On the FI-homology of the injective cogenerators

TL;DR

This paper investigates the FI-homology of injective cogenerators in FI-modules, providing explicit calculations in degree zero and conjectures for higher degrees, using symmetric group representation theory.

Contribution

It offers the first detailed calculation of FI-homology for injective cogenerators and proposes conjectural descriptions for higher degrees, advancing understanding in FI-module theory.

Findings

Complete calculation of FI-homology in degree zero.

Conjectural description of FI-homology in higher degrees.

Reduction of proof to symmetric group representation theory.

Abstract

The purpose of this paper is to give information on the FI-homology of the standard injective cogenerators of the category of FI-modules, where FI is the category of finite sets and injections. Working over a field k of characteristic zero, a full calculation is given in homological degree zero and a conjectural description in higher homological degree. The proof of the main theorem reduces to a calculation in representation theory of the symmetric groups, exploiting the Young orthonormal basis.