On the coefficients in the Jones-Wenzl idempotent

2024/06/10 by Joseph Baine, Baine, J.
Mathematics · #FOS: Mathematics #Functional Equations Stability Results #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2406.06333

openalex publication_date 2024/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• p-Jones-Wenzl idempotents
• The Jones quotients of the Temperley-Lieb algebras
• Categorification of the Kazhdan-Lusztig basis of the Temperley-Lieb algebra by bimodules
• Evaluable Jones-Wenzl idempotents at root of unity and modular\n representation on the center of U_qsl(2)
• Seminormal forms for the Temperley-Lieb algebra