Polynomial Invariants for Triangulated Categories with Exceptional Sequences
Given two triangulated categories, it is desirable to decide, whether they are equivalent as triangulated categories. Essentially there are two aspects, a combinatorial aspect and a geometric aspect. The first one corresponds to an isomorphism on the level of the Grothendieck group together with its Euler form. The solution for triangulated categories of finite dimensional (hereditary or even quasi-hereditary) wild algebras with three vertices is known, the only free parameter is the largest eigen value of the Coxeter transformation (for hereditary algebras), or equivalently, its trace (that works also for quasi-hereditary algebras). This idea was used also for cluster algebras with three vertices (in a joint work with Beineke and Brüstle) to decide, whether it is acyclic or not (with some exceptions). It is classically known as an equation between the ranks of eceptional sequences on the projective plane (Drezet, le Potier and later also Rudakov).
For the general problem, triangulated categories with a full exceptional sequence of length n we determine a finite set of polynomials, called polynomial invariants, so that we can generically solve this problem (there are some exceptions one should consider in more detail) in terms of the values of these polynomials, they generalize the Markov equation for n=3.
In this talk we review some of the history, formulate the problem over arbitrary fields and solve it using so-called polynomial invariants. We also discuss, what can be decided using polynomial invariants and what is the remaining open problem, in particular, what to expect for the geometric aspects of the question.