A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring.

A fusion ring is given by a finite set of integer matrices checking some axioms (see here p 22).
A solution of its pentagon equation is the main condition for having a structure of fusion category,
in which it encodes associativity (see here). For example, the irreducible complex representations
of a finite group, equipped with $\oplus$ and $\otimes$, generate a fusion ring and a fusion category.

The problem is that, in practice the pentagon equations we meet are huge, for example, a system of $2000000$ polynomial equations with $50000$ variables (of degree $3$ with integer coefficients), so that proving the existence of a solution (and a fortiori finding one) is very hard.
Note that there are pentagon equations without solution.

Question: Are there workable algebraic geometry approaches ?
(for proving the existence of a solution or for finding one)

In fact, the pentagon equation is more structured than just a system of scalar equations, it's a system of several invertible (unitary) matrix equations of the form $$A_1 A_2 A_3 = A_4 A_5$$ (and $A_i^* A_i = I$), such that $A_i = \tau_i B_i \tau'_i$, with $\tau_i$,$\tau'_i$ fixed permutation matrices and $B_i$ a block diagonal matrix (each block is a matrix variable), so that there are many holes (see here p 29-30).
Note that in general $\nu_1\nu_2\nu_3 \neq \nu_4\nu_5$ (with $\nu_i = \tau_i\tau'_i$), so that $B_i =I$ can't be a solution.

I'm interested in the equation given by this fusion ring. There are $16227$ matrix equations of dimensions from $1$ to $91$, and $2097$ invertible (unitary) matrix variables of dimensions from $1$ to $17$ (see here p 31).

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    $\begingroup$ It isn't necessarily true that the pentagon equations are unitary matrix equations. Given that $\tau_i$ are permutation matrices, this would then imply that the $B_i$ are direct sums of unitary matrices, and so that all of the fusion matrices are unitary. Off hand and going through my notes though I cannot think of a Grothendieck ring which does not admit a unitary categorification. $\endgroup$ – Matthew Titsworth Jul 18 '14 at 22:25
  • $\begingroup$ @MatthewTitsworth: so the matrix variables are "in theory" non-necessarily unitary (just invertible), but "in practice" there are unitary, isn't it? Moreover, the equation for unitarity ($AA^* = I$) is perhaps much more workable than the equation for invertibility ($det(A) \neq 0$). Does the existence of a solution with invertible matrices implies a solution with unitary matrices ? Anyway, my final goal is the subfactor theory, and in this case, the matrices are effectively unitary. $\endgroup$ – Sebastien Palcoux Jul 18 '14 at 22:50
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    $\begingroup$ If I remember correctly, all fusion categories coming from subfactors are unitary (I believe something like this was mentioned at BIRS, though I cannot speak with authority). It is not true that for an arbitrary solution to the pentagon equations there exists a basis such that the F-Matrices are unitary (Yang-Lee is the usual example), so your second question seems equivalent to asking whether or not all Grothendieck rings admit a unitary categorification. I don't know the answer to this. $\endgroup$ – Matthew Titsworth Jul 18 '14 at 22:58
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    $\begingroup$ I should have remembered the answer to this question: Going back through Rowell, Theorem 4.5 explicitly states that no fusion category Grothendieck equivalent to $\mathcal C(so_{2p+1},l,q)$ where $q^2$ is a primitive $l$'th root of unity, $l$ is odd, and $p$ is such that $2(2p+1)<l$ is unitarizable. $\endgroup$ – Matthew Titsworth Aug 10 '14 at 21:01
  • $\begingroup$ @MatthewTitsworth: Thank you for this reference! More precisely I read that there is no "braided" fusion category (...) unitarizable. But, perhaps such a fusion category is necessarily braided, I don't know. $\endgroup$ – Sebastien Palcoux Aug 10 '14 at 21:44

I'm not an expert in algebraic geometry, but I can say something about methods for solving pentagon equations that will hopefully be of use.

The primary way to determine whether or not there is a solution to the pentagon equations is to use Groebner basis methods. However, these begin to break down very quickly for algebraic varieties of the size that we deal with in the pentagons. However, the Fusion variety has a lot of structure that can be taken advantage of:

  1. We can fix bases on the $Hom(a \otimes b, c)$ spaces by fixing certain 10j-symbols in the pentagon equations. Ocneanu rigidity then implies that there are only finitely many solutions to the pentagon equations. When there's no multiplicity (i.e. all $Hom(a \otimes b, c)$ spaces are one dimensional), this is relatively easy. The big problem here is in trying to figure out whether or not a given 10j-symbol is identically zero or can be fixed to 1.
  2. In the case where there is multiplicity this gets much more difficult because the action of the gauge group on the $Hom(a\otimes b \otimes c, d)$ spaces is no longer one dimensional and so its extension to the $Hom(a\otimes b\otimes c, d)$ spaces can no longer be treated as multiplication by non-zero scalars. Ideally then the thing to do is find a basis for the multidimensional hom-spaces that fixes certain F-Matrices so that the gauge group only acts by scalar matrices.

  3. Some of the structure of the pentagons can be deduced from arm-bending, i.e. using the rigidity and leads to redundancy which can be summarized when solving things by hand. For solving things using computer algebra systems, this redundancy can be helpful.

  4. The pentagon equations also have the structure of a tower of varieties. By this I mean that If we consider only the matrix equations of dimension <=n, our variables can only be those 10j-symbols coming from F-Matrices of dimension <=n. Given this, one way to attack things is by finding admissible values for smaller dimensional equations and then using these as input to simplify larger ones. For example, there exists a subset of pentagon equations which are one dimensional matrix equations. By necessity, these can only include F-matrices which are themselves one dimensional, and so they form a sub-variety of the fusion variety. Solving these by themselves can sometimes be done, and this can then be used to simplify equations with the next dimension up.

  5. It's also not necessarily true that partitioning equations by dimension is the smallest division we can come up with. Given a set of solutions to equations with dimension $<n$, it may be that the equations with dimension $=n$ split into two or more disjoint sets of equations and variables. In this way, things also get easier.

  6. From a different direction, one thing we can also do is first find solutions for sub-categories and then use this as input in other ways. When solving things by hand this can be useful, but my experience has been that for larger categories the majority of the 10j-symbols and pentagon equations come from tensor products with and of objects not in the subcategory.

In the end though, once all is said and done, the polynomial equations one obtains (usually) still have to be simplified before Grobner basis algorithms can be effective. The only real way I know of to do this is then to implement heuristics which perform polynomial arithmetic to reduce the number of equations and variables down to a manageable level.

Beyond this, here are some thoughts on things I'm aware of from algebraic geometry I'd like to entertain as maybe some day being helpful:

  1. All of our equations are cubic, and from an algebro-geometric perspective these varieties should (hopefully) have some connection to the theory of algebraic curves and there are (hopefully) techniques in this amenable to helping find solutions.
  2. What would be great is if the F-Matrices and pentagon equations could be presented in such a way that techniques from non-commutative algebraic geometry (such as non-commutative Grobner basis methods) could be leveraged. From what I understand (which is admittedly not much) there is some difficulty in using NCGB methods and guaranteeing that they will terminate, however the advantage would (presumably) be that we get a gigantic reduction in the size of our varieties: 16227 equations in 2097 variables vs. ~2000000 in ~50000. What also seems difficult is obtaining such a presentation for the F-Matrices/pentagons.
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    $\begingroup$ There is a reference to a comment on page 12 here about the algebraic data for a fusion category being an atlas for an "algebraic stack," but I do not feel qualified to say anything more. $\endgroup$ – Matthew Titsworth Jul 27 '14 at 3:37

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