How to find the invariant forms of a finite group

Question

Let $G\subset GL(n,\mathbb{Z})$. I am looking for an algorithm that finds all symmetric matrices $F$ left invariant by G, ie $$g^TFg=F\quad \forall g\in G.$$ I have found lists of these invariants for 2 and 3 dimensions in a crystallographic context, but I would like a method that can be ideally extended up to 22 dimensions.

Answer 1

The obvious way to do this is to let the entries in $F$ be a set of $n^2$ unknowns to be solved for. It is necessary and sufficient for the equations to be satisfied for all $g$ in a generating set $S$ of $G$. So you get a system of $|S|n^2$ linear equations in $n^2$ unknown to solve.

That should work up to dimension $22$. It depends a bit on the field. If you are working over the rationals, it would probably be better to solve the equations modulo some primes $p$, and then lift the solution using the Chinese Remainder Theorem.

You can also regard the equations as representing an isomorphism between the $FG$-module defined by $G$ and its dual. If this module is absolutely irreducible, then if $F$ exists at all it is unique modulo scalars. There are some more sophisticated methods available in that case, such as MeatAxe algorithm over finite fields.