Let $G$ be the nilpotent Lie group consisting of matrices
$$
\begin{pmatrix}
1 &amp; a_{12} &amp; \cdots &amp; a_{1,n}\\
0 &amp; 1 &amp; \ddots &amp; \vdots\\
\vdots &amp; \ddots &amp; \ddots &amp; a_{n-1,n}\\
0 &amp; \cdots &amp; 0 &amp; 1
\end{pmatrix}
$$
where $a_{ij}\in\mathbb R$.

I would like to find a presentation of the group $\Gamma=G\cap\mathrm{GL}_n\mathbb Z$.

The entries on the superdiagonal play a crucial role, in that the $n-1$ matrices with a single $1$ on the superdiagonal ($1$s on the diagonal and $0$s elsewhere) generate all of $\Gamma$.

Trying to write down a presentation, I would start by choosing $n-1$ generators, $a_{12},\dotsc,a_{n-1,n}$ (named after the matrix entries). It is also clear that we need commutation relations, like
$$
[\cdots[[[a_{12},a_{23}],[a_{23},a_{34}]],\ldots]\cdots]=\cdots=e.
$$
(If $n=3$, this would just be $[[a_{12},a_{23}],a_{12}]=[[a_{12},a_{23}],a_{23}]=e$.)
I am just not sure, though, if I explicitly need to require that, say,
$$
[a_{12},a_{34}]=e,
$$
which seems to be directly related to the embedding I am thinking of. I would like a presentation in which any generators can be mapped to any of the matrices with a $1$ in the $(i,i+1)$ position. Is this possible?


Question: Is there a way to query Magento Database for part OR all of an unserialized string?
Answer: Unfortunately, no, you can't do this any better/efficiently than your current solution.  

The algorithm used to serialize a variable in PHP is proprietary to PHP.  MySQL (or the other RDMS) know nothing about it, so there's no query tools other than a raw string LIKE that work with the fields. 

In theory it would be possible (via object listeners) to save the serialized data into join tables, and then alter your grid collections to pull the data you want from that join table — but this approach is complicated, fraught with problems and would require testing/iterations to work out potential bugs.