A restricted model for the bounded derived category of gentle algebras

TL;DR

This paper introduces a specialized model for the bounded derived category of gentle algebras, enabling enumeration of certain silting objects and connecting to combinatorial numbers.

Contribution

It provides a new restricted model that encodes indecomposables and extensions, facilitating counting of silting objects in specific algebraic settings.

Findings

The model encodes indecomposables and positive extensions.

Number of d-term silting objects in linearly oriented A_n is given by Pfaff-Fuss-Catalan numbers.

The approach links algebraic structures to combinatorial enumeration.

Abstract

We present a restricted model for the bounded derived category of gentle algebras that encodes the indecomposable objects and positive extensions between them. The model is then used to count the number of $d$-term silting objects for linearly oriented $A_n$, recovering the result that they are counted by the Pfaff-Fuss-Catalan numbers.