ScalingStacks

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 $h\in{\mathcal{H}}_{S\setminus s}$.