Unitriangular matrix group

Definition

Definition as matrix group

Suppose $R$ is a commutative unital ring and $n$ is a natural number. The unitriangular matrix group, denoted $U(n,R)$, $UT(n,R)$, or $UL(n,R)$, is the group, under multiplication, with $1$s on the diagonal, $0$s below the diagonal, and arbitrary entries above the diagonal.

Note that the symbol $U$ is also used for the unitary group, hence we use $UT$ or $UL$to avoid confusion.

Explicitly:

$UT(n,R)=\left\{{\begin{pmatrix}1&*&*&\dots &*\\0&1&*&\dots &*\\0&0&1&\dots &*\\\dots &\dots &\dots &\dots &\dots \\0&0&0&\dots &1\\\end{pmatrix}}\mid {\mbox{all star-marked entries vary arbitrarily over }}R\right\}$

Definition by presentation

The presentation given here is similar to the presentation used for the Steinberg group over a unital ring. Specifically, we use only those generators and relations that correspond to upper triangular matrices and discard the rest. We denote by $e_{ij}(\lambda )$ the matrix with 1s on the diagonal, $\lambda$ in the $(i,j)^{th}$ entry, and zeros elsewhere. Then:

• The following is a generating set: For every element $\lambda \in R$ and for $1\leq i<j\leq n$, include the element $e_{ij}(\lambda )$.
• We can get a presentation of the group using this generating set, by including the following relations:
  • $e_{ij}(\lambda )e_{ij}(\mu )=e_{ij}(\lambda +\mu )$ (note that this in particular implies that $e_{ij}(0)$ is the identity element for all $i,j$).
  • $[e_{ij}(\lambda ),e_{jk}(\mu )]=e_{ik}(\lambda \mu )$ (the Steinberg group version also imposes the condition $i\neq k$, but this constraint follows automatically in the upper triangular case because $i<j$ and $j<k$).
  • $[e_{ij}(\lambda ),e_{kl}(\mu )]=1$ (i.e., is the identity element) for $i\neq l,j\neq k$.

Note that this presentation can be trimmed quite a bit. In fact, if $A$ is a generating set for the additive group of $R$, the set:

$\{e_{i,i+1}(\lambda )\mid 1\leq i\leq n-1,\lambda \in A\}$

is a generating set for $UT(n,R)$, and we can work out a presentation in terms of this generating set using the relations above.

Case of a field

When $F$ is a field, the unitriangular matrix group $UT(n,F)$ can also be described as a maximal unipotent subgroup of the general linear group $GL(n,F)$. It is also a maximal unipotent subgroup of the special linear group $SL(n,F)$.

Case of a finite field

When $F$ is a finite field with $q$ elements and characteristic $p$ (so $q$ is a power of $p$), then $UT(n,F)$ is also denoted $UT(n,q)$, and is a $p$-Sylow subgroup of $GL(n,F)=GL(n.q)$.

Particular cases

Value of $n$	$n(n-1)/2$ (the "dimension" of $UT(n,R)$, the size is $|R|^{n(n-1)/2}$)	Nature of unitriangular matrix group $UT(n,R)$ over a ring $R$
1	0	trivial group
2	1	isomorphic to the additive group of $R$
3	3	see unitriangular matrix group of degree three
4	6	see unitriangular matrix group of degree four

Arithmetic functions

We give here the arithmetic functions for $UT(n,R)$.

Function	Value	Similar groups	Explanation
order	$|R|^{n(n-1)/2}$
nilpotency class	$n-1$
derived length	$\lceil \log _{2}n\rceil$ (the smallest integer greater than or equal to $\log _{2}n$
Frattini length	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. (depends both on the nature of $R$ and on $n$)