- TakeMto be a set of 3 elementsx 1 ,x 2 ,x 3. SoF(M) =C^3. Forf∈
F(M),fis a vector inC^3 , with components (f(x 1 ),f(x 2 ),f(x 3 )). - TakeG=S 3 , the group of permutations of 3 elements. This group has
3! = 6 elements. - TakeGto act onMby permuting the 3 elements
(g,xj)→g·xj- This group action provides a representation ofGonF(M) by the linear
maps
(π(g)f)(xj) =f(g−^1 ·xj)
Taking the standard basis ofF(M) =C^3 , thej’th basis element will correspond
to the functionfthat takes value 1 onxj, and 0 on the other two elements.
With respect to this basis theπ(g) give six 3 by 3 complex matrices, which
under multiplication of matrices satisfy the same relations as the elements of
the group under group multiplication. In this particular case, all the entries of
the matrix will be 0 or 1, but that is special to the permutation representation.
A common source of confusion is that representations (π,V) are sometimes
referred to by the mapπ, leaving implicit the vector spaceVthat the matrices
π(g) act on, but at other times referred to by specifying the vector spaceV,
leaving implicit the mapπ. One reason for this is that the mapπmay be the
identity map: oftenGis a matrix group, so a subgroup ofGL(n,C), acting
onV 'Cnby the standard action of matrices on vectors. One should keep
in mind though that just specifyingV is generally not enough to specify the
representation, since it may not be the standard one. For example, it could very
well be the trivial representation onV, where
π(g) = (^1) n
i.e., each element ofGacts onVas the identity.
1.3.3 Unitary group representations
The most interesting classes of complex representations are often those for which
the linear transformationsπ(g) are “unitary”, preserving the notion of length
given by the standard Hermitian inner product, and thus taking unit vectors to
unit vectors. We have the definition:
Definition(Unitary representation).A representation(π,V)on a complex vec-
tor spaceV with Hermitian inner product〈·,·〉is a unitary representation if it
preserves the inner product, i.e.,
〈π(g)v 1 ,π(g)v 2 〉=〈v 1 ,v 2 〉for allg∈Gandv 1 ,v 2 ∈V.