New bases of some Hecke algebras via Soergel bimodules

TL;DR

This paper constructs a new basis for certain Hecke algebras using explicit Soergel bimodules, which approximates the Kazhdan-Lusztig basis and satisfies positivity, advancing algebraic understanding for large Coxeter systems.

Contribution

It introduces a natural set of Soergel bimodules for large Coxeter systems that yields a new basis of the Hecke algebra close to the Kazhdan-Lusztig basis.

Findings

Constructed explicit Soergel bimodules for large Coxeter systems

Established the basis's positivity property

Demonstrated the basis's proximity to Kazhdan-Lusztig basis

Abstract

For extra-large Coxeter systems (m(s,r)>3), we construct a natural and explicit set of Soergel bimodules D={D_w}_{w\in W} such that each D_w contains as a direct summand (or is equal to) the indecomposable Soergel bimodule B_w. When decategorified, we prove that D gives rise to a set {d_w}_{w\in W} that is actually a basis of the Hecke algebra. This basis is close to the Kazhdan-Lusztig basis and satisfies a ``positivity condition''.