Unitriangular matrix group

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 $$1 \le i <j \le n$$, 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 $$i \ne k$$, 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 $$i \ne l, j \ne k$$.

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:

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

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 defining ingredient::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 $$GL(n,F) = GL(n.q)$$.

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