I am curious about which properties of 'the simplest polyhedron' that represents the monster group symmetries are known.
I have of course noticed that there have been quite a few similar questions around. Like about how to construct the monster, or this very general question here and quite a few closely related questions like this on MO with very interesting information on the generators of the monster group.
However, I think I have more concrete questions on it, that I hope justify a separate question. I am interested in the/a simplest polyhedron representing of monster group symmetry.
My first question is:
- Is there a unique such object and does this have a name (e.g. like "icosahedron")?
I suppose there could be several possibilities (like the icosahedron and its dual the dodecahedron that represent the same point group symmetry $*532$ (in orbifold notation)).
Also, we know for example that it can be embedded in the 196882 dimensional Euklidian space. As it cannot be a simplex of that dimension (since then it should have the symmetry of the fully symmetric group) it must be a different polyhedron. My second question would be
- (Do we know) How many vertices does it have?
From the high symmetric polyhedra in 3D-Euclidian space we know that some are constructed/bound by (3-1) dimensional simplices (like the tetrahedron, octahedron, or the icosahedron) and some are not (like the cube and the dodecahedron). How about this 196882 dimensional "monster polyhedron"?
- Is the monster polyhedron constructed from $(196882-1=)196881$ dimensional simplices?
As it appears the group can be represented by 12 generators (and 80 relators) I ask myself
- Is it possible to give an explicit form for the cartesian coordinates of one generating vertex point (a vector with 196882 entries)?
That would depend as far as I understand on the explicit matrix form (definition and orientation) of the 12 generator elements. That would be 12 $196882\times196882$ matrices, which should be completely feasible with current computational resources (for example in quantum chemistry one diagonalizes such matrices in standard computations).
A very obvious question about its shape would also be:
- Are its facets (that is the $196882-1$-faces of the convex hull of its vertices) are $196882-1$-simplices?