Linear representation theory of unitriangular matrix group of degree three over a finite field

This article describes the linear representation theory of the unitriangular matrix group of degree three over a finite field of size $$q$$, where $$q$$ is a prime power $$p^r$$ with underlying prime $$p$$. $$p$$ is the characteristic of the field.

Related information

 * Linear representation theory of unitriangular matrix group:UT(3,p)
 * Linear representation theory of unitriangular matrix group of degree three over a finite discrete valuation ring
 * Linear representation theory of unitriangular matrix group over a finite field

Kirillov orbit method for character computation
The analysis is pretty similar both for the case $$p = 2$$ and for odd $$p$$, but there are a few minor differences, so the cases are presented separately.

Case of odd prime: orbit analysis
In this case, the Lazard Lie ring is the niltriangular matrix Lie ring:NT(3,q) of strictly upper triangular matrices over $$\mathbb{F}_q$$. Explicitly, the bijection is given by matrix exponentiation:

$$X \mapsto I + X + \frac{X^2}{2}$$

where the addition and multiplication are carried out as matrices. Note that higher powers of $$X$$ do not appear because $$X^3$$ becomes zero.

The additive group of $$NT(3,q)$$ is a $$\mathbb{F}_q$$-vector space of dimension three, hence also a $$\mathbb{F}_p$$-vector space of dimension $$3r$$. Modulo a suitable choice of basis for $$\mathbb{F}_q$$ over $$\mathbb{F}_p$$, the dual space to $$NT(3,q)$$ as a $$\mathbb{F}_q$$-vector space is isomorphic to the dual space to $$NT(3,q)$$ as a $$\mathbb{F}_p$$-vector space (the isomorphism itself is as $$\mathbb{F}_p$$-vector spaces). The latter in turn can be identified with the group of characters of $$NT(3,q)$$ once we identify $$\mathbb{F}_p$$ with the group of $$p^{th}$$ roots of unity. Thus, in order to study the group of characters of $$NT(3,q)$$, we can instead study the dual vector space to $$NT(3,q)$$ over $$\mathbb{F}_q$$.

The dual vector space of $$NT(3,q)$$ is a three-dimensional vector space, comprising the linear functionals on $$NT(3,q)$$. Any nonzero element of this vector space is a linear functional with kernel a subspace of codimension one, which in this case means an additive subgroup of index $$q$$ that is invariant under the $$\mathbb{F}_q$$-multiplication. Further, for each subspace of codimension one, there are $$q - 1$$ possible linear functionals to $$\mathbb{F}_q$$.

The action of $$UT(3,q)$$ by conjugation cannot send any linear functional to a different linear functional with the same kernel, because the group is a $$p$$-group and the set of linear functionals with a given kernel has size $$q - 1$$. Moreover, there is another action, namely post-multiplication by scalars, that commutes with the $$UT(3,q)$$-action. Thus, the nature of orbits of all the different linear functionals with the same kernel is the same.

If the kernel is an ideal, then it is also invariant under the conjugation action of the Lazard Lie group, so the orbit size is 1, and we thus get just a one-dimensional character of the group, coinciding with the corresponding character of the additive group. If the kernel is not an ideal, then it is not invariant under the conjugation action of the Lazard Lie group, so the orbit size is bigger than 1, and the corresponding irreducible representation of the group is not one-dimensional.

Further, note that the kernel of a linear functional is a subspace, but it may or may not be a subalgebra. It is a subalgebra if and only if it is an ideal, which in our case happens if and only if it is a subgroup of index $$q$$ that contains the center of $$NT(3,q)$$.

With this in mind, we can classify the orbits.

Case of odd prime: character computation
We can use the explicit construction of the induced representation for the Kirillov orbit method, as well as the corresponding character formula, to compute the characters of the irreducible representations of $$UT(3,q)$$.

Case of the prime two
The procedure outlined above works for the prime two, with the following caveat: the unitriangular matrix group is not a Lazard Lie group, so the niltriangular matrix Lie ring $$NT(3,2)$$ is not its Lazard Lie ring. However, we can still consider the action and use the variant of the Kirillov orbit method for inner-Lazard Lie groups.