Note on decomposing functions into partner functions
Steven G. Johnson, MIT Course 18.369
February 22, 2016
In the representation-theory handout for 18.369, it says that any function ψ(~x) can be
decomposed into a sum of partner functions of the different irreps of any symmetry group
G. Recall that for a coordinate transformation g (a rotation or translation), I denote the
corresponding transformation of functions ψ by Ôg .1 What follows is a brief proof of that.
1. Consider the set S = {Ôg ψ for all g ∈ G}. Form a basis ψi of S, for i ∈ {1, . . . , d}
where d is the dimension of the subspace spanned by S (the number of linearly independent functions in S).
Pd
2. By construction, Ôg ψj ∈ S for any j ∈ {1, . . . , d}, g ∈ G. Hence Ôg ψj = i=1 ψi Dij (g)
where Dij (g) are some coefficients depending on i, j, and g.
3. The matrices D(g) with entries Dij (g) form a representation of G. Proof:
Ôg1 Ôg2 ψj
= Ôg1 g2 ψj =
d
X
ψi Dij (g1 g2 )
i=1
= Ôg1
d
X
ψk Dkj (g2 ) =
k=1
=
d
X
i=1
"
ψi
" d
d
X
X
k=1
d
X
#
ψi Dik (g1 ) Dkj (g2 )
i=1
#
Dik (g1 )Dkj (g2 ) .
k=1
Comparing the first and last lines, which must be true for any i, j, we find Dij (g1 g2 ) =
Pd
k=1 Dik (g1 )Dkj (g2 ), which is exactly the formula for a matrix multiplication, so
D(g1 g2 ) = D(g1 )D(g2 ). Hence D is a representation.
4. D must be reducible into one or more irreps D(α) of G, i.e. we can perform a change
of basis to D̃ = S −1 DS that block-diagonalize D̃ into irreps. Perform
P the same
change of basis on ψi to obtain the corresponding basis functions ψ̃j = i ψi Sij . By
construction, the ψ̃j are partners of D̃, and hence they are partners of the irreps that
D̃ reduces into.
5. ψ ∈ S since the identityPE ∈ G, so ψ is in the span of the basis functions ψi and
hence of ψ̃i . Hence ψ = i ci ψ̃i for some coefficients ci , which from above is a sum of
partner functions of one or more of various irreps of G. (Note it is easy to show that
the partner functions of an irrep form a vector space: summing two partners of the
same irrep or multipling them by scalars ci yields another partner function.) Q.E.D.
1 Some authors just use g interchangeably for rotations of the coordinate space or rotations of the Hilbert
space, but for vector fields it is confusing if you don’t distinguish the two. In hindsight, maybe I should
have used ĝ instead of Ôg ; oh well.
1