Idempotents, traces, and dimensions in Hecke categories

TL;DR

This paper develops methods to compute idempotents, categorical dimensions, and partial traces in Hecke categories, enabling detailed analysis of their structure, especially for asymptotic cases in finite Coxeter groups.

Contribution

It introduces formulas and recursive methods for calculating idempotents and dimensions in Hecke categories, advancing understanding of their algebraic and categorical properties.

Findings

Formulas for idempotents in Hecke categories

Recursive calculations for partial traces

Partial classification of asymptotic Hecke categories

Abstract

We explain how to compute idempotents that correspond to the indecomposable objects in the Hecke category. Closed formulas are provided for some common coefficients that appear in these idempotents. We also explain how to compute categorical dimensions in the asymptotic Hecke category. In many cases, we reduce this to a computation of a partial trace and give recursive formulas for some common partial traces. In the sequel, we apply this technology and perform additional (computer) calculations to complete the description of the asymptotic Hecke category for finite Coxeter groups in all but three cells.