Class Functions
Our first observation about characters takes our work from last time and spins it in a new direction.
Let’s say 
and 
are conjugate elements of the group 
. That is, there is some 
so that 
. I say that for any -module 
with character 
, the character takes the same value on both and . Indeed, we find that
We see that is not so much a function on the group as it is a function on the set of conjugacy classes 
, since it takes the same value for any two elements in the same conjugacy class. We call such a complex-valued function on a group a “class function”. Clearly they form a vector space, and this vector space comes with a very nice basis: given a conjugacy class 
we define 
to be the function that takes the value 
for every element of and the value 
otherwise. Any class function is a linear combination of these 
, and so we conclude that the dimension of the space of class functions in is equal to the number of conjugacy classes in .
The space of class functions also has a nice inner product. Of course, we could just declare the basis 
to be orthonormal, but that’s not quite what we’re going to do. Instead, we’ll define
The basis isn’t orthonormal, but it is orthogonal. However, we can compute:
Incidentally, this is the reciprocal of the size of the centralizer 
of any 
. Thus if we pick a 
in each we can write down the orthonormal basis 
.
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October 15, 2010 -
Posted by John Armstrong |
Algebra, Group theory, Representation Theory
The Unapologetic Mathematician 
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