Finding an irreducible region of a space given a group of transformations

Question

Given some $d$ dimensional torus, (i.e. just a $d$-dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$ of $\Omega$, I want to find the smallest sub-region in the hyper-torus such that the rest of the torus can be generated by the action of the group elements of $G$.

That is, I want to find the smallest $\tilde{\Omega}$ such that

\begin{equation} \Omega = \bigcup_{g \in G} g (\tilde{\Omega}) \end{equation}

Condensed matter physicists approach this problem quite often in finding an irreducible Brillouin zone, but I cannot find any general procedure for solving this problem that doesn't involve just guessing the solution.

Does anyone have suggestions for how to go about doing this, or references they can point me to?

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There is a related and unanswered question here: https://math.stackexchange.com/questions/1284436/algebraic-determination-of-asymmetric-unit-aka-irreducible-wedge-in-brillouin

Answer 1

I assume that $G$ acts isometrically on the flat torus $T^n$. Then the standard construction of $\tilde\Omega$ proceeds as follows. Pick a point $x\in T^n$ not fixed by any $g\in G$ and consider its $G$-orbit $Gx= \{gx: g\in G\}$. Take the Voronoi tiling of $T^n$ corresponding to this subset, let $D_x$ be the tile "centered" at $x$: $$ D_x=\{y\in T^n: \forall G \setminus \{1\}, d(x,y)\le d(gx,y)\}. $$ (Here $d$ is the distance function on the torus corresponding to the flat Riemannian metric.) This will be your $\tilde\Omega$. Indeed, since $gD_x= D_{gx}$, $G$ will permute simply-transitively the tiles $D_{gx}, g\in G$.

It is a nice exercise to prove that if $E\subset D_g$ is a subset whose closure is not the entire $D_x$, then $GE\ne T^n$, where $$ GE= \bigcup_{g\in G} gE. $$