432 X A Character Study
Decomposing the characterχinto its irreducible components by means of (7),
we obtainχ= 2 χ 1 + 2 χ 3. Since the irreducible representations ofS 3 are all real,
this means that the configuration spaceR^6 is the direct sum of four irreducible invari-
ant subspaces, two of dimension 1 and twoof dimension 2. Knowing what to look
for, we may verify that the one-dimensional subspaces spanned byr 1 +r 2 +r 3 and
α 23 +α 31 +α 12 are invariant. Also, the two-dimensional subspace formed by all vec-
torsμ 1 r 1 +μ 2 r 2 +μ 3 r 3 withμ 1 +μ 2 +μ 3 =0 is invariant and irreducible, and
so is the two-dimensional subspace formed by all vectorsv 1 α 23 +v 2 α 31 +v 3 α 12 with
v 1 +v 2 +v 3 =0. Hence we can find a real non-singular matrixTsuch that
T−^1 B−^1 CT=
⎛
⎜
⎜
⎝
λ 1 I 1 000
0 λ 2 I 1 00
00 λ 3 I 2 0
000 λ 4 I 2⎞
⎟
⎟
⎠.
This shows that the ammonia moleculeNH 3 has two nondegenerate normal frequen-
cies and two doubly degenerate normal frequencies.
8 Generalizations
During the past century the character theory of finite groups has been extensively gen-
eralized to infinite groups with a topological structure. It may be helpful to give an
overview here, without proofs, of this vast development. The reader wishing to pursue
some particular topic may consult the references at the end of the chapter.
Atopological groupis a groupGwith a topology such that the map(s,t)→st−^1
ofG×GintoGis continuous. Throughout the following discussion we will assume
thatGis a topological group which, as a topological space, islocally compact and
Hausdorff, i.e. any two distinct points are contained in open sets whose closures are
disjoint compact sets. (A closed setEin a topological space iscompactif each open
cover ofEhas a finite subcover. In a metric space this is consistent with the definition
of sequential compactness in Chapter I,§4.)
LetC 0 (G)denote the set of all continuous functions f:G → Csuch that
f(s)=0forallsoutside some compact subset ofG(which may depend on f).
AmapM:C 0 (G)→Cis said to be anonnegative linear functionalif
(i) M(f 1 +f 2 )=M(f 1 )+M(f 2 )for allf 1 ,f 2 ∈C 0 (G),
(ii) M(λf)=λM(f)for allλ∈Candf∈C 0 (G),
(iii)M(f)≥0iff(s)≥0foreverys∈G.
It is said to be aleft(resp.right)Haar integralif, in addition, it is nontrivial, i.e.
M(f)=0forsomef∈C 0 (G), and left (resp. right) invariant, i.e.
(iv)M(tf)=M(f)for everyt∈Gandf∈C 0 (G),wheretf(s)=f(t−^1 s), (resp.
M(ft)=M(f)for everyt∈Gandf∈C 0 (G),whereft(s)=f(st)).
It was shown by Haar (1933) that a left Haar integral exists on any locally
compact group; it was later shown to be uniquely determined apart from a positive
multiplicative constant. By definingM∗(f)=M(f∗),wheref∗(s)= f(s−^1 )for
everys∈G, it follows that a right Haar integral also exists and is uniquely determined
apart from a positive multiplicative constant.