Dual bases in Temperley-Lieb algebras, quantum groups, and a question of Jones (Q1633851)

Summary: We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra TL\(_k(d)\), converging for all complex loop parameters \(d\) with \(| d| > 2\cos(\frac{\pi}{k+1})\). In particular, this yields a new formula for the structure coefficients of the Jones-Wenzl projection in TL\(_k(d)\). The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation in terms of a certain graph structure we place on non-crossing pairings, and these coefficients turn out to have the remarkable property that they either always positive integers or always negative integers. As an application, we answer affirmatively a question of Vaughan Jones, asking whether every Temperley-Lieb diagram appears with non-zero coefficient in the expansion of each dual basis element in TL\(_k(d)\), when \(d \in \mathbb R \backslash [-2\cos(\frac{\pi}{k+1}),2\cos(\frac{\pi}{k+1})]\). Specializing to Jones-Wenzl projections, this result gives a new proof of a result of \textit{A. Ocneanu} [Contemp. Math. 294, 133--159 (2002; Zbl 1193.81055)], stating that every Temperley-Lieb diagram appears with non-zero coefficient in a Jones-Wenzl projection. Our methods establish a connection with the Weingarten calculus on free quantum groups, and yield as a byproduct improved asymptotics for the free orthogonal Weingarten function.