Dyck Paths, Configuration Spaces and Polytopes For Linear Nakayama algebras

TL;DR

This paper develops a combinatorial framework linking Dyck paths to configuration spaces and polytopes for linear Nakayama algebras, providing explicit constructions and elementary proofs.

Contribution

It introduces a new combinatorial model connecting Dyck paths with algebras, varieties, and polytopes specific to linear Nakayama algebras, with natural morphisms between these objects.

Findings

Established explicit combinatorial constructions for the model.

Connected Dyck paths to algebraic and geometric objects.

Provided elementary proofs for the properties of these constructions.

Abstract

We present a combinatorial model of configuration spaces and polytopes associated to the quotients of $\mathbb{C} A_n$, the path algebra of the linearly oriented $A_n$ quiver, i.e. the algebra of upper triangular matrices. These quotient algebras are known as linear Nakayama algebras. Such configuration spaces were recently introduced for more general algebras by the second author and collaborators. In this special setting, we provide elementary proofs and explicit combinatorial constructions. From a Dyck path we define three related objects: a finite-dimensional algebra, an affine algebraic variety, and a polytope. Moreover, our constructions are natural: each relation in the poset of Dyck paths gives a morphism between the corresponding objects.