In the second step we are using the fact that elements of the dual basis transform
as the dual representation. This is what is needed to ensure the relation
(e′j)∗(e′k) =δjk
The change of basis formula shows that if two matricesL 1 andL 2 are related
by conjugation by a third matrixA
L 2 =A−^1 L 1 Athen they represent the same linear transformation, with respect to two different
choices of basis. Recall that a finite dimensional representation is given by a set
of matricesπ(g), one for each group element. If two representations are related
by
π 2 (g) =A−^1 π 1 (g)A
(for allg,Adoes not depend ong), then we can think of them as being the
same representation, with different choices of basis. In such a case the represen-
tationsπ 1 andπ 2 are called “equivalent”, and we will often implicitly identify
representations that are equivalent.
4.4 Inner products
An inner product on a vector spaceV is an additional structure that provides
a notion of length for vectors, of angle between vectors, and identifiesV∗'V.
In the real case:
Definition(Inner product, real case).An inner product on a real vector space
Vis a symmetric ((v,w) = (w,v)) map
(·,·) :V×V→Rthat is non-degenerate and linear in both variables.
Our real inner products will usually be positive-definite ((v,v)≥ 0 and
(v,v) = 0 =⇒ v= 0), with indefinite inner products only appearing in the
context of special relativity, where an indefinite inner product on four dimen-
sional space-time is used.
In the complex case:
Definition(Inner product, complex case). A Hermitian inner product on a
complex vector spaceVis a map
〈·,·〉:V×V→Cthat is conjugate symmetric
〈v,w〉=〈w,v〉non-degenerate in both variables, linear in the second variable, and antilinear in
the first variable: forα∈Candu,v,w∈V
〈u+v,w〉=〈u,w〉+〈v,w〉, 〈αu,v〉=α〈u,v〉