Incidence category of the Young lattice, injections between finite sets, and Koszulity

TL;DR

This paper characterizes the algebraic structure of the category of injections between finite sets, revealing its quadratic self-duality and providing explicit linear resolutions for simple modules.

Contribution

It introduces the Gabriel quiver with relations for the category of injections, proves its quadratic self-duality, and constructs linear resolutions for simple modules.

Findings

Gabriel quiver with defining relations described

Proves quadratic self-duality of the algebra

Constructs linear resolutions for simple modules

Abstract

We describe the Gabriel quiver with defining relations of the category of injections between finite sets, show that it is quadratic self-dual, and construct linear resolutions for its simple modules.