We will describe how the character table of is related to the expansion of the power-sum symmetric polynomials in terms of monomials. Recall:
where exactly since is not defined.
Now each irrep of is of the form for some ; we will denote its character by Each conjugancy class of is also given by where and has cycle structure E.g. if we can take as a representative.
We will calculate the character value by looking at the characters of For that, we need:
Lemma. If the finite group acts on finite set , then the trace of is: , the number of fixed points of in
Taking elements of as a natural basis of , the matrix representing is a permutation matrix. Its trace is thus the number of ones along the main diagonal, which is the number of fixed points of ♦
Hence we have:
Theorem. Let be the trace of on Expressing each as a sum of monomials, we get:
Let us compute the coefficient of in the expansion of For illustration let us take and ; pick the representative To obtain terms with product here is one possibility:
The corresponding fixed point is given by and
In the general case, each contribution of corresponds to a selection of: such that
This corresponds to a partition defined by running through all the cycles of and putting the elements of the c-th cycle into the set for Such a partition is invariant under ♦
Thus is the number of ways of “merging” terms to form the partition upon sorting. For example, taking and from above, the coefficient of in is 7:
Writing the above vectorially, we have where is the trace of on Thus, we have
which gives us where so X is the transpose of the character table for Hence, we have:
For d=4 we obtain:
where the rows and columns are indexed by partitions 4, 31, 22, 211 and 1111. One checks that is the character table for :
Finally, orthonormality of irreducible characters translates to orthogonality of power-sum polynomials.
Proposition. The polynomials form an orthogonal set, and is the order of the centralizer of any with cycle structure
Since and the are orthonormal,
which is entry of . This is the dot product between columns and of the character table. By standard character theory, it is where is the conjugancy class containing Now apply ♦
Under the Frobenius map, the power-sum symmetric polynomial corresponds to: