On the coefficients in the Jones-Wenzl idempotent

TL;DR

This paper derives a closed formula for the Jones-Wenzl idempotent in the Temperley-Lieb algebra using categorification, revealing that coefficients are graded ranks of indecomposable Soergel modules or ratios of Kazhdan-Lusztig polynomials.

Contribution

It provides a new categorification-based formula for Jones-Wenzl idempotents and links their coefficients to Soergel modules and Kazhdan-Lusztig polynomials.

Findings

Coefficients are graded ranks of indecomposable Soergel modules.

Coefficients can be expressed as ratios of Kazhdan-Lusztig polynomials.

Results extend to generalized Jones-Wenzl idempotents in other types.

Abstract

By studying a categorification of the antisymmetriser quasi-idempotent in the Hecke algebra, we derive a closed formula for the Jones-Wenzl idempotent in the Temperley-Lieb algebra. In particular, we show that when the idempotent is expressed in terms of the monomial basis, the coefficients are the graded ranks of certain indecomposable Soergel modules. Equivalently, the coefficients can be expressed as a ratio of certain Kazhdan-Lusztig polynomials. Similar results are obtained for generalised Jones-Wenzl idempotents in other types.