Stability patterns in representation theory

TL;DR

This paper develops a comprehensive theory of stable representation categories for classical and symmetric groups, revealing deep equivalences and producing new results in representation theory and related fields.

Contribution

It introduces a unified framework for stable representation categories and establishes multiple equivalences with other mathematical categories, advancing understanding of their structure.

Findings

Construction of injective resolutions of simple objects

Duality between orthogonal and symplectic theories

Canonical derived auto-equivalence of the general linear theory

Abstract

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results, e.g., the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, a canonical derived auto-equivalence of the general linear theory, etc.