The global dimension of the algebra of the monoid of all partial functions on an $n$-set as the algebra of the EI-category of epimorphisms between subsets

TL;DR

This paper determines the global dimension of the algebra of all partial functions on an n-set, showing it equals n-1, and connects it to the representation theory of symmetric groups using homological and combinatorial methods.

Contribution

It establishes the exact global dimension of the algebra of partial functions and relates it to the category of epimorphisms, providing new insights into their homological properties.

Findings

Global dimension of the algebra is n-1 for all n≥1.

Provides a partial description of the Cartan matrix.

Connects algebraic properties to symmetric group representation theory.

Abstract

We prove that the global dimension of the complex algebra of the monoid of all partial functions on an n-set is $n-1$ for all $n\geq 1$. This is also the global dimension of the complex algebra of the category of all epimorphisms between subsets of an $n$-set. In our proof we use standard homological methods as well as combinatorial techniques associated to the representation theory of the symmetric group. As part of the proof, we obtain a partial description of the Cartan matrix of these algebras.