I assume that when you are using $a_{ij}$ to denote an element of $\Gamma$, you mean the identity matrix with an extra 1 in the $(i,j)$ position.

It is easy to write down a presentation of $\Gamma$ on the $m := n(n-1)/2$ generators $a_{ij}$ with $i&lt;j$. Just use the $m(m-1)/2$ commutator relations
$[a_{ij},a_{jk}] = a_{ik}$, and $[a_{ij},a_{kl}]=1$ when $j \ne k$, $i \le k$,
and $j&lt;l$ when $i=k$. Of course many of these relations are redundant.

To get a presentation on the generators $a_{i,i+1}$, you can then use the equations $[a_{ij},a_{jk}] = a_{ik}$ to eliminate all of the other generators. Doing that with $n=3$ gives the presentation you have written down.

Although many relations in this presentation are redundant, those like $[a_{12},a_{34}]=1$ between the commuting generators $a_{i,i+1}$ are definitely not redundant, because if you left one of those out, then you could make the group bigger by putting that commutator equal to a new central generator.

Presentation of discrete upper triangular group
I made the following graphic with Mathematica 9.0:



Does anybody know how to put a title on the top of the whole graph? (not in each single plot).

Basically I used 

GraphicsRow[ListLinePlot[]ListLinePlot[]]


Thanks.
