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).