On the coefficients of the Jones-Wenzl idempotent

Abstract

The Jones-Wenzl idempotent is an element of the Temperley-Lieb algebra which plays a crucial role in numerous fields: the representation theory of SL2, knot theory, Soergel bimodules, and more. A long-standing problem has been to express this idempotent in terms of the diagrammatic basis of the Temperley-Lieb algebra. Various recursive and algorithmic approaches to this problem have appeared over the years. In this talk I will explain how the coefficients that appear have deep representation-theoretic and geometric significance. This follows from categorifying a lift of the Jones-Wenzl idempotent to the Hecke algebra. I will then explain how certain structural properties of these coefficients are expected to hold more generally.