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

\cup -> \bigcup

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edited Jul 18, 2020 at 21:21

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Suppose I haveGiven some $d$ dimensional torus, (i.e. just a d dimensional$d$-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 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 = \cup_{g \in G}~ g (\tilde{\Omega}) \end{equation}\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?

Suppose I have some $d$ dimensional torus, (i.e. just a d dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$, I want to find the smallest sub-region 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 = \cup_{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?

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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edited Jul 17, 2020 at 20:55

Mason

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Suppose I have some $d$ dimensional torus, (i.e. just a d dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$, I want to find the smallest sub-region 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 = \cup_{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?

Suppose I have some $d$ dimensional torus, (i.e. just a d dimensional hypercube with periodic boundary conditions) I'll call $\Omega$, and a group of transformations $G$, I want to find the smallest sub-region 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 $\tilde{\Omega}$ such that