Unitriangular matrix group:UT(4,p)

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Definition

The cases $p=2$ (see unitriangular matrix group:UT(4,2)) and $p=3$ (see unitriangular matrix group:UT(4,3)) differ somewhat from the cases of other primes. This is noted at all places in the page where it becomes significant.

As a group of matrices

Given a prime $p$, the group $UT(4,p)$ is defined as the unitriangular matrix group of degree four over the prime field ${\mathbb{F}}_p$. Explicitly, this is described as the following group under matrix multiplication:

$\{\left(\begin{array}{cccc}1& a_{12}& a_{13}& a_{14}\\ 0& 1& a_{23}& a_{24}\\ 0& 0& 1& a_{34}\\ 0& 0& 0& 1\\ \end{array}\right)\mid a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}\in {\mathbb{F}}_p\}$

The multiplication of matrices $A=(a_{ij})$ and $B=(b_{ij})$ gives the matrix $C=(c_{ij})$ where:

• $c_{12}=a_{12}+b_{12}$
• $c_{13}=a_{13}+b_{13}+a_{12}{b}_{23}$
• $c_{14}=a_{14}+b_{14}+a_{12}{b}_{24}+a_{13}{b}_{34}$
• $c_{23}=a_{23}+b_{23}$
• $c_{24}=a_{24}+b_{24}+a_{23}{b}_{34}$
• $c_{34}=a_{34}+b_{34}$

The identity element is the identity matrix, i.e., the matrix where all off-diagonal entries are zero and all diagonal entries are 1.

The inverse of a matrix $A=(a_{ij})$ is the matrix $M=(m_{ij})$ where:

• $m_{12}=-a_{12}$
• $m_{13}=-a_{13}+a_{12}{a}_{23}$
• $m_{14}=-a_{14}+a_{12}{a}_{24}+a_{13}{a}_{34}-a_{12}{a}_{23}{a}_{34}$
• $m_{23}=-a_{23}$
• $m_{24}=-a_{24}+a_{23}{a}_{34}$
• $m_{34}=-a_{34}$

In coordinate form

We may define the group as the set of ordered 6-tuples $(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})$ over ${\mathbb{F}}_p$ (the coordinates are allowed to repeat), with the multiplication law, identity element, and inverse operation given by:

$(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})(b_{12},b_{13},b_{14},b_{23},b_{24},b_{34})=$

$(a_{12}+b_{12},a_{13}+b_{13}+a_{12}{b}_{23},a_{14}+b_{14}+a_{12}{b}_{24}+a_{13}{b}_{34},a_{23}+b_{23},a_{24}+b_{24}+a_{23}{b}_{34},a_{34}+b_{34})$

$\text{Identity element}=(0,0,0,0,0,0)$

$(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}{)}^{-1}=(-a_{12},-a_{13}+a_{12}{a}_{23},-a_{14}+a_{12}{a}_{24}+a_{13}{a}_{34}-a_{12}{a}_{23}{a}_{34},-a_{23},-a_{34},-a_{24}+a_{23}{a}_{34})$

The matrix corresponding to the 6-tuple $(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})$ is:

$\left(\begin{array}{cccc}1& a_{12}& a_{13}& a_{14}\\ 0& 1& a_{23}& a_{24}\\ 0& 0& 1& a_{34}\\ 0& 0& 0& 1\\ \end{array}\right)$

This definition clearly matches the earlier definition, based on the rules of matrix multiplication.

Families

These groups fall in the more general family $UT(n,p)$ of unitriangular matrix groups. The unitriangular matrix group $UT(n,p)$ can be described as the group of unipotent upper-triangular matrices in $GL(n,p)$, which is also a $p$-Sylow subgroup of the general linear group $GL(n,p)$. This further can be generalized to $UT(n,q)$ where $q$ is the power of a prime $p$. $UT(n,q)$ is the $p$-Sylow subgroup of $GL(n,q)$.

Element structure

Further information: element structure of unitriangular matrix group:UT(4,p)

Summary

Item	Value
number of conjugacy classes	$2p^3+p^2-2p$ equals number of irreducible representations. See number of irreducible representations equals number of conjugacy classes, linear representation theory of unitriangular matrix group of degree four over a finite field
order	$p^6$ Follows from the general formula, order of $UT(n,q),n=4,q=p$ is $q^{n(n-1)/2}=p^6$
conjugacy class size statistics	1 ($p$ times), $p$ ($p^2-1$ times), $p^2$ ($p^3+p^2-2p$ times), $p^3$ ($p^3-p^2-p+1$ times)
order statistics	Case $p=2$: order 1 (1 element), order 2 (27 elements), order 4 (36 elements) Case $p=3$: order 1 (1 element), order 3 (512 elements), order 9 (216 elements) Case $p\ge 5$: order 1 (1 element), order $p$ ($p^6-1$ elements)
exponent	$p^2$ if $p<5$ $p$ if $p\ge 5$

Conjugacy class structure

For the right-most column for the type of matrix, we use the (row number, column number) notation for matrix entries. Explicitly, the matrix under consideration is:

$\left(\begin{array}{cccc}1& a_{12}& a_{13}& a_{14}\\ 0& 1& a_{23}& a_{24}\\ 0& 0& 1& a_{34}\\ 0& 0& 0& 1\\ \end{array}\right)$

The subgroups mentioned in the table below are:

Subgroup	Visual description	Condition	Order
center	$\left(\begin{array}{cccc}1& 0& 0& a_{14}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right)$	$a_{12}=a_{13}=a_{23}=a_{24}=a_{34}=0$	$p$
derived subgroup	$\left(\begin{array}{cccc}1& 0& a_{13}& a_{14}\\ 0& 1& 0& a_{24}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right)$	$a_{12}=a_{23}=a_{34}=0$	$p^3$
unique abelian subgroup of maximum order	$\left(\begin{array}{cccc}1& 0& a_{13}& a_{14}\\ 0& 1& a_{23}& a_{24}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\\ \end{array}\right)$	$a_{12}=a_{34}=0$	$p^4$

Nature of conjugacy class	Jordan block size decomposition	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Order of elements in each such conjugacy class	Type of matrix (constraints on $a_{ij},i<j$)
identity element	1 + 1 + 1 + 1	$x-1$	1	1	1	1	all the $a_{ij},i<j$ are zero
non-identity element, but central (has Jordan blocks of size 1,1,2 respectively)	2 + 1 + 1	$(x-1{)}^2$	1	$p-1$	$p-1$	$p$	$a_{14}\ne 0$, all the others are zero
non-central but in derived subgroup, has Jordan blocks of size 1,1,2	2 + 1 + 1	$(x-1{)}^2$	$p$	$2(p-1)$	$2p(p-1)$	$p$	$a_{12}=a_{23}=a_{34}=0$ Among $a_{13}$ and $a_{24}$, exactly one of them is nonzero. $a_{14}$ may be zero or nonzero
non-central but in derived subgroup, Jordan blocks of size 2,2	2 + 2	$(x-1{)}^2$	$p$	$(p-1{)}^2$	$p(p-1{)}^2$	$p$	$a_{12}=a_{23}=a_{34}=0$ Both $a_{13}$ and $a_{24}$ are nonzero. $a_{14}$ may be zero or nonzero
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2	2 + 1 + 1	$(x-1{)}^2$	$p^2$	$p-1$	$p^2(p-1)$	$p$	$a_{12}=a_{34}=a_{14}=0$ $a_{23}$ is nonzero $a_{13}$ and $a_{24}$ are arbitrary
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2	2 + 2	$(x-1{)}^2$	$p^2$	$(p-1{)}^2$	$p^2(p-1{)}^2$	$p$	$a_{12}=a_{34}=0$ $a_{23}$ and $a_{14}$ are both nonzero $a_{13}$ and $a_{24}$ are arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2	2 + 1 + 1	$(x-1{)}^2$	$p^2$	$2(p-1)$	$2p^2(p-1)$	$p$	Two subcases: Case 1: $a_{12}=a_{23}=a_{13}=0$, $a_{34}$ nonzero, $a_{14},a_{24}$ arbitrary Case 2: $a_{23}=a_{24}=a_{34}=0$, $a_{12}$ nonzero, $a_{13},a_{14}$ arbitrary
outside abelian subgroup of maximum order, Jordan blocks of size 2,2	2 + 2	$(x-1{)}^2$	$p^2$	$(p-1{)}^2$	$p^2(p-1{)}^2$	$p$	$a_{12},a_{34}$ both nonzero $a_{23}=0$ $a_{14},a_{24}$ arbitrary $a_{13}$ uniquely determined by other values
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order $q^4$	3 + 1	$(x-1{)}^3$	$p^2$	$(p-1{)}^2(p+1)$	$p^2(p-1{)}^2(qp+1)$	$p$ if $p$ odd 4 if $p=2$	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order $q^3$	3 + 1	$(x-1{)}^3$	$p^3$	$2(p-1{)}^2$	$2p^3(p-1{)}^2$	$p$ if $p$ odd 4 if $p=2$	Two subcases: Case 1: $a_{12},a_{23}$ nonzero, $a_{34}=0$, other entries arbitrary Case 2: $a_{23},a_{34}$ nonzero, $a_{12}=0$, other entries arbitrary
Jordan block of size 4	4	$(x-1{)}^4$	$p^3$	$(p-1{)}^3$	$p^3(p-1{)}^3$	$p^2$ if $p<5$ $p$ if $p\ge 5$	$a_{12},a_{23},a_{34}$ nonzero $a_{13},a_{14},a_{24}$ arbitrary
Total (--)	--	--	--	$2p^3+p^2-2p$	$p^6$	--	--

Linear representation theory

Further information: linear representation theory of unitriangular matrix group:UT(4,p)

Summary

Item	Value
number of conjugacy classes (equals number of irreducible representations over a splitting field)	$2p^3+p^2-2p$. See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree four over a finite field
degrees of irreducible representations	1 (occurs $p^3$ times), $p$ (occurs $p^3-p$ times), $p^2$ (occurs $p^2-p$ times)
sum of squares of degrees of irreducible representations	$p^6$ (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group
lcm of degrees of irreducible representations	$p^2$

GAP implementation

We assume $p$ is assigned a prime number value beforehand.

Description	Functions used
SylowSubgroup(GL(4,p),p)	SylowSubgroup, GL
SylowSubgroup(SL(4,p),p)	SylowSubgroup, SL
SylowSubgroup(PGL(4,p),p)	SylowSubgroup, PGL
SylowSubgroup(PSL(4,p),p)	SylowSubgroup, PSL