Brought to you by Hendrik Roehm and Leo Margolis. Code available at github, contributions are welcome.
Notation: Let the blue corner correspond to 1, the green to 2 and the red to 3. Then we denote \(e = id\), \(s = (1,2)\), \(t = (1,2,3)\), \(tt = (1,3,2)\), \(st = (1,3)\) and \(stt = (2,3)\).
Type in an element of \(\mathbb{Z}S_3\) and press enter.
The symmetric group \(S_3\) acting on three elements is the symmetry group of the equilateral triangle, fixing the center of this triangle. Extending this action linearly on the integral group ring \(\mathbb{Z}S_3\) we obtain an action on the hexagonal lattice spanned by this triangle. This description was used in an article by Marciniak and Sehgal to describe the unit group of \(\mathbb{Z}S_3\) explicitly. See also an article of Hertweck where the same is described and illustrated in more detail.
The action of \(\mathbb{Z}S_3\) on this lattice can be used to give an elegant and explicit description of the unit group of \(\mathbb{Z}S_3\) - something which is very comlicated to achieve in general for integral group rings.