Algebra Seminar: Combinatorics and Geometry of n-representation infinite algebras of type Ã

Abstract: n-representation infinite (= n-RI) algebras are an analog of hereditary representation infinite algebras in Iyama's higher Auslander-Reiten theory. An n-RI algebra is of type à if its higher preprojective algebra is a skew-group algebra R*G, where G < SL_{n+1} is a finite abelian group acting on the polynomial ring R in n+1 variables. However, there is no perfect correspondence between n-RI algebras of type à and abelian subgroups G, so describing all n-RI of type à is a non-trivial problem.

The theme of this talk is a solution to this problem. I will present a detailed summary of the well-known classical case (n=1), and use it to set up the necessary combinatorics and geometry, which come from the algebra R*G and the quotient singularity R^G respectively. Then we will see that the higher case (n>1) works in essentially the same way. In particular, I want to present an equivalence between n-RI algebras of type à and so-called height functions on a lattice, and show that higher tilting classes of these algebras are naturally parametrised by lattice points in some polytope. In terms of toric geometry, this polytope is the junior simplex of R^G, and I want to explain what the higher tilting classes have to do with exceptional crepant divisors in a relative minimal model of R^G.

This is based on joint work with Oleksandra Gasanova.

This is a seminar in our algebra seminar series.