Conjugacy class size formula in general linear group over a finite field
Case of semisimple elements
Elements diagonalizable over
Suppose is diagonalizable over , with eigenvalues having multiplicities respectively (the s are all distinct). Note that .
Then, the centralizer of the diagonal representative of this conjugacy class is isomorphic to:
In fact, if the diagonal entries are arranged so that all the s occur first, then the s, and so on, then the centralizer is the set of invertible block diagonal matrices with blocks of sizes .
The size of the conjugacy class is thus:
This is the same as the -analogue of the multinomial coefficient:
In all cases, this simplifies to a polynomial in , and its degree is .
Some particular cases for the partition of as a sum of s, and the corresponding sizes, are given below.
|Partition of||Size of||Size of centralizer of diagonal element||Degree of polynomial giving size||Size of conjugacy class||Degree of polynomial giving size|
|2||1 + 1||2||2|
|3||2 + 1||5||4|
|3||1 + 1 + 1||3||6|