| When the Temperley-Lieb algebra is semi-simple, the Jones-Wenzl idempotent can be expressed as a linear combination of words in the generators. We review the relevance of this algebra to subfactors, and prove that for a large class of these words, the formula of Ocneanu produces correct coefficients.; The theory of planar algebras was developed by Jones primarily as a tool for studying subfactors. A planar algebra is an algebra over the colored operad of tangles, objects that are represented by planar diagrams. By using tangles with just one "input" and one "output"---annular tangles---it is possible to realize any planar algebra as a Hilbert Temperley-Lieb module. We review the definition of a planar algebra module, and present and prove a positivity result extending a theorem of Jones that enables us to identify the irreducible Hilbert modules in the non-generic case. Our proof makes use of the Jones-Wenzl idempotent coefficients we have verified.; We then construct planar algebras from the Coxeter graphs A m, m ≥ 3 and Dm, m ≥ 4 and, using our positivity result, decompose them completely into irreducible Temperley-Lieb planar algebra modules. |
