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.

We put $\operatorname{HH}\nolimits_{i}=\operatorname{HH}\nolimits_{i}^{(W,S)}=\operatorname{Tor}\nolimits_{i}^{P^{{\mathrm{en}}}}(P,-):P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits\to P\operatorname{\!-modgr}\nolimits$. This gives rise to functors $\operatorname{HH}\nolimits_{i}:\operatorname{Ho}\nolimits^{b}(P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits)\to\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-modgr}\nolimits)$ and to a functor $\operatorname{HH}\nolimits_{*}:\operatorname{Ho}\nolimits^{b}(P^{{\mathrm{en}}}\operatorname{\!-modgr}\nolimits)\to\operatorname{Ho}\nolimits^{b}(P\operatorname{\!-modgr}\nolimits)^{{\mathbf{Z}}}$.

Given $I\subset S$, we have a functor $\rho_{I}^{*}:D^{b}(P_{I}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}\to D^{b}(P\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$. This defines a functor from ${\mathcal{C}}{ox}$ to graded triangulated categories $(W,S)\mapsto D^{b}(P_{S}\operatorname{\!-modgr}\nolimits)^{\mathbf{Z}}$. Our grading here is the one coming from $P_{S}\operatorname{\!-modgr}\nolimits$.

The following theorem is a consequence of Theorem 5.1 below.

Passing to homology, we obtain the following result.

The construction of §4.1.3 provides a Markov trace, recovering a result of Lasy [La1, La2]. Note that Webster and Williamson [WeWi] have shown that for finite Weyl groups, this is actually Gomi’s trace, as conjectured by J. Michel. That has been shown to hold also in type $I_{2}(n)$ by Lasy [La1].

By shifting suitably the invariants, we can get rid of the automorphisms $T_{S,s,\pm}$, but we lose functoriality (it would be interesting to see if functoriality with respect to an appropriate notion of cobordisms can be implemented). In order to do this, we need to use $\frac{1}{2}{\mathbf{Z}}$-complexes.

Given ${\mathcal{A}}$ an additive category, the category of $\frac{1}{2}$-complexes in ${\mathcal{A}}$ has objects $(C^{i},d^{i})_{i\in\frac{1}{2}{\mathbf{Z}}}$ where the differential has degree $1$, and morphisms are $\frac{1}{2}{\mathbf{Z}}$-graded maps commuting with the differential. Its homotopy category is denoted by $\operatorname{Ho}\nolimits_{\frac{1}{2}}({\mathcal{A}})$ and, when ${\mathcal{A}}$ is an abelian category, its derived category by $D_{\frac{1}{2}}({\mathcal{A}})$.

For a more intrinsic approach to this section, cf [De, §1.1].

Let ${\mathcal{A}}$ be an additive category and $n\geq 0$. We denote by $\operatorname{Comp}\nolimits({\mathcal{A}})$ the category of complexes of objects of ${\mathcal{A}}$. The category $n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$ of $n$-fold complexes is defined inductively by $n\operatorname{\!-Comp}\nolimits({\mathcal{A}})=\operatorname{Comp}\nolimits((n-1)\operatorname{\!-Comp}\nolimits({\mathcal{A}}))$ and $0\operatorname{\!-Comp}\nolimits({\mathcal{A}})={\mathcal{A}}$. Its objects are families $(X,d_{1},\ldots,d_{n})$ where $X$ is an object of ${\mathcal{A}}$ graded by ${\mathbf{Z}}^{n}=\bigoplus_{i=1}^{n}{\mathbf{Z}}e_{i}^{*}$, $d_{i}$ is a graded map of degree $e_{i}$ and $d_{i}^{2}=[d_{i},d_{j}]=0$ for all $i,j$.

Given $X$ an $n$-complex and $i\in\{1,\ldots,n\}$, we define $Y=X[e_{i}]$ as the $n$-complex given by $Y^{b}=X^{e_{i}+b}$ and differentials $\partial_{i}^{b}=(-1)^{\delta_{ij}}d_{j}^{e_{i}+b}$.

Let $f:\{1,\ldots,n\}\to\{1,\ldots,m\}$ be a map. It induces a map $\sigma:{\mathbf{Z}}^{n}\to{\mathbf{Z}}^{m}$ and gives by duality a map ${\mathbf{Z}}^{m}\to{\mathbf{Z}}^{n}$. This provides a functor from ${\mathbf{Z}}^{n}$-graded objects to ${\mathbf{Z}}^{m}$-graded objects of ${\mathcal{A}}$. Let $X$ be an $n$-complex. We have a corresponding ${\mathbf{Z}}^{m}$-graded object $X^{\prime}$. We define a structure of $m$-complex by

where $b=\sum_{i}b_{i}e_{i}$.

Note that when $f$ is an injection, the sum above has only positive signs. When $f$ is a bijection, then $\operatorname{Tot}\nolimits^{f}$ is a self-equivalence of $n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$. We write $\operatorname{Tot}\nolimits=\operatorname{Tot}\nolimits^{f}$ when $m=1$.

We have defined an additive functor

Let $g:\{1,\ldots,m\}\to\{1,\ldots,p\}$ be a map and $\tau:{\mathbf{Z}}^{m}\to{\mathbf{Z}}^{p}$ the associated morphism. Let $X\in n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$. The ${\mathbf{Z}}^{p}$-graded objects underlying $\operatorname{Tot}\nolimits^{gf}(X)$ and $\operatorname{Tot}\nolimits^{g}(\operatorname{Tot}\nolimits^{f}(X))$ have their component of degree $a$ equal to $\bigoplus_{c\in(\tau\sigma)^{-1}(a)}X^{c}$. We define an isomorphism between these $p$-complexes by multiplication by $(-1)^{\varepsilon(c)}$ on $X^{c}$, where

This gives an isomorphism of functors

Let $k$ be a commutative ring and $A$, $B$ and $C$ be three $k$-algebras. Let $X\in n\operatorname{\!-Comp}\nolimits((A\otimes B)\operatorname{\!-Mod}\nolimits)$ and $Y\in m\operatorname{\!-Comp}\nolimits((B\otimes C)\operatorname{\!-Mod}\nolimits)$. Then $X\otimes_{B}Y$ defines an object of $(n+m)\operatorname{\!-Comp}\nolimits((A\otimes C)\operatorname{\!-Mod}\nolimits)$. We have $(X\otimes_{B}Y)^{(a_{1},\ldots,a_{n+m})}=X^{(a_{1},\ldots,a_{n})}\otimes_{B}Y^{(a_{n+1},\ldots,a_{n+m})}$.

Assume ${\mathcal{A}}$ is an abelian category. Let $X\in n\operatorname{\!-Comp}\nolimits({\mathcal{A}})$. Let $r\in\{1,\ldots,n\}$ and $Y=\ker d_{r}/\operatorname{im}\nolimits d_{r}$. This is an $n$-complex with $d_{r,Y}=0$. Let $i\in{\mathbf{Z}}$. Consider the map $f:{\mathbf{Z}}^{n-1}\to{\mathbf{Z}}^{n},\ (a_{1},\ldots,a_{n-1})\mapsto(a_{1},\ldots,a_{r-1},i,a_{r},\ldots,a_{n-1})$. We put $H^{i}_{d_{r}}(X)=\bigoplus_{a\in{\mathbf{Z}}^{n-1}}Y^{f(a)}$. This defines a functor

Let $g:\{1,\ldots,n\}\to\{1,\ldots,m\}$ be a map and let $r\in\{1,\ldots,m\}$ such that $f^{-1}(r)=\{s\}$ for some $s\in\{1,\ldots,n\}$. Define maps

and

Then, we have a canonical isomorphism

Let $k$ be a commutative ring and $A$ a $k$-algebra. The $k$-linear functor $H^{0}:\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\to A\operatorname{\!-Mod}\nolimits$ restricted to the full subcategory of complexes $M$ with $H^{i}(M)=0$ for $i\not=0$ is an equivalence. Let $C=C_{A}:A\operatorname{\!-Mod}\nolimits\to\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)$ be an inverse, composed with the inclusion functor. By construction the resolution functor $C$ is fully faithful. It induces a functor, still denoted by $C$,

We view $\operatorname{Ho}\nolimits^{b}\bigl(\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\bigr)$ as a triangulated category with the canonical structure on $\operatorname{Ho}\nolimits^{b}({\mathcal{C}})$, where ${\mathcal{C}}$ is the additive category $\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)$. The functor $C$ is a fully faithful triangulated functor.

Assume $A$ is projective as a $k$-module. Let $X=C_{A^{\mathrm{en}}}(A)$ be a projective resolution of $A$ as an $A^{\mathrm{en}}$-module. The functor

composed with the canonical functor $2\operatorname{\!-Comp}\nolimits(A\operatorname{\!-Proj}\nolimits)\to\operatorname{Ho}\nolimits\bigl(\operatorname{Ho}\nolimits(A\operatorname{\!-Proj}\nolimits)\bigr)$ is isomorphic to $C$: we have $M\otimes_{A^{\operatorname{opp}\nolimits}}X\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{A}(M)$ for $M\in\operatorname{Ho}\nolimits^{b}(A\operatorname{\!-Mod}\nolimits)$.

Let $i\in{\mathbf{Z}}$. The functor $H^{i}:\operatorname{Ho}\nolimits^{-}(A\operatorname{\!-Proj}\nolimits)\to A\operatorname{\!-Mod}\nolimits$ induces a functor

It extends the functor $H^{i}_{d_{2}}:2\operatorname{\!-Comp}\nolimits(A\operatorname{\!-Mod}\nolimits)\to\mathrm{Comp}(A\operatorname{\!-Mod}\nolimits)$.

Let $B,B^{\prime}$ be two $k$-algebras, projective as $k$-modules. Let $L\in\operatorname{Ho}\nolimits^{b}\bigl((A\otimes B^{\operatorname{opp}\nolimits})\operatorname{\!-Mod}\nolimits\bigr)$ and $M\in\operatorname{Ho}\nolimits^{b}((B\otimes B^{\prime{\operatorname{opp}\nolimits}})\operatorname{\!-Mod}\nolimits)$. Assume the components of $M$ are projective right $B^{\prime}$-modules. We deduce from (1) an isomorphism

Note that when $A$ is coherent, then $A\operatorname{\!-Mod}\nolimits$ can be replaced by the abelian category $A\operatorname{\!-mod}\nolimits$ and $A\operatorname{\!-Proj}\nolimits$ by $A\operatorname{\!-proj}\nolimits$. If $A$ is graded, we can replace these categories by the corresponding categories of graded modules.