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 $UT(4,R)$ 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.

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