Triangulated categories of matrix factorizations for regular systems of weights with $\epsilon=-1$

TL;DR

This paper constructs a special collection in the category of matrix factorizations for certain weighted polynomials, revealing connections to root systems and Coxeter elements, advancing understanding in algebraic geometry and representation theory.

Contribution

It introduces a full strongly exceptional collection in the triangulated category of graded matrix factorizations for regular systems of weights with , linking algebraic structures to root systems and Coxeter elements.

Findings

Defines a root basis of a generalized root system of sign (l,0,2)

Establishes a Coxeter element of finite order with a regular eigenvector

Connects matrix factorizations to geometric structures in type IV domains

Abstract

We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a non-degenerate regular system of weights whose smallest exponents are equal to -1. In the associated Grothendieck group, the strongly exceptional collection defines a root basis of a generalized root system of sign (l,0,2) and a Coxeter element of finite order, whose primitive eigenvector is a regular element in the expanded symmetric domain of type IV with respect to the Weyl group.