Linear representation theory of unitriangular matrix group:UT(4,p)

This article describes the linear representation theory of unitriangular matrix group:UT(4,p): the unitriangular matrix group of degree four over aprime field of size equal to a prime $$p$$. $$p$$ is the characteristic of the field.

For a slightly more general version, see linear representation theory of unitriangular matrix group of degree four over a finite field.

Kirillov orbit method for character computation
The analysis is pretty similar both for the case $$p \ge 5$$ and the special cases $$p = 2$$ and $$p = 3$$, but the prima facie applicability of the analysis is only for $$p \ge 5$$.

Case of $$p \ge 5$$
In this case, the Lazard Lie ring is the niltriangular matrix Lie ring:NT(4,p) (for more information, see Baer correspondence between NT(4,p) and UT(4,p)). Explicitly, the bijection is given by matrix exponentiation:

$$X \mapsto I + X + \frac{X^2}{2} + \frac{X^3}{6}$$

where the addition and multiplication are carried out as matrices. Note that higher powers of $$X$$ do not appear because $$X^4$$ becomes zero.

The additive group of $$NT(4,p)$$ is a $$\mathbb{F}_p$$-vector space, i.e., it is an elementary abelian group with exponent $$p$$ (the dimension of this vector space is 6). Thus, the characters of this all map to the group of $$p^{th}$$ roots of unity. Modulo a choice of isomorphism between the additive group of $$\mathcal{F}_p$$ and the multiplicative group of $$p^{th}$$ roots of unity, the Pontryagin dual can be identified with the dual vector space to $$NT(4,p)$$ as a $$\mathbb{F}_p$$-vector space. Since the dual vector space is easier to understand, we will carry out our analysis in these terms.

The dual vector space of $$NT(4,p)$$ is a six-dimensional vector space, comprising the linear functionals on $$NT(4,p)$$. 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 $$p$$. Further, for each subspace of codimension one (i.e., each additive subgroup of index $$p$$) there are $$p - 1$$ possible linear functionals to $$\mathbb{F}_p$$ (and, once we have fixed the isomorphism to the $$p^{th}$$ roots of unity, there are $$p - 1$$ characters with that subgroup as kernel).

The action of $$UT(4,p)$$ 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 $$p - 1$$. Moreover, there is another action, namely post-multiplication by scalars, that commutes with the $$UT(4,p)$$-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 an additive subgroup, but it may or may not be a subring of the Lie ring. It is a subring if and only if it is an ideal, which in our case happens if and only if it is a subgroup of index $$p$$ that contains the derived subring of $$NT(4,p)$$ (a three-dimensional subring).

In order to compute the number of possible subspaces containing a given subspace of codimension $$s$$, we use the equivalence of definitions of size of projective space to conclude that this number is $$p^{s-1} + p^{s-2} + \dots + p + 1$$. Further, the number of possible subspaces containing a subspace of codimension $$s$$ but not containing a subspace of codimension $$t$$ that contains the former space is $$p^{s-1} + \dots + p^{t+1}$$. The set of all subspaces is just the set of subspaces containing the zero subspace.

With this in mind, we can classify the orbits.

Case of the prime two
The procedure outlined above works for the prime two, with the following caveat: the unitriangular matrix group $$UT(3,2)$$, which is isomorphic to dihedral group:D8, 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.