The number of times each irreducible representation in permutation matrix representation

Question

I am studying a code. One of key part is

Return a list representing the number of times each IRR is present in the permutation matrix representation.

Suppose we have:

1. character table
2. number of inequivalent irreducible representations (just find the length of character table)
3. group order
4. conjugacy classes matrices

The code is the following:

https://sourceforge.net/p/networksym/code/ci/master/tree/networkIRR.py
(def get_IRR_degeneracies(self), particularly)

My question is the following key formula:

total=total +float(len(matricies[j])/group_order)* np.conj(np.complex(characters[i][j]))* np.trace(matricies[j][0])

What is this formula? Where can I find this formula?

Answer 1

I think that the problem is not that hard once you understand some key ideas. One key idea is that the character table has orthogonal rows and columns. Each row is the character of an irreducible representation and since they are orthogonal they form a basis. As usual, given any character, to find its coefficients relative to the orthonomal basis of irreducibles, you calculate the dot product. This means that if you have any representation, say using permutation matrices, you find the trace of the matrix representing each conjugacy class, and then dot product this character with any irreducible character. The result is the number of times that irreducible character appears in the representation.