# Group theory - symmetry inequivalent sqaure matrices with PBC

I would like to obtain the symmetry inequivalent possible occupations of indistinguishable objects in a square lattice with periodic boundary conditions.

Examples:

• 1 object in a 4 by 4 square lattice has 16 equivalent positions -> 1 inequivalent position.
• 2 objects in a 3 by 3 square lattice has 2 inequivalent positions.

000    000
000    010
110    100

• 2 objects in a 4 by 4 square lattice has 5 inequivalent positions.

0000    0000    0000    0000    0000
0000    0000    0000    0000    0010
0000    0000    0100    0010    0000
1100    1010    1000    1000    1000


I can search for the inequivalent positions with a bruteforce approach by applying all possible symmetries to all possible initial configurations of 1 or 2, 3 objects and so on. Anyway, I was wondering if there is a way to do this with group theory in Mathematica.

I tried starting from this SO question but I didn't manage to get translations working.

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I started from that question and added all the other symmetries when trying the "bruteforce" method. I was wondering if it's possible to use a different strategy. –  Pie86 Sep 25 '13 at 11:12
When are two positions "equivalent"? –  Hector Sep 25 '13 at 11:14
Consider an infinite 2D lattice made of one of the 4x4 matrices with 1 object. You can put the object anywhere in the 16 positions, the distance between the objects in the infinite lattice will always be the same. I hope this helps clarifying what I mean by equivalent. –  Pie86 Sep 25 '13 at 12:36