Categorical diagonalization

TL;DR

This paper develops a categorical framework for diagonalization, extending linear algebra tools to the realm of categories, enabling the construction of idempotent functors that project onto eigencategories, with applications to Hecke algebras.

Contribution

It introduces the concept of categorical diagonalization, including eigenmaps and idempotent functors, to analyze endofunctors in triangulated categories, advancing categorical representation theory.

Findings

Constructed idempotent functors projecting to eigencategories.

Proved convolution of idempotent functors is isomorphic to the identity.

Laid groundwork for categorified Young symmetrizers in Hecke algebras.

Abstract

This paper lays the groundwork for the theory of categorical diagonalization. Given a diagonalizable operator, tools in linear algebra (such as Lagrange interpolation) allow one to construct a collection of idempotents which project to each eigenspace. These idempotents are mutually orthogonal and sum to the identity. We categorify these tools. At the categorical level, one has not only eigenobjects and eigenvalues but also eigenmaps, which relate an endofunctor to its eigenvalues. Given an invertible endofunctor of a triangulated category with a sufficiently nice collection of eigenmaps, we construct idempotent functors which project to eigencategories. These idempotent functors are mutually orthogonal, and a convolution thereof is isomorphic to the identity functor. In several sequels to this paper, we will use this technology to study the categorical representation theory of Hecke algebras. In particular, for Hecke algebras of type A, we will construct categorified Young symmetrizers by simultaneously diagonalizing certain functors associated to the full twist braids.