Conjugacy class sizes of unitriangular matrix group of fixed degree over a finite field are all powers of field size and number of occurrences of each is polynomial function of field size

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Statement

Denote by UT(n,q) the unitriangular matrix group of degree n over a finite field of size q, where q is a prime power.

Then, the following is true about the conjugacy class size statistics of UT(n,q):

  • All the conjugacy class sizes are powers of q.
  • For any nonnegative integer d, there exists a univariate polynomial g_{n,d} with integer coefficients (dependent on n and d but independent of q) such that the number of conjugacy classes in UT(n,q) of size q^d equals g_{n,d}(q). Further, we have, based on the fact that the sum of all conjugacy class sizes must equal the order of the group, that the following is true as a polynomial identity in q:

\sum_{d=0}^\infty q^{d}g_{n,d}(q) = q^{n(n-1)/2}

Note that the summation is actually finite, because all but finitely many of the g_{n,d} are zero polynomials.

Related facts

More information on element structure

Full information on the linear representation theory of these groups, along with explicit polynomial formulas for the degrees, is available at element structure of unitriangular matrix group over a finite field.

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