CHAPTER VII Representations of GL(n, C) and its continuous subgroups integer n 2 ≥ VII.1 Low-dimensional irreducible representations of GL(n, C) VII.1.1 Representations of dimension 1 ::::::::VII.1.1 a ::::::::::::::::::::::Trivial representation As always, the mapping which maps an arbitrary element M GL(n, C) to the complex number 2 1 constitutes the trivial representation of the group GL(n, C), denoted 1. ::::::::VII.1.1 b:::::::::::::::::::::::::::::Determinant representation Another representation of dimension 1 consists of the mapping which maps an given element M GL(n, C) to its determinant det M C 0 . 2 2 \{ } In the case of the subgroups of GL(n, C) consisting of matrices with determinant 1, like SL(n, C), SL(n, R),SU(n) or SO(n), the determinant representations obviously coincides with the trivial representation. VII.1.2 Vector representation Definition VII.1. The defining representation of GL(n, C), i.e. the automorphism of GL(n, C) which maps every matrix M GL(n, C) to itself, is called vector representation and denoted n. 2 The latter notation corresponds to the dimension of the representation, which comes from the fact that the matrices of GL(n, C) operate on complex vectors of dimension n, i.e. on the vector space Cn. Since there is no proper subspace of Cn which is invariant under all elements of GL(n, C), the vector representation is irreducible. VII.1.3 Complex conjugate representation Theorem & Definition VII.2. The mapping which to every matrix M GL(n, C) associates the 2 matrix M GL(n, C) whose entries are complex conjugate to those of M is a faithful irreducible ⇤ 2 representation of GL(n, C) of dimension n,calledthecomplex conjugate representation of the vector representation and denoted n¯. In the case of the subgroups of GL(n, C) consisting of real matrices, i.e. of GL(n, R) and its subgroups, the complex conjugate representation n¯ coincides with the vector representation n. 92 Representations of GL(n) and its continuous subgroups VII.1.4 Dual representation Theorem & Definition VII.3. The mapping which to every matrix M GL(n, C) associates the 2 matrix (M 1)T GL(n, C) is a faithful irreducible representation of GL(n, C) of dimension n, − 2 called the dual or contragredient representation of the vector representation and denoted n⇤. Considering that the matrix M represents a regular linear map V W in given bases, where 1 ! V and W are n-dimensional complex vector spaces, then M − represents a map W V,and 1 T ! (M − ) amapV ⇤ W ⇤, where V ⇤ and W ⇤ are the dual spaces to V and W,respectively. ! Property VII.4. In the case of the unitary group U(n) and its subgroups SU(n),O(n),SO(n)..., the dual representation coincides with the complex conjugate representation. 1 T T AunitarymatrixU U(n) obeys UU† = U †U = 1 ,i.e.(U − ) =(U †) = U ⇤. 2 n 2 Eventually, yet another irreducible representation of dimension n of the group GL(n, C) is the dual of the complex conjugate, denoted n¯⇤, which maps M GL(n, C) to the Hermitian conjugate 1 2 of its inverse, (M − )†. VII.2 Tensor representations of GL(n, C) In this section... VII.2.1 Tensor products of the vector representation In Sec. VII.1.2, we introduced the vector representation n of GL(n, C): the corresponding n n- ⇥ matrices operate on the vector space Cn according to n n x C x0 = Mx C , (VII.5a) 2 ! 2 i.e. component-wise n i i i j x x0 = M x i 1,...,n , (VII.5b) ! j 8 2{ } Xj=1 i i i where the x resp. x are the components of x resp. x0 and M the entries of the matrix M, { } { 0 } j where as usual the left resp. right index stands for the line resp. column of the matrix. Starting from the vector space Cn, one constructs for every p N the tensor-product space 2 ⇤ n p n n (C )⌦ C C , ⌘ ⌦···⌦ p copies whose elements are tensors of order p, hereafter| generically{z denoted} T, with components (in a given basis) traditionally characterized by p indices: T i1i2...ip with i ,i ,...,i 1,...,n . 1 2 p 2{ } Generalizing Eq. (VII.5), one defines the operation of M GL(n, C) on such tensors: 2 Theorem & Definition VII.6. The mapping from GL(n, C) into the group of regular linear applica- n p i tions of the tensor-product space (C )⌦ into itself which associates to a matrix M with entries M j p the linear operator M ⌦ such that p M ⌦ : T T0 =(M M)T, (VII.6a) ! ⌦···⌦ p copies corresponding in terms of components to | {z } VII.2 Tensor representations of GL(n, C) 93 n n i1i2...ip i1i2...ip i1 i2 ip j1j2...jp T T 0 = M M M T , (VII.6b) ! ··· j1 j2 ··· jp jX1=1 jXp=1 is a representation of GL(n, C) of dimension np, called the tensor-product representation n n. (VII.6c) ⌦···⌦ p copies This representation is generally reducible when| p>{z 1.} VII.2.2 Operations on the indices of a tensor ::::::::VII.2.2 a ::::::::::::::::::::::::::::::::::::::::Operation of a permutation on a tensor Definition VII.7. Let σ Sp be a permutation of p elements. One associates with it an operator σˆ 2 n p acting on the tensor-product space (C )⌦ such that for every tensor T of order p, the tensor σˆ(T) has the components i i ...i σT 1 2 p = T iσ(1)iσ(2)...iσ(p) . (VII.7) That is, the permutation acts on the tensor indices. n p Remark: One can actually show that this definition is independent of the choice of basis in (C )⌦ . Example VII.8. Letting p =2and considering the transposition ⌧ = (1 2) S , the associated 12 2 2 operator ⌧ˆ12 is such that for every second-order tensor T, the tensor ⌧ˆ12(T) has the components ij ⌧ T = T ji for all i, j 1,...,n . (VII.8) 12 2{ } Property VII.9. The operator σˆ associated to a permutation σ S is linear. 2 p Theorem VII.10. For every permutation σ Sp and every matrix M GL(n, C),thelinear p 2 n p 2 operators σˆ and M ⌦ on the tensor-product space (C )⌦ commute. That is, denoting the image by p M ⌦ with a 0 as in Eq. (VII.6a),onehas σˆ(T0)= σˆ(T) 0 (VII.10) for every tensor T of order p. ::::::::VII.2.2 b::::::::::::::::::Young operators n p Any linear combination of linear operators on the tensor-product space (C )⌦ is again a linear operator. Accordingly, one can associate such operators to the elements of the group algebra CSp in the same manner as in definition VII.7. In particular, to each Young element Y introduced in n p § III.5.1 b corresponds a linear operator Yˆ on (C )⌦ , generically called Young operators . For instance, one associates to the symmetrizer (III.55a) and to the antisymmetrizer (III.55b) for p elements the operators 1 Sˆ σˆ (VII.11a) ⌘ p! σ Sp X2 1 Aˆ "(σ)ˆσ, (VII.11b) ⌘ p! σ Sp X2 where we recall that "(σ) denotes the signature of the permutation σ. Invoking Eq. (VII.7), their respective actions on a tensor T (Cn) p of order p read 2 ⌦ 94 Representations of GL(n) and its continuous subgroups i i ...i 1 S T 1 2 p = T iσ(1)iσ(2)...iσ(p) (VII.12a) p! σ Sp X2 i i ...i 1 A T 1 2 p = "(σ)T iσ(1)iσ(2)...iσ(p) (VII.12b) p! σ Sp X2 for every p-tuple (i1,i2,...ip). The tensor Sˆ(T) resp. Aˆ(T) is then symmetric resp. antisymmetric under the exchange of any two indices. In the case of order-2 tensors, these expressions become ij 1 ij 1 S T = T ij + T ji and A T = T ij T ji (VII.13) 2 2 − for every i, j 1,...,n . 2{ } Remark: While the operators σˆ associated to permutations are regular, this is no longer the case of those associated to Young elements. For instance, the operation of the antisymmetrizer (VII.11b) (40) n p on any fully symmetric tensor — i.e. on an element of the image of (C )⌦ by Sˆ — is the zero tensor. VII.2.3 Irreducible representations built from the vector representation ::::::::VII.2.3 a ::::::::::::::::::::::::::::::::::::::::::::::::::Irreducible representations and Young operators The theorem (VII.10) extends to the Young operators Yˆ, which all commute with every operator on (Cn) p of the form M p with M GL(n, C): ⌦ ⌦ 2 n p p p for every T (C )⌦ , Yˆ M ⌦ (T) = M ⌦ Yˆ(T) . 2 One deduces thereof at once that the image of every Young operator Yˆ is a subspace of the n p p tensor-product space (C )⌦ which is invariant under all operators M ⌦ of the tensor-product representation n n. ⌦···⌦ n p Consider T im Yˆ:thereexistsS (C )⌦ such that T = Yˆ(S). Then for any M GL(n, C) 2 2 2 one may write p p p M ⌦ (T)=M ⌦ Yˆ(S) = Yˆ M ⌦ (S) , p which shows that M ⌦ (T) im Yˆ. 2 2 Accordingly, the tensors of order p belonging to the image of any Young operator Yˆ span n p p a subspace of (C )⌦ on which the restrictions of the linear operators M ⌦ constitute a linear representation of GL(n, C).