Subsection 4.1.1. (0P2K) · ScalingStacks

4.1.1. Markov traces

Let ${\mathcal{C}}{ox}$ be the poset of finite Coxeter groups, viewed as a category. The objects are Coxeter groups $(W,S)$ and $\operatorname{Hom}\nolimits((W,S),(W^{\prime},S^{\prime}))$ is the set of injective maps $f:S\to S^{\prime}$ such that $m_{f(s),f(t)}=m_{st}$ for all $s,t\in S$ . Given $s\in S$ , we denote by $i_{s}:(W_{S\setminus s},S\setminus s)\to(W,S)$ the inclusion.

Let ${\mathcal{F}}$ be a full subposet of ${\mathcal{C}}{ox}$ closed below.

Let ${\mathcal{H}}_{(W,S)}={\mathbf{Z}}[q^{\pm 1}]B_{(W,S)}/((T_{s}-1)(T_{s}+q))_{s\in S}$ be the Hecke algebra of $(W,S)$ .

0P1W

Definition 4.1. Let $R$ be a ${\mathbf{Z}}[t_{-},t_{+},q^{\pm 1}]$ -module. A Markov trace on ${\mathcal{F}}$ is the data of a family of ${\mathbf{Z}}[q^{\pm 1}]$ -linear maps $\tau_{(W,S)}:{\mathcal{H}}_{(W,S)}\to R$ for $(W,S)\in{\mathcal{F}}$ such that

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$\tau_{S}(hh^{\prime})=\tau_{S}(h^{\prime}h)$ for $h,h^{\prime}\in{\mathcal{H}}_{S}$
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$\tau_{S}(hT_{s}^{\pm 1})=t_{\pm}\tau_{S\setminus s}(h)$ for all $s\in S$ and $h\in{\mathcal{H}}_{S\setminus s}$ .

Markov traces, with a possibly more general definition, have been studied by Jones and Ocneanu in type $A$ [Jo], Geck-Lambropoulou in type $B$ [GeLa], Geck in type $D$ [Ge], and Kihara in type $I_{2}(n)$ [Ki]. Gomi has provided a general construction of Markov traces for Weyl groups, using Lusztig’s Fourier transform [Go]. More recently, Lasy has studied Markov traces in relation with Gomi’s definition and Soergel bimodules [La1].

Raphaël Rouquier

Original source: arXiv:1203.5065v1