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
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