Generating Representations

One way of generating representations is to take some representations one knows and combine them in some fashion.There are three standard methods of obtaining new representations from old, namely, direct sum of representations, tensor products of representations, and dual representations.

Direct Sum of Representations

Let G be a matrix Lie group and let Π₁, Π₂, ..., Π_m be representations of G acting on vector spaces V₁, V₂, ..., V_m. Then, the direct sum of  Π₁, Π₂, ..., Π_m is a representation Π₁ ⊕ Π₂ ⊕ ...⊕ Π_m of G acting on the space V₁ ⊕ V₂ ⊕ ... ⊕ V_m, defined by [Π₁ ⊕ Π₂ ⊕ ... ⊕ Π_m (A)] (v₁, v₂, ..., v_m) = (Π₁(A)v₁, Π₂(A)v₂, ..., Π_m(A)v_m) for all A ∈ G. 

Similarly, if g is a Lie algebra, and π₁, π₂, ..., π_m are representations of g acting on V₁, V₂, ..., V_m, then we define the direct sum of  π₁, π₂, ..., π_m , acting on  V₁ ⊕ V₂ ⊕ ... ⊕ V_m by [π₁ ⊕ π₂ ⊕ ...⊕ π_m (X)] (v₁, v₂, ..., v_m) = (π₁(X)v₁, π₂(X)v₂, ..., π_m(X)v_m) for all X ∈ g.

Tensor Products of Representations

Consider an element u of U and v of V, the "product" of these two are denoted by u⊗v. The space U⊗V is then the space of linear combination of such products. The space of elements is in the form of a₁u₁⊗v₁ + a₂u₂⊗v₂ + ... + a_nu_n⊗v_n . The product is not necessary commutative (v⊗u is in different space) but is bilinear ((u₁+ au₂)⊗v = u₁⊗ v + au₂ ⊗ v, u ⊗ (v₁+ av₂) = u ⊗ v₁ + au ⊗ v₂). 

If U and V are finite-dimensional real or complex vector spaces, then tensor product (W, ϕ) of U and V is a vector space W, together with a bilinear map ϕ : U x V --> W with the following property: If ψ is any bilinear map of UxV into a vector space X, then there exists a unique linear map ψ of W into X. Bilinear maps on UxV turn into linear maps on W. Suppose e₁, e₂ , e_n is a basis for U and f₁, f₂ , f_m  is a basis for V. Then {ϕ(e_i, f_j) | 1 ≤ i ≤ n, 1≤ j ≤ m} is a basis for W. In this case, {e_i⊗f_j | 1 ≤ i ≤ n, 1≤ j ≤ m} is a basis for U⊗V and dim (U⊗V) = (dim U)(dim V). 

The tensor product (W, ϕ)  is unique up to canonical isomorphism. That is, if (W₁, ϕ₁) and (W₂, ϕ₂) are two tensor products, then there exists a unique vector space isomorphism Φ : W₁ --> W₂ . 

The defining property of U⊗V is called the universal property of tensor products. Suppose that ψ(u,v) is some bilinear expression in (u,v). Then , the universal property says precisely that there is a unique linear map T(= ψ ) such that T(u⊗v) = ψ(u,v). Let A : U --> U and B : V --> V be linear operators. Then, there exists a unique linear operator from U⊗V to U⊗V, denoted by A⊗B, such that (A⊗B)(u⊗v) = (Au)⊗(Bv) for all u ∈ U and v ∈ V. Moreover, (A₁⊗B₁)(A₂⊗B₂) = (A₁A₂)⊗(B₁B₂). 

There are two approaches to define tensor of representations. 

(A) Starts with a representation of a group G acting on a space V and a representation of another group H acting on space U and produces a representation of the product group GxH acting on space U⊗V.

(B) Starts with two different representations of the same group G, acting on spaces U and V, and produces a representation of G acting on U⊗V.

(A) Let G and H be matrix Lie groups. Let Π₁ be a representation of G acting on a space U and let Π₂ be a representation of G acting on a space V. Then, the tensor product of Π₁ and Π₂ is a representation Π₁⊗Π₂ of GxH acting on U⊗V defined by  Π₁⊗Π₂ (A,B) =  Π₁(A)⊗Π₂(B) for all A ∈ G and B ∈ H. Let  π₁⊗ π₂ denote the associated representation of the Lie algebra of GxH, namely g⊕h. Then, for all X ∈ g and Y ∈ h, π₁⊗ π₂(X,Y) = π₁(X) ⊗ I + I ⊗ π₂(Y).  

(B) Let G be a matrix Lie groups and let Π₁ and Π₂ be representation of G acting on a space V₁ and V₂. Then, the tensor product of Π₁ and Π₂ is a representation of G acting on V₁ ⊗ V₂ defined by Π₁⊗Π₂ (A) =  Π₁(A)⊗Π₂(A) for all A ∈ G. The associated Lie algebra g satisfies π₁⊗ π₂(X) = π₁(X) ⊗ I  + I ⊗ π₂(X) all X ∈ g. 

Suppose Π₁ and Π₂ are irreducible representations of a group G. If regard Π₁⊗Π₂ as representation of G, it may no longer be irreducible. If it is not irreducible, one can attempt to decompose it as a direct sum of irreducible representations. This process is called the Clebsch-Gordan theory. In the physics, the problem of analyzing tensor products of representations of SU(2) is called "addition of angular momentum."

Dual Representations

A linear functional on a vector space V is a linear map of V into C. If v₁, v₂, ..., v_n is a basis for V, then for each set of constants a₁, a₂, ..., a_n, there is a unique linear functional ϕ such that ϕ(v_k) = a_k. If V is a finite-dimensional complex vector space, then the dual space to V, denoted by V^*, is the set of all linear functionals on V. This is also a vector space and its dimension is the same as that of V.  If A is a linear operator on V, let A^tr denote the dual or transpose operator on V^*, (A^trϕ)(v) = ϕ(Av) for all ϕ ∈ V^*, v  ∈ V. Note that the matrix of A^tr is the transpose of the matrix A and not the conjugate transpose. If A and B are linear operators on V, then (AB)^tr = B^tr A^tr . 

Suppose G is a matrix Lie group and Π is a representation of G acting on a finite-dimensional vector space V. Then, the dual representation Π^* to Π is the representation of G acting on V^* given by Π^*(g) = [Π(g^-1)]^tr . Similarly, if π is a representation of a Lie algebra g acting on a finite-dimensional vector space V, then π^* is the representation of g acting on V^* given by  π^*(g) = -π(X)^tr. The dual representation is also called contragredient representation. Note that the transpose is an order-reversing operation, we cannot simply define Π^*(g) = Π(g)^tr. This would not be a representation.