32 2. Basic notions of representation theory

and a typical element of E is

N

i1,...,in,j1,...,jm=1

Tji1...jm 1...in

ei1 ⊗ · · · ⊗ ein ⊗

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