Unitriangular matrix group of degree four

As a group of matrices
Suppose $$R$$ is a unital ring. The unitriangular matrix group of degree three over $$R$$, denoted $$UT(4,R)$$ or $$UL(4,R)$$, 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:

$$\begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24} \\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \\\end{pmatrix}$$

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


 * $$c_{12} = a_{12} + b_{12}$$
 * $$c_{13} = a_{13} + b_{13} + a_{12}b_{23}$$
 * $$c_{14} = a_{14} + b_{14} + a_{12}b_{24} + a_{13}b_{34}$$
 * $$c_{23} = a_{23} + b_{23}$$
 * $$c_{24} = a_{24} + b_{24} + a_{23}b_{34}$$
 * $$c_{34} = a_{34} + b_{34}$$

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 the ring $$R$$ (the coordinates are allowed to repeat), with the multiplication law, identity element, and inverse operation given by:

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

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

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

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

$$\begin{pmatrix} 1 & a_{12} & a_{13} & a_{14}\\ 0 & 1 & a_{23} & a_{24} \\ 0 & 0 & 1 & a_{34}\\ 0 & 0 & 0 & 1 \\\end{pmatrix}$$

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