Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones

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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\big(\frac{\pi}{k+1}\big) . 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 \big[-2\cos\big(\frac{\pi}{k+1}\big),2\cos\big(\frac{\pi}{k+1}\big)\big] . Specializing to Jones–Wenzl projections, this result gives a new proof of a result of Ocneanu [27], 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.