Monoid algebras of projection functors

TL;DR

This paper investigates the structure of monoids formed by projection functors related to simple modules in finite dimensional algebras, establishing relations and isomorphisms with known algebraic structures like parking functions and incidence algebras.

Contribution

It characterizes the monoid of projection functors, finds relations in specific cases, and establishes isomorphisms with parking function monoids and incidence algebras.

Findings

Monoid of projection functors for certain algebras is explicitly described.

Isomorphism between monoid of projection functors and non-decreasing parking functions.

Explicit algebraic isomorphism with an incidence algebra, independent of the base field.

Abstract

We study the monoid of so called projection functors $\p{S}$ attached to simple modules $S$ of a finite dimensional algebra, which appear naturally in the study of torsion pairs. We determine defining relations in special cases of path algebras. For the linearly oriented Dynkin quiver of Type $A$, we get an isomorphism to the monoid of non-decreasing parking functions. Moreover we give an explicit isomorphism between the monoid algebra of non-decreasing parking functions and a certain incidence algebra independent of the field.