), projection operators, LCAO molecular orbitals, and electron counting ideas. additionally includes many new workouts and illustrations.

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**Example text**

Therefore, T (g)−1 σT (g) = σ, and so σT (g) = T (g)σ. Now, by Schur’s Lemma, σ = cε for some c ∈ C. But then f = cf0 , as we needed to show. Theorem 9. Let T : G → GL(V ) be a unitary representation. Let U, W ⊂ V TW . Then U and W are be minimal invariant subspaces such that TU orthogonal with respect to any invariant inner product in V . Proof. Fix an invariant inner-product in V and denote the corresponding orthogonal projection of the subspace W onto U by p. It is easy to see that p is a morphism of the representation TW into TU .

Let T be an irreducible ﬁnite-dimensional representation. Then T is irreducible. Proof. Let U ⊂ V be a T -invariant subspace. Consider its annihilator U 0 = { x ∈ V | f (x) = 0 for all f ∈ U } ⊂ V. It is a T -invariant subspace. In fact, for any g ∈ G, x ∈ U 0 , and f ∈ U we have f (T (g)x) = (T (g)−1 f )(x) = 0, because T (g)−1 f ∈ U . It is known from the theory of systems of linear equations that dim U 0 = dim V − dim U . Since T is irreducible, U 0 is equal to 0 or V , and correspondingly U is equal to V or 0.

Given any linear representation T : G → GL(V ), we deﬁne in a canonical manner the contragredient or dual representation T : G → GL(V ) in the dual space V of V . ) Definition. (T (g)f )(x) = f (T (g)−1 x) (g ∈ G, f ∈ V , x ∈ V ). 9). 3. Basic Operations on Representations 31 For a ﬁnite-dimensional representation T , the contragredient representation can be described in terms of matrices as follows. Let (e) = (e1 , . . , en ) and (ε) = (ε1 , . . , εi (ej ) = δij . Let T(e) (g) = [aij ] and T(ε) (g) = [bij ].