Inner products on the Hecke algebra of the braid group

We point out that the Homfly polynomial (that is to say, Ocneanu’s trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin’s braid group. These bear a close connection to the Morton–Franks–Williams inequality. In these structures, the sets of positive, respectively negative permutation braids become orthonormal bases. In the second case, many inner products can be geometrically interpreted through Legendrian fronts and rulings. 1 The Hecke algebra In this note we make a few observations on the Hecke algebra Hn(z). Our main reference is Jones’s seminal paper [6]. As an algebra, Hn(z) is generated by the same symbols σ1, . . . , σn−1 as Artin’s braid group Bn. In addition to the standard relations of Bn (that is, σiσj = σjσi for |i − j| ≥ 2 and σiσi+1σi = σi+1σiσi+1 for all i) we also impose σi − σ −1 i = z for all i. (1) For topologists, the Hecke algebra is significant because of its role in the original definition of the Homfly and Jones polynomials [6] (see also [9] for a concise survey). Namely, using Ocneanu’s trace Tr: Hn(z) → Z[z, T ], the framed Homfly polynomial H of a braid β ∈ Bn is obtained as Hβ(v, z) = ( v − v z )n−1 · Tr(β) ∣∣∣∣ T= z 1−v2 . (2) In words: for β ∈ Bn, take its natural representation β ∈ Hn(z) (a common abuse of notation) and its trace, substitute T = z/(1− v) in it, and normalize suitably. If β has exponent sum w, then Pb β(v, z) = v Hβ(v, z) is an expression for the Homfly polynomial of the oriented link β̂ which is the closure of β. This is