Unitriangular matrix group of degree four

Definition

Suppose $R$ is a unital ring. The unitriangular matrix group of degree three over $R$, denoted $U(3,R)$ or $UT(4,R)$, is defined as the unitriangular matrix group of $4\times 4$ matrices over $R$. Explicitly, it can be described as the group of upper triangular matrices with 1s on the diagonal, and entries over $R$ (with the group operation being matrix multiplication).

Each such matrix $(a_{ij})$ can be described by the six entries $a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}$, each of which varies freely over $R$. The matrix looks like:

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

Elements

Further information: element structure of unitriangular matrix group of degree four over a finite field

Linear representation theory

Further information: linear representation theory of unitriangular matrix group of degree four over a finite field