Derived equivalences between one-branch extensions of "rectangles"

TL;DR

This paper explores derived equivalences among incidence algebras from one-branch extensions of rectangles, providing explicit realizations and revealing new relationships between Nakayama algebras.

Contribution

It establishes that all four types of one-branch extensions of rectangles have derived equivalent incidence algebras and introduces explicit tilting complex realizations.

Findings

All four extension types yield derived equivalent incidence algebras.

Explicit Coxeter polynomial formulas for certain Nakayama algebras are provided.

An unexpected derived equivalence between specific Nakayama algebras is discovered.

Abstract

In this paper we investigate the incidence algebras arising from one-branch extensions of "rectangles". There are four different ways to form such extensions, and all four kinds of incidence algebras turn out to be derived equivalent. We provide realizations for all of them by tilting complexes in a Nakayama algebra. As an application, we obtain the explicit formulas of the Coxeter polynomials for a half of Nakayama algebras (i.e., the Nakayama algebras $N(n,r)$ with $2r\geq n+2$). Meanwhile, an unexpected derived equivalence between Nakayama algebras $N(2r-1,r)$ and $N(2r-1,r+1)$ has been found.