Characters of Permutation Representations
Let’s take to be a permutation representation coming from a group action on a finite set that we’ll also call . It’s straightforward to calculate the character of this representation.
Indeed, the standard basis that comes from the elements of gives us a nice matrix representation:
On the left is the matrix of the action on , while on the right it’s the group action on the set . Hopefully this won’t be too confusing. The matrix entry in row and column is if sends to , and it’s otherwise.
So what’s the character ? It’s the trace of the matrix , which is the sum of all the diagonal elements:
This sum counts up for each point that sends back to itself, and otherwise. That is, it counts the number of fixed points of the permutation .
As a special case, we can consider the defining representation of the symmetric group . The character counts the number of fixed points of any given permutation. For instance, in the case we calculate:
In particular, the character takes the value on the identity element , and the degree of the representation is as well. This is no coincidence; will always be the degree of the representation in question, since any matrix representation of degree must send to the identity matrix, whose trace is . This holds both for permutation representations and for any other representation.