ScalingStacks

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 $(W,S)$.

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].

One can ask in addition that the functors $T_{S,s,\pm}$ are invertible. On the other hand, one can get a more general definition by dropping the functoriality of ${\mathcal{C}}$ and by requiring the existence of functors $D_{S,s,\pm}:{\mathcal{C}}_{S\setminus s}\to{\mathcal{C}}_{S}$ such that $M_{S}(\gamma_{S\setminus s}(?)\cdot F_{s}^{\pm 1})\simeq D_{S,s,\pm}M_{S\setminus s}(?)$.

The first “trace” condition, once formulated in the appropriate homotopical setting, leads to a universal solution (“abelianization”) that can be described explicitly [BeNa]. It would be interesting to find a suitable formulation of the second condition in this setting leading to a universal solution. It would also be interesting to study functoriality with respect to cobordism in type $A$, along the lines of [KhTh] and [ElKr].

Let ${\mathcal{S}}{oe}$ be the category of Soergel bimodules: this is the full subcategory of $P^{\mathrm{en}}\operatorname{\!-modgr}\nolimits$ whose objects are direct summands of direct sums of objects of the form $\theta_{s_{1}}\cdots\theta_{s_{n}}\langle r\rangle$, for some $s_{1},\ldots,s_{n}\in S$ and $r\in{\mathbf{Z}}$.

There is a ${\mathbf{Z}}[q^{\pm 1}]$-algebra morphism ${\mathcal{H}}\to K_{0}({\mathcal{S}}{oe})$ given by $T_{s}\mapsto[F_{s}]$, and that morphism is actually an isomorphism (cf [Soe, Theorem 1] and [Li, Théorème 2.4]).

We consider now a Markov $2$-trace in the following setting. Assume the functor ${\mathcal{C}}$ takes values in graded triangulated categories and $M_{(W,S)}$ is the restriction of a graded triangulated functor $M_{(W,S)}:\operatorname{Ho}\nolimits^{b}({\mathcal{S}}{oe}_{(W,S)})\to{\mathcal{C}}_{(W,S)}$. In particular, it induces a ${\mathbf{Z}}[q^{\pm 1}]$-linear map ${\mathcal{H}}_{(W,S)}\to K_{0}({\mathcal{C}}_{(W,S)})$. Let $R=\operatorname{colim}\nolimits_{(W,S)\in{\mathcal{F}}}K_{0}({\mathcal{C}}_{(W,S)})$, a ${\mathbf{Z}}[q^{\pm 1}]$-module. Assume there are commuting endomorphisms $t_{\pm}$ of $R$ compatible with the action of $[T_{S,s,\pm}]$ on $K_{0}({\mathcal{C}}_{(W,S)})$, for all $(W,S)\in{\mathcal{F}}$ and $s\in S$, via the canonical maps $\iota_{(W,S)}:K_{0}({\mathcal{C}}_{(W,S)})\to R$.

Define $\tau_{(W,S)}:B_{(W,S)}\to R$ by $\tau(b)=\iota_{S}([M_{S}(F_{b})])$. We have the following immediate proposition.