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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 Poincare 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 afficionadoaficionado, 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.)

(Edit Jan 13: adding two new tags, both coming from subfactor-related occurrences of the FC numbers, with of course the hope that this will give a new life to the question. My revised, downgraded question would concern an AG-OA connection not necessarily via subfactors, but perhaps via quantum groups or random matrices. But, IMO, the NC Hilbert scheme rather reminds a subfactor.)

2 added 386 characters in body; edited tags

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 Poincare 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 afficionado, 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.)

(Edit Jan 13: adding two new tags, both coming from subfactor-related occurrences of the FC numbers, with of course the hope that this will give a new life to the question. My revised, downgraded question would concern an AG-OA connection not necessarily via subfactors, but perhaps via quantum groups or random matrices. But, IMO, the NC Hilbert scheme rather reminds a subfactor.)

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# Fuss-Catalan algebras and non-commutative Hilbert schemes

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 Poincare 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 afficionado, 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.)