#### Sin-H

##### Member

recently, I've been (quite randomly) wondering:

How does representation theory of the Rubik's Cube group look like? Have there been constructed nontrivial representations, are they irreducible, what are their dimensionalities?

Before I go into more detail, let's do a quick introduction to representations:

If G is a finite group and ρ: G -> GL(n,ℂ) is a group homomorphism mapping from G to the space of invertible linear transformations on some n-dimensional ℂ-vector space, then ρ is called a n-dimensional representation of G.

Basically, having a representation makes working with groups conceptually very easy because we are used to working with matrices and linear operators.

There is of course always the trivial representation, which maps the whole group to the identity transformation on ℂ^n, but that is not really an interesting one.

Now, a representation is called reducible if there exists a non-trivial invariant subspace U \subset ℂ^n so that ρ(U) \subset U and irreducible if {0} and ℂ^n are the only invariant subspaces.

The irreducible representations are the really interesting ones, and you can write any representation as a direct sum of their irreducible subrepresentations.

Now why not try to actually find a neat representation of the Rubik's Cube group? Or maybe even classify all its irreducible subrepresentation (probably way too much work to do)? I reckon it might turn out to be really hard to find such a thing, mainly because the Rubik's Cube group is kinda huge. There exist formulas which constrain the possible dimensionalities of irreducible representations, but there's still a lot of stuff that can happen.

I didn't really find any papers about this, although Axel tells me that Prof. Plesken of RWTH Aachen might have worked a bit on something related.

Now I don't know what the mathematicians and physicists in here are interested in, but maybe some are interested in this more abstract stuff.

I have lots of other stuff to work on this summer as well but it might be fun to try to mess around a little. It is probably not worth the effort, but hey. I just thought I'd share this thought with you and maybe it's worth a little discussion.