An application of Nakayama functor in representation stability theory

TL;DR

This paper leverages the Nakayama functor to establish a categorical equivalence, advancing the understanding of representation stability in categories like FI$_G$ and VI$_q$ and resolving an open question in the field.

Contribution

It introduces a novel application of the Nakayama functor to connect module categories and addresses an open problem in representation stability theory.

Findings

Established a categorical equivalence using the Nakayama functor.

Applied the theoretical result to categories FI$_G$ and VI$_q$.

Provided a positive answer to Nagpal's open question.

Abstract

Using the Nakayama functor, we construct an equivalence from a Serre quotient category of a category of finitely generated modules to a category of finite-dimensional modules. We then apply this result to the categories FI$_G$ and VI$_q$, and answer positively an open question of Nagpal on representation stability theory.