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

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 $$\mathbb{C}$$):


 * All the degrees of irreducible representations are powers of $$q$$, hence the group $$UT(n,q)$$ is a q-power degree group.
 * For any nonnegative integer $$d$$, there exists a univariate polynomial $$f_{n,d}$$ with integer coefficients (dependent on $$n$$ and $$d$$ but independent of $$q$$) such that the number of irreducible representations of $$UT(n,q)$$ of degree $$q^d$$ equals $$f_{n,d}(q)$$. Further, we have, based on the fact that sum of squares of degrees of irreducible representations equals order of group, that the following is true as a polynomial identity in $$q$$:

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

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:

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

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.

Similar facts

 * 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
 * Number of conjugacy classes in unitriangular matrix group of fixed degree over a finite field is polynomial function of field size

Opposite facts

 * The analogous statement is not completely true for other maximal unipotent subgroups in the finite groups of Lie type. For instance, a look at the linear representation theory of maximal unipotent subgroup of symplectic group of degree four over a finite field shows that in the characteristic two case, there exist irreducible representations of degree $$q/2$$ which is not a power of $$q$$ for $$q > 2$$. For odd characteristics, however, the statement still continues to hold.