Degrees of irreducible representations 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 degrees of irreducible representations of UT(n,q) over a splitting field (such as \overline{\mathbb{Q}} or \mathbb{C}):

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

Note that the summation is actually finite, because all but finitely many of the f_{n,d} are zero polynomials. It seems to be the case that f_{n,d} is nonzero for 0 \le d \le m, where:

m = \left \lfloor \frac{(n - 1)^2}{4} \right \rfloor

Related facts

More information on linear representation theory

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

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