32 2. Basic notions of representation theory and a typical element of E is N i1,...,in,j1,...,jm=1 T i1...in j1...jm ei 1 ⊗ · · · ⊗ ei n ⊗ ej1 ⊗ · · · ⊗ ejm, where T is a multidimensional table of numbers. Physicists define a tensor as a collection of such multidimensional tables TB attached to every basis B in V , which change according to a certain rule when the basis B is changed (derive this rule!). Here it is important to distinguish upper and lower indices, since lower indices of T correspond to V and upper ones to V ∗ . The physicists don’t write the sum sign, but remember that one should sum over indices that repeat twice — once as an upper index and once as lower. This convention is called the Einstein summation, and it also stipulates that if an index appears once, then there is no summation over it, while no index is supposed to appear more than once as an upper index or more than once as a lower index. One can also define the tensor product of linear maps. Namely, if A : V → V and B : W → W are linear maps, then one can define the linear map A ⊗ B : V ⊗ W → V ⊗ W given by the formula (A ⊗ B)(v ⊗ w) = Av ⊗ Bw (check that this is well defined!). The most important properties of tensor products are summarized in the following problem. Problem 2.11.3. (a) Let U be any k-vector space. Construct a natural bijection between bilinear maps V × W → U and linear maps V ⊗ W → U (“natural” means that the bijection is defined without choosing bases). (b) Show that if {vi} is a basis of V and {wj} is a basis of W , then {vi ⊗ wj} is a basis of V ⊗ W . (c) Construct a natural isomorphism V ∗ ⊗ W → Hom(V, W ) in the case when V is finite dimensional. (d) Let V be a vector space over a field k. Let SnV be the quotient of V n (n-fold tensor product of V ) by the subspace spanned by the tensors T − s(T ) where T ∈ V n and s is a transposition. Also let ∧nV be the quotient of V n by the subspace spanned by the tensors T such that s(T ) = T for some transposition s. These spaces are

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