Appendix

Operations on linear representations

Let G be a group, k a commutative field, E_1, E_2, ..., E_n vector spaces over k and π_i a linear representation of G on E_i (1 ≤ i ≤ n). The mapping

is a linear mapping of G into the vector space E_1 ⊗ ... ⊗ E_n, called the tensor product of π_1, ..., π_n and denoted by π_1 ⊗ ... ⊗ π_n.

Let E be a vector space over k and π a linear representation of G on E. For all g ∈ G, let τ(g) (resp. σ(g), ε(g)) be the unique automorphism of the algebra T(E) (resp. S(E), ∧(E)) which extends π(g) (Algebra, Chapter III, § 5, no. 2, § 6, no. 2 and § 7, no. 2). Then τ (resp. σ, ε) is a linear representation of G on T(E) (resp. S(E), ∧(E)) denoted by T(π) (resp. S(π), ∧(π)). The subrepresentation of T(π) (resp. S(π), ∧(π)) defined by T^n(E) (resp. S^n(E), ∧^n(E)) is called the n-th tensor (resp. symmetric, exterior) power of π and is denoted by T^n(π) (resp. S^n(π), ∧^n(π)). Then

(n factors). The representations S(π), ∧(π) are quotient representations of T(π) and hence S^n(π), ∧^n(π) are quotient representations of T^n(π).

Let g be a Lie algebra over k. The tensor product of a finite number of representations of g has already been defined in Chapter I, § 3, no. 2; it is denoted by π_1 ⊗ ... ⊗ π_n. Let E be a vector space over k and π a representation of g on E. For all x ∈ g, let τ'(x) (resp. σ'(x), ε'(x)) be the unique derivation of the algebra T(E) (resp. S(E), ∧(E)) which extends π(x) (Algebra, Chapter III, § 10, no. 9, Example 1). Then τ' (resp. σ', ε') is a linear representation of g on T(E) (resp. S(E), ∧(E)) by Algebra, loc. cit., formula (35), denoted by $T(\pi)$ (resp. $S(\pi)$, $\Lambda(\pi)$). The subrepresentation of $T(\pi)$ (resp. $S(\pi)$, $\Lambda(\pi)$) defined by $T^n(E)$ (resp. $S^n(E)$, $\Lambda^n(E)$) is denoted by $T^n(\pi)$ (resp. $S^n(\pi)$, $\Lambda^n(\pi)$). The representation $T^n(\pi)$ is the tensor product on $n$ representations identical with $\pi$. The representations $S(\pi)$, $\Lambda(\pi)$ are quotient representations of $T(\pi)$ and hence $S^n(\pi)$, $\Lambda^n(\pi)$ are quotient representations of $T^n(\pi)$.