Categorical diagonalization

Abstract

This is an expository chapter in Springer’s “Introduction to Soergel Bimodules”. In classical linear algebra, given a diagonalizable operator on a vector space, Lagrange interpolation produces an idempotent decomposition of the identity corresponding to the projections to eigenspaces. In this chapter, we explain a categorical analogue of this procedure due to Elias and Hogancamp. Given a “diagonalizable” functor acting on a monoidal homotopy category, we produce idempotent functors which project to “eigencategories”. The main application is to the full twist Rouquier complex acting on the homotopy category of Soergel bimodules.