On the ideals and essential algebras of shifted functors of linear representations

TL;DR

This paper investigates the structure of ideals and essential algebras in shifted functors related to linear representations, providing new characterizations, criteria, and equivalences in the context of biset and rational representation functors.

Contribution

It offers a novel characterization of ideals, a vanishing criterion for essential algebras, and an equivalence between module categories of shifted functors and semisimple algebras.

Findings

Characterization of lattices of ideals for biset functors

Criterion for vanishing of essential algebras

Equivalence of module categories for shifted functors

Abstract

We present a study on the Yoneda-Dress construction of biset functors of linear representations over a field of characteristic zero. We give a characterization of their lattices of ideals and we provide a criterion of vanishing for their essential algebras. We provide a parametrization for a family of simple modules over the functor of rational representations. Finally, we give an equivalence between the category of modules over the shifted functor of complex class functions and the category of modules over a semisimple algebra.