Unitriangular matrix group of degree four

Definition

As a group of matrices

Suppose $R$ is a unital ring. The unitriangular matrix group of degree three over $R$ , denoted $U T (4, R)$ or $U L (4, R)$ , 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 $(a_{i j})$ 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:

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

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

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

$Identity element = (0, 0, 0, 0, 0, 0)$

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

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

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

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

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