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

This article gives specific information, namely, linear representation theory, about a family of groups, namely: unitriangular matrix group of degree three.
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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

Summary

Item	Value
number of conjugacy classes (equals number of irreducible representations over a splitting field)	$q^{2}+q-1$ . See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree three over a finite field
degrees of irreducible representations	1 (occurs $q^{2}$ times), $q$ (occurs $q-1$ times)
sum of squares of degrees of irreducible representations	$q^{3}$ (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group
lcm of degrees of irreducible representations	$q$
condition for a field (characteristic not equal to $p$ ) to be a splitting field	The polynomial $x^{p}-1$ should split completely. For a finite field of size $s$ , this is equivalent to $s\equiv 1{\pmod {p}}$ .
field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero)	Field $\mathbb {Q} (\zeta )$ where $\zeta$ is a primitive $p^{th}$ root of unity. This is a degree $p-1$ extension of the rationals.
unique minimal splitting field (characteristic $c\neq 0,p$ )	The field of size $c^{r}$ where $r$ is the order of $c$ mod $p$ .
degrees of irreducible representations over the rational numbers	1 (1 time), $p-1$ ( $(q^{2}-1)/(p-1)$ times), $p(p-1)$ ( $(q-1)/(p-1)$ times)
Orbits over a splitting field under the action of the automorphism group	Case $p=2$ : Orbit sizes: 1 (degree 1 representation), $2(q-1)$ (degree 1 representations), $(q-1)^{2}$ (degree 1 representations), $q-1$ (degree $q$ representations) Case odd $p$ : Orbit sizes: 1 (degree 1 representation), $q^{2}-1$ (degree 1 representations), $q-1$ (degree $q$ representations) number: 4 (case $p=2$ ), 3 (case odd $p$ )
Orbits over a splitting field under the multiplicative action of one-dimensional representations	Orbit sizes: $q^{2}$ (degree 1 representations), and $q-1$ orbits of size 1 (degree $q$ representations)

Irreducible representations

Description of collection of representations	Parameter for describing each representation	How the representation is described	Degree of each representation	Number of representations	Sum of squares of degrees
trivial representation	--	trivial representation, sends everything to $(1)$	1	1	1
representation with kernel a maximal subgroup	maximal subgroup and a nontrivial homomorphism from the quotient, identified with the additive group of $\mathbb {F} _{p}$ , to $\mathbb {C} ^{\ast }$ . Equivalently, a linear functional on a $2r$ -dimensional vector space over $\mathbb {F} _{p}$	one-dimensional representation with kernel the maximal subgroup and homomorphism as described on the quotient.	1	$q^{2}-1$	$q^{2}-1$
representation induced from a nontrivial one-dimensional representation on the subgroup $\{{\begin{pmatrix}1&&\\0&1&0\\0&0&1\\\end{pmatrix}}$ that is nontrivial on a subgroup of order $p$ in the center.	The original representation restricted to the center, which is effectively specifying a nontrivial character of the additive group of $\mathbb {F} _{q}$	Induce the representation.	$q$	$q-1$	$q^{3}-q^{2}$
Total (3 rows)	--	--	--	$q^{2}+q-1$	$q^{3}$

Kirillov orbit method for character computation

Further information: Kirillov orbit method for finite Lazard Lie group

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.

Type of linear functional (and hence, type of character of additive group)	Size of orbit	Degree of corresponding irreducible representation (equals square root of size of orbit)	Number of orbits of kernels	Number of orbits of functionals (equals $q-1$ times number of orbits of kernels, except in the zero functional case)	Total number of kernels	Total number of such linear functionals (or characters of the additive group) (equals $q-1$ times number of orbits of kernels, except in the zero functional case)
zero functional	1	1	1	1	1	1
nonzero functional whose kernel is a subring (and hence an ideal) in $NT(3,q)$ . In our case, this is equivalent to the requirement that the kernel contain the center of $NT(3,q)$ , which is a subring of order $q$ .	1	1	$q+1$	$q^{2}-1$	$q+1$	$q^{2}-1$
nonzero functional whose kernel is not a subring	$q^{2}$	$p$	1	$q-1$	$q^{2}$	$q^{3}-q^{2}$
Total	--	--	--	$q^{2}+q-1$	--	$q^{3}$

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 $UT(3,q)$ .

Zero functional case

Item	Value
Size of orbit	1
Description of polarizing subring	whole ring $NT(3,q)$ is the only polarizing subring. It has index 1.
Induced representation	Trivial representation of the whole group.

Nonzero functional whose kernel is an ideal

Item	Value
Size of orbit	1
Description of polarizing subring	whole ring $NT(3,q)$ is the only polarizing subring. It has index 1.
Induced representation	Essentially, the same representation of $UT(3,q)$ as the sole one-dimensional character in the coadjoint orbit.
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit	1
Number of such coadjoint orbits	$q^{2}-1$

Nonzero functional whose kernel is not an ideal/subring

Item	Value
Size of orbit	$p^{2}$
Description of polarizing subring	Any $\mathbb {F} _{q}$ -subalgebra of order $q^{2}$ (which must therefore also be an ideal) is a polarizing subring for any representative of the orbit. Each polarizing subring has index $q$ .
Induced representation	Induce from any $\mathbb {F} _{q}$ -subalgebra of order $q^{2}$ any one-dimensional character that is nontrivial on the center (an order $q$ subring of it).
Degree of irreducible representation obtained = index of polarizing subring = square root of size of orbit	$q$
Number of such coadjoint orbits	$q-1$

Case of the prime two

Further information: Kirillov orbit method for finite inner-Lazard Lie group

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.