Explicit diagonalization of the Markov form on the Temperley-Lieb algebra

2005/10/31 by Josh Genauer, Genauer, Josh, Neal W. Stoltzfus +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.math/0511003

openalex publication_date 2005/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a triangular change of basis in which the form is diagonal and explicitly compute the diagonal entries of this matrix as products of quotients of Chebyshev polynomials, corroborating the determinant computation of Ko and Smolinsky. The proof employs a recursive method for defining the required orthogonal basis elements in the Temperley-Lieb algebra, similar in spirit to Jones' and Wenzl's recursive formula for a family of projectors in the Temperley-Lieb algebra. In short, we define a partial order on the non-crossing chord diagram basis and find the orthogonal basis using a recursive formula over this partial order. Finally we relate this orthogonal basis to bases using the calculus of trivalent graphs developed by Kauffman and Lins.

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