Hello, this is a question regarding Reineke's paper "Cohomology of non-commutative Hilbert schemes", <http://arxiv.org/abs/math/0306185>, and more precisely the formula on page 4 there (at $n=1$), namely:
$$\chi(H\_{d,1}^{(m)})=\frac{1}{(m-1)d+1}\binom{md}{d}$$


Here at left we have the cohomological Euler characteristic of the non-commutative Hilbert scheme $H\_{d,1}^{(m)}$, and at right we have the Fuss-Catalan numbers.


The point now is that the Fuss-Catalan numbers appear as well in a key place in subfactor theory, namely as coefficients of the Poincaré series of the Fuss-Catalan algebra of Bisch and Jones (for a quick introduction to this algebra, see e.g. Bisch's survey paper <http://arxiv.org/abs/math/0304340>).


* Question: is there any serious relation between Reineke's formula and the Bisch-Jones formula?


(Note: regarding my own motivation, some 10 years ago I happened to attend a talk by Reineke, and, as a subfactor aficionado, when I saw that numbers on the blackboard I jumped on my seat! I asked him after his talk, he didn't know, and then for some time I kept asking people around, either in AG or in subfactors. But, so far, couldn't find anyone knowing well enough both subjects, in order to answer.) 



Fuss-Catalan algebras and non-commutative Hilbert schemes
Not really an answer, but too long for a comment. The question is certainly quite interesting, theoretically speaking, but, in practice, who might be *really interested* in doing that? It's a cat-mouse game. What you're asking for is to connect an extremely down-to-earth subarea of OA - the one centered around the TL and FC algebras - to one of the most abstract branches of AG. Mission impossible, IMO :)


Yet another comment, if you're interested in such connections, then.. better look at good old algebraic varieties! These are technically speaking at about the same abstraction level as the OA stuff that you mention, i.e. lots of combinatorics and fun. You can check for instance matrix varieties $M\subset M\_n(\mathbb C)$, by doing inside some spin model statistics you might perhaps get into FC stuff?


