Definition(Dual or contragredient representation).The dual or contragredient
representation onV∗is given by taking as linear operators
(π−^1 )t(g) :V∗→V∗ (4.2)These satisfy the homomorphism property since
(π−^1 (g 1 ))t(π−^1 (g 2 ))t= (π−^1 (g 2 )π−^1 (g 1 ))t= ((π(g 1 )π(g 2 ))−^1 )tOne way to characterize this representation is as the action onV∗such that
pairings between elements ofV∗andV are invariant, since
l(v)→((π−^1 (g))tl)(π(g)v) =l(π(g)−^1 π(g)v) =l(v)For a given representation operatorπ(g), acting (with respect to a chosen
basis) onVby
v 1
v 2vn
→P
v 1
v 2vn
for a matrixP, we will have actions on the dual spaceV∗by
(
l 1 l 2 ··· ln)
→
(
l 1 l 2 ··· ln)
(P−^1 )T
or, in terms of column vectors by
l 1
l 2ln
→P−^1
l 1
l 2ln
4.3 Change of basis
Any invertible transformationAonVcan be used to change the basisejofV
to a new basise′jby taking
ej→e′j=Aej
The matrix for a linear transformationLtransforms under this change of basis
as
Ljk=e∗j(Lek)→(e′j)∗(Le′k) =(Aej)∗(LAek)
=(AT)−^1 (e∗j)(LAek)
=e∗j(A−^1 LAek)
=(A−^1 LA)jk