# Topological quantum field theories from Hecke algebras

Vladimir Fock<EMAIL_ADDRESS>IRMA, UMR 7501, Université de Strasbourg et
CNRS
7 rue René Descartes 67000 Strasbourg, France Valdo Tatitscheff
<EMAIL_ADDRESS>IRMA, UMR 7501, Université de Strasbourg et
CNRS
7 rue René Descartes 67000 Strasbourg, France Alexander Thomas a.thomas@mpim-
bonn.mpg.de Max-Planck-Institut für Mathematik Bonn, Vivatsgasse 7, 53111
Bonn, Germany

###### Abstract

We construct one-dimensional non-commutative topological quantum field
theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter
system. These TQFTs associate an invariant to each ciliated surface, which is
a Laurent polynomial for punctured surfaces. There is a graphical way to
compute the invariant using minimal colored graphs. We give explicit formulas
in terms of the Schur elements of the Hecke algebra and prove positivity
properties for the invariants when the Coxeter group is of classical type, or
one of the exceptional types $H_{3}$, $E_{6}$ and $E_{7}$.

##  1 Introduction

Iwahori–Hecke algebras (referred to as Hecke algebras in the sequel) are
remarkable associative non-commutative deformations of Coxeter groups
depending on a parameter $q$. We use these Hecke algebras to construct
topological invariants of ciliated surfaces. This construction behaves nicely
under gluing along the boundary of the surfaces and hence defines a
topological quantum field theory (TQFT).

The origin of this construction comes from the study of the space of all
functions on character varieties of surface groups. Using a triangulation of
the surface, one is led to the function space of the representation ring of a
Lie group. The Satake correspondence gives a description of this space in
terms of an affine Hecke algebra. The question arose what happens if one uses
a finite Hecke algebra. The result are the invariants presented in this paper.

Let $G$ be a finite-dimensional simple Lie group over $\mathbb{F}_{q}$, where
$q=p^{\alpha}$ is the power of a prime number. Let $H$ be a Cartan subgroup,
$B$ a corresponding Borel subgroup and $W\simeq\mathrm{Norm}(H)/H$ the Weyl
group. The Hecke algebra $\mathcal{H}_{G}^{q}$ is the algebra of functions on
$G$ invariant under left and right shift by $B$, or equivalently the algebra
of functions of the double coset $B\backslash G/B$, with product given by the
convolution divided by the order of $B$. Since as a set the Weyl group $W$ is
in bijection with $B\backslash G/B$, the Hecke algebra $\mathcal{H}_{G}^{q}$
is a deformation of the group algebra $\mathbb{C}[W]$.

The structure constants in the corresponding Hecke algebra
$\mathcal{H}_{G}^{q}$ are related to the counting of $\mathbb{F}_{q}$-points
of flag varieties associated to the corresponding algebraic group (see e.g.
[Cur88]). In a judiciously chosen basis the (modified) structure constants of
$\mathcal{H}_{G}^{q}$ are Laurent polynomial in $q$ invariant under cyclic
permutations of their indices. Subsequently, one can associate such a
structure constant to an oriented topological triangle with oriented sides
labeled with the corresponding basis elements of $\mathcal{H}_{G}^{q}$,
without the need to fix an initial edge. This construction extends naturally
to the Hecke algebras corresponding to finite Coxeter systems $(W,S)$ which
are not Weyl groups.

Now one can consider gluing two triangles along one of their edges if its
orientation and labels coincide. Summing over all possible labels for the edge
along which the gluing is performed, that is over all basis vectors of the
Hecke algebra, yields a Laurent polynomial that only depends on the labels on
the exterior edges. Repeating this procedure one obtains a way to associate a
Laurent polynomial $P\in\mathbb{Z}[v^{\pm 1}]$ to any triangulated ciliated
surface $\Sigma$ whose boundary is labeled by elements of the Weyl group $W$:

$P_{\Sigma,W}(v)=\sum\prod_{f}c_{f}(v)\leavevmode\nobreak\ ,$ (1.1)

where the sum runs over all labelings of the inner edges of the triangulation
by elements of $W$, the product runs over all faces $f$, and
$c_{f}(v)\in\mathbb{Z}[v^{\pm 1}]$ denotes the structure constant associated
to a triangle $f$ (see Section 3 for the precise definition).

The definition of the polynomial $P$ as a state sum uses a triangulation of
the surface. Associativity in the Hecke algebra gives (see 3.4):

######  Theorem A.

The polynomial invariant is independent of the triangulation. Hence it is a
topological invariant of the ciliated surface.

In Section 4, we introduce a diagrammatic way to compute the product in the
Hecke algebra when $W$ is a Weyl group. From this we get a graphical calculus
of our invariants using graphs with edges labeled by simple reflections in
$W$. These graphs are called higher laminations. They emerge from the
interpretation of the structure constants in the Hecke algebra in terms of
configurations of triples of flags. Figure 1.1 gives two examples of graphs
for $W=\mathfrak{S}_{3}$.

Figure 1.1: Examples of graphs for $W=\mathfrak{S}_{3}$

Counting higher laminations with some weight gives a diagrammatic way to
compute the polynomial invariant (see 4.19):

######  Theorem B.

The polynomial invariant for a ciliated surface $\Sigma$ and a Weyl group $W$
is given by

$P_{\Sigma,W}(Q)=\sum_{\Gamma}Q^{\operatorname{ram}(\Gamma)}$

where the sum runs over all higher laminations $\Gamma$ of type $W$ and
$Q^{\operatorname{ram}(\Gamma)}$ is the weight associated to $\Gamma$.

As a consequence we get a symmetry in the invariant that for closed surfaces
(see 4.21):

######  Corollary C.

For a closed surface, the invariant is a polynomial in $q$, where
$Q=q^{1/2}-q^{-1/2}$, invariant under the transformation $q\mapsto q^{-1}$.

In Section 5, we extend our construction to ciliated surfaces whose boundary
is labeled by elements of the Hecke algebra. Our main result is then the
following (see 5.5):

######  Theorem D.

Our construction satisfies the axioms of a $1$-dimensional TQFT as defined in
[Ati88].

Hence we construct a family of TQFTs, one for each finite Coxeter system. The
interesting feature of these TQFTs is that they are non-commutative. This
ultimately comes from the fact that the cobordisms we consider are ciliated
surfaces, whose boundary is a disjoint union of segments joining adjacent
cilia.

When the ciliated surface has only punctures and no cilium one can show that
the Laurent polynomials given by our TQFTs lie in the center of the
corresponding Hecke algebra. Since the structure of the latter is well
understood we can derive an explicit expression for the invariant associated
with a punctured surface in terms of the Schur elements of the Hecke algebra
(see 6.8):

######  Theorem E.

The polynomial invariant for punctured surfaces is given by

$P_{g,k,W}(q)=\sum_{\lambda}(\dim V_{\lambda})^{k}s_{\lambda}(q)^{2g-2+k}$

where the sum is taken over all irreducible representations $V_{\lambda}$ of
the Hecke algebra and where $s_{\lambda}$ are the corresponding Schur
elements.

For ciliated surfaces, we also give an explicit expression in terms of Schur
elements and irreducible characters, see 6.9.

From these explicit expressions and a thorough analysis of the Schur elements,
carried out in Appendix B, we can derive positivity properties of the
invariant (6.10 and 6.13):

######  Theorem F.

The polynomial invariant has positive coefficients for all Coxeter groups of
classical type and in the exceptional types $H_{3},E_{6}$ and $E_{7}$. For all
other types, the polynomial can have negative coefficients.

In type $A$ and for boundary labels with positive coefficients in the
Kazhdan–Lusztig basis, the polynomial invariant has positive coefficients.

##### Notations.

We write Hecke algebra for an Iwahori–Hecke algebra for a finite Coxeter
system. When we speak about graphs, we always mean fat graphs, i.e. graphs
embedded in some surface (planar graphs for example). We say that a polynomial
is positive if all its coefficients are positive. Furthermore, we use the
following notations:

$\Sigma_{g,k}$ |  ciliated surface (see Section 2.3)
---|---
$(W,S)$ |  Coxeter system with simple reflections $S$
$\mathcal{H}_{(W,S)},\mathcal{H}$ |  Iwahori–Hecke algebra of $(W,S)$
$v,q,Q$ |  formal parameters in the Hecke algebra linked by
|  $q=v^{-2}$ and $Q=q^{1/2}-q^{-1/2}$

##### Acknowledgments.

We warmly thank Maria Chlouveraki for her various inputs about Schur elements,
Sebastian Manecke for his contribution for the positivity, Malte Lackmann for
showing us SageMathCell, Wille Liu and Daniel Tubbenhauer for discussions. A.
Thomas acknowledges support from the Max-Planck-Institute for Mathematics in
Bonn.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Hecke algebras and their standard basis
    2. 2.2 The Kazhdan–Lusztig basis
    3. 2.3 Ciliated surfaces
  3. 3 Definition of the polynomial and first properties
    1. 3.1 On the standard structure constants in Hecke algebras
    2. 3.2 An invariant for surfaces with punctures
    3. 3.3 Gluing surfaces
    4. 3.4 Invariants of punctured surfaces at $q=1$
    5. 3.5 The case of $(W,S)=(\mathfrak{S}_{2},\\{s\\})$
  4. 4 Graphical calculus
    1. 4.1 Structure constants and flag counting
    2. 4.2 Finite higher laminations for triangles
      1. 4.2.1 Definition
      2. 4.2.2 Relations
      3. 4.2.3 Existence
      4. 4.2.4 Product in the Hecke algebra
    3. 4.3 Higher laminations for ciliated surfaces
  5. 5 Hecke topological quantum field theory
    1. 5.1 Triangle invariants and gluing revisited
    2. 5.2 Hecke topological quantum field theories
    3. 5.3 Traces in the Hecke algebra and invariants of $n$-gons
  6. 6 Explicit expression and positivity
    1. 6.1 Schur elements and Wedderburn decomposition
    2. 6.2 Central elements
    3. 6.3 Explicit expression
    4. 6.4 Positivity properties
  7. 7 Perspectives
  8. A Computation using Sage and CHEVIE
    1. A.1 Sage
    2. A.2 CHEVIE
  9. B Positivity for Schur elements

##  2 Preliminaries

We recall the definition of the Hecke algebra of a Coxeter system and
introduce its standard and Kazhdan–Lusztig bases following [EW16, Lib19], and
then set our conventions and notations for ciliated surfaces as in section 2
of [FG07].

###  2.1 Hecke algebras and their standard basis

Let $(W,S)$ be a finite Coxeter system. Given two reflections $s,t\in S$ let
$m_{st}\in\mathbb{N}\cup\\{\infty\\}$ denote the order of $st$. With this
notation111By definition of a Coxeter system, for all $s\in S$ one has
$m_{ss}=1$., $W$ can be presented as:

$W=\left\langle s\in S\leavevmode\nobreak\ |\leavevmode\nobreak\
(st)^{m_{st}}=\operatorname{id}\ \forall s,t\in S\right\rangle\ .$ (2.1)

Let $l:W\rightarrow\mathbb{N}_{\geq 0}$ and $\geq$ be respectively the length
function and the Bruhat order on $W$.

Let $v$ be an indeterminate. The Hecke algebra $\mathcal{H}_{(W,S)}$
corresponding to $(W,S)$ as defined in [Iwa64] is the associative
$\mathbb{Z}[v^{\pm 1}]$-algebra with generators $\\{h_{s}\\}_{s\in S}$ and
relations of two types, the quadratic ones:

$h_{s}^{2}=(v^{-1}-v)h_{s}+1$ (2.2)

for all $s\in S$, and the braid ones:

$h_{s}h_{r}\dots=h_{r}h_{s}\dots$ (2.3)

for all $s,t\in S$ such that $m_{st}<\infty$, in which case there are $m_{st}$
terms on each side.

Except when the explicit subscript $(W,S)$ is needed for clarity or in
explicit examples, we will drop it in what follows and simply refer to the
Hecke algebra corresponding to $(W,S)$ as $\mathcal{H}$.

######  Remark 2.1.

An equivalent, but different, way to describe the Hecke algebra is to use
generators $(T_{s})_{s\in S}$ which satisfy the same braid relations and where
the quadratic relation is

$T_{s}^{2}=(q-1)T_{s}+q\ .$ (2.4)

for another indeterminate $q$. Apart from Section 4.1, we always use the the
“normalized” version of the Hecke algebra where the quadratic relations take
the form of Equation 2.2 with $v=q^{-1/2}$.

Let $\bar{w}=s_{1}\dots s_{k}$ be a reduced expression for some $w\in W$ in
terms of $s_{1},\dots,s_{k}\in S$, and set:

$h_{\bar{w}}=h_{s_{1}}\dots h_{s_{k}}\ .$ (2.5)

By the famous result of Matsumoto [Mat64] that every reduced expression for
$w$ can be obtained from $\bar{w}$ using braid relations only, the element
$h_{\bar{w}}$ does not depend on the choice of a reduced expression for $w$
and one can define $h_{w}:=h_{\bar{w}}$. Let also $h_{e}:=1$.

######  Lemma 2.2.

The set $\\{h_{w}\\}_{w\in W}$ is a basis of $\mathcal{H}$ as a free
$\mathbb{Z}[v^{\pm 1}]$-module. It is called the standard basis of
$\mathcal{H}$.

Let $s\in S$ and $w\in W$. The multiplication in $\mathcal{H}$ can be
rewritten as:

$h_{s}h_{w}=\left\\{\begin{array}[]{lcc}h_{sw}&\mathrm{if}&w\leq sw\\\
(v^{-1}-v)h_{w}+h_{sw}&\mathrm{if}&sw\leq w\end{array}\right.\ .$ (2.6)

######  Example 2.3.

The Coxeter system of type $A_{1}$ is $(\mathfrak{S}_{2},\\{s\\})$, where
$\mathfrak{S}_{2}$ is the group of permutations of a set of two elements, and
$s$ is its generating involution. The corresponding Hecke algebra
$\mathcal{H}_{(\mathfrak{S}_{2},\\{s\\})}$ has basis $(h_{e},h_{s})$ as
$\mathbb{Z}[v^{\pm 1}]$-module, and the relations $h_{s}h_{e}=h_{e}h_{s}$ and
$h_{s}^{2}=(v^{-1}-v)h_{s}+1$ hold.

######  Example 2.4.

The Coxeter system of type $A_{2}$ is $(\mathfrak{S}_{3},\\{s,t\\})$, where
$s$ and $t$ are two transpositions generating $\mathfrak{S}_{3}$. The standard
basis of the Hecke algebra $\mathcal{H}_{(\mathfrak{S}_{3},\\{s,t\\})}$ as a
$\mathbb{Z}[v^{\pm 1}]$-module is $(h_{e},h_{s},h_{t},h_{st},h_{ts},h_{sts})$.
However as a $\mathbb{Z}[v^{\pm 1}]$-algebra,
$\mathcal{H}_{(\mathfrak{S}_{3},\\{s,t\\})}$ is generated by $h_{e},h_{s}$ and
$h_{t}$ only, where $h_{e}$ is in the center and with the quadratic relations
of Equation 2.2 for $h_{s}$ and $h_{t}$, and the braid relation:

$h_{s}h_{t}h_{s}=h_{t}h_{s}h_{t}\ .$ (2.7)

######  Example 2.5.

The Coxeter system of type $B_{2}$ is $(\mathbb{D}_{4},\\{s,t\\})$, where $s$
and $t$ are two transpositions which generate the dihedral group
$\mathbb{D}_{4}$. The standard basis of the corresponding Hecke algebra
$\mathcal{H}_{(\mathbb{D}_{4},\\{s,t\\})}$ is
$(h_{e},h_{s},h_{t},h_{st},h_{ts},h_{sts},h_{tst},h_{stst})$ as a
$\mathbb{Z}[v^{\pm 1}]$-module. The multiplication is such that $h_{e}$
commutes with $h_{s}$ and $h_{t}$, both $h_{s}$ and $h_{t}$ satisfy the
quadratic relation of Equation 2.2, and there is the following braid relation:

$h_{s}h_{t}h_{s}h_{t}=h_{t}h_{s}h_{t}h_{s}\ .$ (2.8)

###  2.2 The Kazhdan–Lusztig basis

Let $s\in S$. One can check easily that the inverse of $h_{s}$ is
$h_{s}+v-v^{-1}\in\mathcal{H}$. Since the set of all $h_{s}$ for $s\in S$
generates $\mathcal{H}$ as a $\mathbb{Z}[v^{\pm 1}]$-algebra, it follows that
for every $w\in W$, the corresponding $h_{w}$ also admits an inverse.

The morphism of $\mathbb{Z}$-modules defined by

$\iota:\left\\{\begin{array}[]{rcl}\mathcal{H}&\rightarrow&\mathcal{H}\\\
v&\mapsto&v^{-1}\\\ h_{w}&\mapsto&(h_{w^{-1}})^{-1}\end{array}\right.$ (2.9)

is a ring automorphism of $\mathcal{H}$. A cornerstone of Kazhdan–Lusztig
theory is the following theorem (see [KL79, Theorem 1.1]):

######  Theorem 2.6 (Kazhdan–Lusztig).

For all $w\in W$ there exists a unique $\iota$-self-dual element of
$\mathcal{H}$ of the form

$b_{w}=h_{w}+\sum_{z\leq w}h_{z,w}h_{z}$ (2.10)

where $h_{z,w}\in v\mathbb{Z}[v]$. Moreover the set $\\{b_{w}\\}_{w\in W}$ is
a basis of $\mathcal{H}$ as $\mathbb{Z}[v^{\pm 1}]$-module.

The Kazhdan–Lusztig polynomial $p_{z,w}$ is defined as:

$p_{z,w}=v^{l(w)-l(z)}h_{z,w}\ .$ (2.11)

######  Example 2.7.

For $s\in S$, the element $b_{s}=h_{s}+v\in\mathcal{H}$ is always self-dual
under $\iota$ and moreover it is of the form of Equation 2.10. It is clear
that $b_{e}=h_{e}=1$. Hence the Kazhdan–Lusztig basis of
$\mathcal{H}_{(\mathfrak{S}_{2},\\{s\\})}$ is the pair $(b_{e}=1,b_{s})$.

##### Positivity.

The Kazhdan–Lusztig basis enjoys numerous positivity properties which have
been shown to be a consequence of a combinatorial Hodge theory in the category
of Soergel bimodules [Wil16]. For example the Kazhdan–Lusztig polynomials
$p_{z,w}$ of Equation 2.11 are positive (meaning that
$p_{z,w}\in\mathbb{Z}_{\geq 0}[v]$), and the structure constants of the Hecke
algebra are positive when expressed in the Kazhdan–Lusztig basis: if one sets

$b_{x}b_{y}=\sum\tensor{\mu}{{}_{x}{}_{y}^{z}}b_{z}\leavevmode\nobreak\ ,$
(2.12)

then $\tensor{\mu}{{}_{x}{}_{y}^{z}}\in\mathbb{Z}_{\geq 0}[v^{\pm 1}]$.

###  2.3 Ciliated surfaces

A ciliated surface is an oriented topological surface obtained as follows:
remove $n$ disjoint open disks labeled $1,\dots,n$ out of the oriented surface
of genus $g$ with $k$ punctures, for $g,k,n\geq 0$. On the $i$-th boundary
circle add $p_{i}\geq 1$ marked points called cilia, for $i=1,\dots,n$.

Topologically, a ciliated surface is determined by its genus, the integer $k$
and the set $\\{p_{1},...,p_{n}\\}$. We will denote such a surface
$\Sigma_{g,k,\\{p_{1},...,p_{n}\\}}$ in general, and simplify the writing to
$\Sigma_{g,k}$ when it has no cilium, and to
$\Sigma_{g,\\{p_{1},...,p_{n}\\}}$ when it has no puncture. Let $c=\sum p_{i}$
be the total number of cilia.

Figure 2.1: The ciliated surface $\Sigma_{2,3,\\{2,2,1,1,3\\}}$.

In the sequel we will only consider ciliated surfaces such that $k+n\geq 1$,
and such that when $g=0$ either $k+n\geq 3$ or $k+n=2$ and $c\geq 1$ or
$k+n=1$ and $c\geq 3$, for such ciliated surfaces can be triangulated. A
triangulation $\mathsf{T}$ of $\Sigma$ is a decomposition of $\Sigma$ into
triangles such that every vertex of a triangle is either a cilium or a
puncture. Edges of $\mathsf{T}$ belonging to the boundary of $\Sigma$ are said
to be external and the others, internal. Let $\\#F(\mathsf{T})$,
$\\#E(\mathsf{T})$, $\\#E_{0}(\mathsf{T})=c$ and $\\#V(\mathsf{T})=k+c$ be
respectively the number of faces, edges (external and internal), internal
edges and vertices of $\mathsf{T}$. The Euler characteristic of the closure
$\overline{\Sigma}$ is

$\\#F(\mathsf{T})-\\#E(\mathsf{T})+\\#V(\mathsf{T})=2-2g+n\leavevmode\nobreak\
,$ (2.13)

and since $\mathsf{T}$ is a triangulation:

$3\\#F(\mathsf{T})=2\\#E(\mathsf{T})-\\#E_{0}(\mathsf{T})\leavevmode\nobreak\
.$ (2.14)

From Equation 2.13 and Equation 2.14 one deduces that:

$\left.\begin{array}[]{l}\\#E(\mathsf{T})=6g-6+2c+3(k+n)\\\
\\#F(\mathsf{T})=4g-4+c+2(k+n)\end{array}\leavevmode\nobreak\ .\right.$ (2.15)

Except when $g=0$ and $(k,n)=(0,n)$ or $(k,n)=(1,n)$ the number of
triangulations of a ciliated surface is infinite. However one can always reach
any triangulation from a reference one in a finite number of flips which
consist of replacing the diagonal of a quadrilateral formed by two adjacent
triangles with the other diagonal.

In what follows we will speak of the boundary of a ciliated surface to refer
to the disjoint union of the boundary segments connecting two adjacent cilia.

##  3 Definition of the polynomial and first properties

###  3.1 On the standard structure constants in Hecke algebras

Let $(W,S)$ be a Coxeter system and $\mathcal{H}$ the corresponding Hecke
algebra. In Section 2.1 we introduced the standard basis $\\{h_{w}\\}_{w\in
W}$ of $\mathcal{H}$ as a free $\mathbb{Z}[v^{\pm 1}]$-module. Let
$\mathcal{H}^{*}$ be the free $\mathbb{Z}[v^{\pm 1}]$-module dual to
$\mathcal{H}$ with standard dual basis $\\{h^{w}\\}_{w\in W}$.

The product in the Hecke algebra $\mathcal{H}$ corresponds to a tensor

$c\in\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes\mathcal{H}\ ,$ (3.1)

whose components in the standard bases are:

$\tensor{c}{{}_{xy}^{z}}=h^{z}(h_{x}\cdot h_{y})\;\in\mathbb{Z}[v^{\pm 1}]$
(3.2)

for $x,y,z\in W$. Let us set:

$c_{xyz}:=\tensor{c}{{}_{xy}^{z^{-1}}}\ .$ (3.3)

The notation of Equation 3.3 can be understood through the standard trace in
the Hecke algebra. A trace on $\mathcal{H}$ is a $\mathbb{Z}[v^{\pm
1}]$-linear map $\operatorname{tr}:\mathcal{H}\rightarrow\mathbb{Z}[v^{\pm
1}]$ such that
$\operatorname{tr}(hh^{{}^{\prime}})=\operatorname{tr}(h^{{}^{\prime}}h)$ for
all $h,h^{{}^{\prime}}\in\mathcal{H}$. A trace is said to be symmetrizing if
the map $h^{\prime}\mapsto\operatorname{tr}(hh^{\prime})$ is non-degenerate
for all $h\neq 0$. Symmetrizing traces are studied in general in [GP00,
Chapter 7]. The map

$\operatorname{tr}\left(\sum_{w\in W}c_{w}h_{w}\right)=h^{e}\left(\sum_{w\in
W}c_{w}h_{w}\right)=c_{e}$ (3.4)

is a symmetrizing trace on $\mathcal{H}$ called the standard trace [GP00,
Proposition 8.1.1]. It is easy to see that the standard trace on $\mathcal{H}$
satisfies:

$\operatorname{tr}(h_{x}h_{y})=\delta_{x,y^{-1}}\leavevmode\nobreak\ .$ (3.5)

######  Proposition 3.1.

For all $x,y,z\in W$, one has

$c_{xyz}=\operatorname{tr}(h_{x}h_{y}h_{z})\ .$ (3.6)

In particular, this implies that $c_{xyz}$ is cyclically symmetric:
$c_{xyz}=c_{yzx}=c_{zxy}$.

###### Proof.

By definition of the structure constants one has:

$h_{x}h_{y}=\sum_{z^{\prime}}c_{xyz^{\prime}}h_{z^{\prime-1}}\leavevmode\nobreak\
.$ (3.7)

Multiplying by $h_{z}$, taking the trace and using Equation 3.5 yields:

$\operatorname{tr}(h_{x}h_{y}h_{z})=\sum_{z^{\prime}}c_{xyz^{\prime}}\delta_{z,z^{\prime}}=c_{xyz}\leavevmode\nobreak\
,$ (3.8)

which concludes the proof. ∎

###  3.2 An invariant for surfaces with punctures

##### Triangles and structure constants.

Let us associate the structure constant $c_{xyz}\in\mathbb{Z}[v^{\pm 1}]$ to
the topological oriented triangle whose boundary is oriented in a counter-
clockwise way, and whose sides are labeled respectively by $x,y,z\in W$ as one
follows the boundary of the triangle consistently with its orientation. 3.1
implies that the result does not depend on the choice of an initial edge for
the labels. This is illustrated on the left of Figure 3.1.

In what follows we will often draw triangles as planar triangles and hence say
that an edge of the boundary oriented counterclockwise with respect to its
triangle is oriented consistently with the orientation of the triangle, and
that it is oriented against the orientation of the triangle otherwise.

In general one associates the structure constant
$c_{x^{a}y^{b}z^{c}}\in\mathbb{Z}[v^{\pm 1}]$ to the triangle with oriented
edges labeled by $x,y,z\in W$ as one follows the boundary counterclockwise,
where $a$ is $1$ (respectively, $-1$) if the edge with label $x$ is oriented
counterclockwise (respectively, clockwise) with respect to the triangle, and
mutatis mutandis for the two other edges. An example is shown on the right of
Figure 3.1.

Figure 3.1: We associate $c_{xyz}$ to the left-most triangle and
$c_{x^{-1}y^{-1}z}$ to the right-most one.

######  Proposition 3.2.

The structure constants satisfy $c_{xyz}=c_{z^{-1}y^{-1}x^{-1}}$. Thus, the
associated quantity to a triangle only depends on the oriented edges, not on
the orientation of the triangle.

###### Proof.

Consider the following map $\sigma$ on $\mathcal{H}$, defined on the standard
basis by

$\sigma(h_{w})=h_{w^{-1}}$ (3.9)

and extended by linearity. We claim that $\sigma$ is an anti-involution of
$\mathcal{H}$, i.e. that
$\sigma(h_{a}h_{b})=\sigma(h_{b})\sigma(h_{a})\;\forall a,b\in W$. Indeed it
is enough to check this on the basic relations: the quadratic relation is
invariant under $\sigma$, and for $w=sw^{\prime}$ where $s\in S$ and
$w^{\prime}$ is a reduced word, we have

$\sigma(h_{s}h_{w^{\prime}})=\sigma(h_{w})=h_{w^{-1}}=h_{w^{\prime-1}}h_{s}=\sigma(h_{w^{\prime}})\sigma(h_{s})\leavevmode\nobreak\
.$ (3.10)

By definition, we have

$h_{x}h_{y}=\sum_{z\in W}c_{xyz}h_{z^{-1}}\leavevmode\nobreak\ .$ (3.11)

Applying $\sigma$ gives

$h_{y^{-1}}h_{x^{-1}}=\sum_{z\in W}c_{xyz}h_{z}\leavevmode\nobreak\ .$ (3.12)

Hence $c_{xyz}=c_{y^{-1}x^{-1}z^{-1}}$. We conclude by cyclicity of $c_{xyz}$.
∎

##### Gluing triangles into ciliated surfaces.

Let $\mathsf{T}$ be a triangulation of the oriented topological surface
$\Sigma_{g,k}$ of genus $g$ and with $k\geq 1$ punctures.

######  Definition 3.3.

Let $P_{g,k,W,\mathsf{T}}$ be the Laurent polynomial:

$P_{g,k,W,\mathsf{T}}(v)=\sum\prod_{f\in\mathsf{T}}c_{f}(v)\in\mathbb{Z}[v^{\pm
1}]\ ,$ (3.13)

where the sum runs over all possible labelings of the edges of $\mathsf{T}$ by
elements of $W$ and the product over all faces $f$ of $\mathsf{T}$, and where
$c_{f}(v)$ is the structure constant associated to the face $f$ of
$\mathsf{T}$ as before.

More generally, we can define such a polynomial for any ciliated surface with
triangulation $\mathsf{T}$, as soon as each boundary component is labeled with
an element of $W$. The sum in Equation 3.13 runs in this case over all
possible labelings of internal edges of $\mathsf{T}$.

In order to compute $P_{g,k,W,\mathsf{T}}$ one has to choose an orientation
for the edges of $\mathsf{T}$ so that the $c_{f}(v)$ are well defined. As
before, an edge can be oriented consistently or against the orientation of a
triangle of which it is a side. The orientation of the faces of $\mathsf{T}$
is the one induced by the orientation of the surface. Since we are summing
over all possible labels of the edges, the polynomial $P_{g,k,W,\mathsf{T}}$
does not depend on the choice of orientation for the edges of $\mathsf{T}$
since changing the orientation amounts to replace each $w\in W$ by its
inverse.

The main point of our construction is the following:

######  Theorem 3.4.

The polynomial invariant does not dependent on the triangulation $\mathsf{T}$,
hence it is a topological invariant of the ciliated surface. Further, it is
preserved under reversing the orientation of the surface (and thus inverting
all the boundary data).

We will denote the invariant by $P_{\Sigma,W}$ for a ciliated surface $\Sigma$
or by $P_{g,k,W}$ in case of a punctured surface $\Sigma_{g,k}$.

###### Proof.

The second part is a direct consequence of 3.2 and the definition of
$P_{g,k,W}$.

The first part is a consequence of the associativity of the product in Hecke
algebras. Let $x,y,z\in W$. Then by definition:

$h_{x}h_{y}=\sum_{w\in W}c_{xyw}h_{w^{-1}}\ ,$ (3.14)

hence

$\sum_{w,v\in
W}c_{xyw}c_{w^{-1}zv}h_{v^{-1}}=(h_{x}h_{y})h_{z}=h_{x}(h_{y}h_{z})=\sum_{t,v\in
W}c_{yzt}c_{xt^{-1}v}h_{v^{-1}}\ ,$ (3.15)

which implies

$\sum_{w\in W}c_{xyw}c_{w^{-1}zv}=\sum_{t\in W}c_{yzt}c_{xt^{-1}v}$ (3.16)

for all $x,y,z,v\in W$. Equation 3.16 can be described graphically as in
Figure 3.2.

Figure 3.2: A consequence of associativity in Hecke algebras.

Now since any two triangulations of $\Sigma_{g,k}$ can be related via a
sequence of flips, Equation 3.16 implies the proposition. ∎

######  Example 3.5.

Let us give some examples of $P_{g,k,W}$. Later on we will come back to them
and see how to compute them.

  * •

$P_{0,3,\mathfrak{S}_{2}}(v)=P_{1,1,\mathfrak{S}_{2}}(v)=v^{2}+2+v^{-2}.$

  * •

$P_{0,4,\mathfrak{S}_{2}}(v)=v^{4}+2v^{2}+2+2v^{-2}+v^{-4}.$

  * •

$P_{0,3,\mathfrak{S}_{3}}(v)=v^{6}+2v^{4}+10v^{2}+10+10v^{-2}+2v^{-4}+v^{-6}.$

  * •

$P_{1,1,\mathfrak{S}_{3}}(v)=v^{6}+2v^{4}+4v^{2}+4+4v^{-2}+2v^{-4}+v^{-6}.$

  * •

$P_{\triangle_{x,y,z},W}(v)=c_{xyz}(v)$ where $\triangle_{x,y,z}$ stands for
the triangle seen as the ciliated surface $\Sigma_{0,0,\\{3\\}}$ with labels
$x,y,z\in W$ on the exterior edges assigned counterclockwisely.

We give additional examples in Appendix A where we explain how to use Sage and
the package CHEVIE of Gap3 to compute these polynomials.

There are three remarkable observations to be done for punctured surfaces from
these examples: the polynomials are functions of $v^{-2}=q$, they are
invariant under $q\mapsto q^{-1}$ and have positive coefficients. The first
two observations are actually properties that we will prove in 4.21, while we
will analyze the positivity in Section 6.4.

######  Remark 3.6.

Jumping ahead a bit, since for punctured surfaces the invariant polynomial
only depends on $v^{-2}=q$ we will use the variable $q$ in this case instead
of $v$ to lighten the notation. For surfaces with cilia we keep the use of
$v$.

###  3.3 Gluing surfaces

The polynomial invariant behaves nicely under gluing of two surfaces
$\Sigma_{1}$ and $\Sigma_{2}$ with same boundary data $D$ as in Figure 3.3.
Schematically:

$P(\Sigma_{1}\cup\Sigma_{2})=\sum_{D}P(\Sigma_{1},D)P(\Sigma_{2},D)\leavevmode\nobreak\
,$ (3.17)

where $P(\Sigma,D)$ denotes the polynomial invariant for the surface $\Sigma$
with boundary data $D$.

Figure 3.3: Gluing ciliated surfaces.

We can use Equation 3.17 to compute recursively the polynomial. For
$\Sigma_{1}=\Sigma_{0,0,{3}}$ a triangle and
$\Sigma_{2}=\Sigma_{g,k,\\{3,...\\}}$ a surface with a triangle boundary (that
is, a circle component of its boundary with three cilia), Equation 3.17
becomes:

$P\left(\Sigma_{g,k,\\{...\\}},D\right)=\sum_{x,y,z\in
W}c_{xyz}P\left(\Sigma_{g,k,\\{3,...\\}},D\cup\\{x,y,z\\}\right)\leavevmode\nobreak\
,$ (3.18)

where $D$ is the boundary data of the ciliated surface
$\Sigma_{g,k,\\{...\\}}$ which may have non-empty boundary, and
$D\cup\\{x,y,z\\}$ is the boundary data of $\Sigma_{g,k,\\{3,...\\}}$. Note
that we used 3.2.

Figure 3.4: Reducing the number of punctures.

For a ciliated surface with a triangle boundary carrying the labels $x,y$ and
$z$, the gluing can be used to reduce the number of punctures. To do so, we
use for $\Sigma_{2}$ a quadrilateral with boundary
$z,x,x^{\prime-1},z^{\prime-1}$. The surface after gluing is depicted in
Figure 3.4. The polynomial invariant of this quadrilateral is given by
$\sum_{w}c_{xx^{\prime-1}w}c_{zw^{-1}z^{\prime-1}}$. Hence we get from the
gluing property:

$P\left(\Sigma_{g,k+1}\backslash\triangle_{x,y,z}\right)=\sum_{x^{\prime},z^{\prime},w\in
W}c_{xx^{\prime-1}w}c_{zw^{-1}z^{\prime-1}}P\left(\Sigma_{g,k}\backslash\triangle_{x^{\prime},y,z^{\prime}}\right)\leavevmode\nobreak\
,$ (3.19)

with the obvious generalizations to the general cases.

Let us now study our invariants in two special contexts which make it easy to
gather some information: first, in the case of punctured surfaces for $q=1$
since the Hecke algebra specializes to the group algebra $\mathbb{C}[W]$ in
that case, and then in the special case $W=\mathfrak{S}_{2}$ since the Hecke
algebra of $\mathfrak{S}_{2}$ is commutative.

###  3.4 Invariants of punctured surfaces at $q=1$

For $q=1$ the Hecke algebra specializes to the group algebra $\mathbb{C}[W]$.

######  Proposition 3.7.

For a punctured surface $\Sigma_{g,k}$ the value at $q=1$ of the polynomial
is:

$P_{g,k,W}(1)=(\\#W)^{k-1}\times\\#\\{\text{solutions in }W\text{ to
}\prod_{i=1}^{g}[a_{i},b_{i}]=1\\}\leavevmode\nobreak\ .$ (3.20)

In particular $P_{0,k,W}(1)=(\\#W)^{k-1}$ which is in accordance with Example
3.5.

######  Remark 3.8.

The right-most term in the right-hand-side of Equation 3.20 can be expressed
in terms of the characters of $W$ using Frobenius’ formula (see [LZ13, theorem
A.1.10 in the Appendix by Don Zagier]). This yields:

$P_{g,k,W}(1)=(\\#W)^{2g-2+k}\times\sum_{\chi}\frac{1}{\chi(1)^{2g-2}}\leavevmode\nobreak\
.$ (3.21)

Our strategy to prove 3.7 is to develop the surface $\Sigma_{g,k}$ as a
$4g$-gon and to count explicitly the contributions.

###### Proof.

We start with the case $k=1$ and $g>0$. The surface $\Sigma_{g,1}$ is obtained
by gluing the sides of a $4g$-gon as described on the left of Figure 3.5 in
the case $g=2$. We also show a triangulation $\mathsf{T}$. Note that the $4g$
vertices of the polygon correspond to a single point in $\Sigma_{g,1}$ which
is the puncture.

Figure 3.5: Gluing of $4g$-gon

The specializations of the structure constants $c_{xyz}$ of the Hecke algebra
at $q=1$ are equal to 1 if $xyz=1$ in $W$ and to zero otherwise. In order to
get non-zero contributions to the polynomial invariant specialized at $q=1$,
the label on a side of a face $f$ of $\mathsf{T}$ is determined by the labels
on the two other sides of $f$. Let now $a_{1},b_{1},...,a_{g},b_{g}\in W$ be
the labels assigned to the edges of the $4g$-gon. Let $\mathsf{T}$ be a
triangulation of the $4g$-gon whose inner edges are all incident to a fixed
vertex as on the left of Figure 3.5. The labels on the inner edges of
$\mathsf{T}$ are completely determined by the $a_{i}$ and $b_{i}$, otherwise
the corresponding contribution to the invariant vanishes.

There is the following additional constraint:

$\prod_{i=1}^{g}[a_{i},b_{i}]=1\leavevmode\nobreak\ ,$ (3.22)

which can be understood as the consequence of the fact that the products of
labels assigned to each triangle whose boundary is oriented counterclockwise
has to be $1$ if one considers all the triangles glued together along edges as
on the left of Figure 3.5.

Every solution of Equation 3.22 has contribution $1$ to the polynomial, and
all other choices of $a_{i}$ and $b_{i}$ do not contribute. Therefore we get
the proposition in the case $k=1$.

For $k>1$, we add in our picture $k-1$ marked points in the middle of the
polygon. We complete to a triangulation as shown on the right of Figure 3.5.
In particular there is a path of length $k-1$ connecting all $k$ punctures
shown in red in Figure 3.5. To each edge of this path, we associate a new
label $x_{i}\in W$. One easily checks that these $x_{i}$ together with the
$a_{j}$ and $b_{j}$ from above uniquely determines the labels assigned to each
edge of the triangulation. The only relation is Equation 3.22 from above.
Having fixed the data on the boundary of the $4g$-gon we have $(\\#W)^{k-1}$
free choices which contribute by $1$ to our polynomial.

Eventually for $g=0$ we represent $\Sigma_{0,k}$ as the gluing of a $2k-2$-gon
with boundary $a_{1},a_{1}^{-1},a_{2},a_{2}^{-1},...,a_{k-1},a_{k-1}^{-1}$
with a triangulation similar as the ones of Figure 3.5. The labels on the
inner edges are determined by the boundary data and the constraint on the
boundary is always satisfied. So we get $P_{0,k,W}(1)=(\\#W)^{k-1}$. ∎

###  3.5 The case of $(W,S)=(\mathfrak{S}_{2},\\{s\\})$

The choice $(W,S)=(\mathfrak{S}_{2},\\{s\\})$ is a very special one since the
Coxeter group in that case is abelian. It has strong consequences on the
polynomial invariant we constructed since the latter then only depends on the
number of faces of the triangulations, that is, $4g-4+2k$.

The Hecke algebra $\mathcal{H}_{(\mathfrak{S}_{2},\\{s\\})}$ is the free
$\mathbb{Z}[v^{\pm 1}]$-module of rank two with standard basis $h_{e},h_{s}$.
The only non-trivial multiplication is the quadratic relation for $h_{s}$
given in Equation 2.2, hence the standard structure constants are:

$\left\\{\begin{array}[]{l}c_{eee}=1\\\ c_{ees}=c_{ese}=c_{see}=0\\\
c_{ess}=c_{sse}=c_{ses}=1\\\
c_{sss}=v^{-1}-v=q^{1/2}-q^{-1/2}\end{array}\right.$ (3.23)

From 3.4 we know that the polynomial invariant associated to a surface
$\Sigma_{g,k}$ is independent of the triangulation of $\Sigma_{g,k}$. For the
sphere with three punctures $\Sigma_{0,3}$ and the one-puncture torus
$\Sigma_{1,1}$ we use the triangulations shown in Figure 3.6.

Figure 3.6: A triangulation of $\Sigma_{0,3}$ on the left and of
$\Sigma_{1,1}$ on the right.

In the case of $\Sigma_{0,3}$, the formula of Equation 3.13 gives:

$P_{0,3,\mathfrak{S}_{2}}=c_{eee}^{2}+3c_{ees}^{2}+3c_{ess}^{2}+c_{sss}^{2}\leavevmode\nobreak\
.$ (3.24)

We have used the fact that the $c_{xyz}$ are invariant under permutations of
the indices and that each element in $\mathfrak{S}_{2}$ is its own inverse.
Eventually we get:

$P_{0,3,\mathfrak{S}_{2}}=q+2+q^{-1}\leavevmode\nobreak\ .$ (3.25)

One also easily computes that:

$P_{1,1,\mathfrak{S}_{2}}=q+2+q^{-1}\leavevmode\nobreak\ .$ (3.26)

These were our first two cases of Example 3.5.

######  Proposition 3.9.

The polynomial $P_{g,k,\mathfrak{S}_{2}}$ does not depend on the pair $(g,k)$
but only on the (absolute value of the) Euler characteristic $2g-2+k$.

###### Proof.

We use triangulation of Figure 3.5. The mismatch between different pairs
$(g,k)$ with fixed value of $2g-2+k$ can then only come from the different
ways to glue the sides of the $4g$-gon, which is irrelevant in the case of
$(W,S)=(\mathfrak{S}_{2},\\{s\\})$ since it is abelian. ∎

######  Proposition 3.10.

One has:

$P_{g,k,\mathfrak{S}_{2}}(q)=(1+q)^{2g-2+k}+(1+q^{-1})^{2g-2+k}\leavevmode\nobreak\
.$ (3.27)

By 3.9 it is sufficient to compute the polynomial for the punctured spheres
$\Sigma_{0,k}$. We prove in fact a more general formula for the sphere with
$h$ punctures and triangle boundary $\Sigma_{0,k,\\{3\\}}$, using induction
and the gluing formula (3.19):

######  Lemma 3.11.

For all $k\geq 3$, we have:

$\displaystyle P(S_{0,k}\backslash\triangle_{e,e,e})$
$\displaystyle=(1+q)^{k-4}+(1+q^{-1})^{k-4}$ (3.28) $\displaystyle
P(S_{0,k}\backslash\triangle_{s,e,e})$
$\displaystyle=q^{1/2}(1+q)^{k-4}-q^{-1/2}(1+q^{-1})^{k-4}$ (3.29)
$\displaystyle P(S_{0,k}\backslash\triangle_{s,s,e})$
$\displaystyle=q(1+q)^{k-4}+q^{-1}(1+q^{-1})^{k-4}$ (3.30) $\displaystyle
P(S_{0,k}\backslash\triangle_{s,s,s})$
$\displaystyle=q^{3/2}(1+q)^{k-4}-q^{-3/2}(1+q^{-1})^{k-4}$ (3.31)

###### Proof.

The proof is by induction on $k$. For $k=3$, we get the structure constants
$c_{xyz}$ which are our polynomial for a triangle $\triangle_{x,y,z}$. For the
heredity, we use the gluing property Equation 3.19. Let us give the prove of
the heredity for Equation 3.28 and Equation 3.31. The other two are similar.

For boundary data $(e,e,e)$ the gluing formula (3.19) gives

$P(S_{0,k+1}\backslash\triangle_{e,e,e})=P(S_{0,k}\backslash\triangle_{e,e,e})+P(S_{0,k}\backslash\triangle_{s,s,e})\leavevmode\nobreak\
.$ (3.32)

Hence by the induction hypothesis, we get

$\begin{split}P(S_{0,k+1}\backslash\triangle_{e,e,e})&=(1+q)^{k-4}+(1+q^{-1})^{k-4}+q(1+q)^{k-4}+q^{-1}(1+q^{-1})^{k-4}\\\
&=(1+q)^{k-3}+(1+q^{-1})^{k-3}\leavevmode\nobreak\ .\end{split}$ (3.33)

For the triangle data $\triangle_{s,s,s}$, the gluing formula (3.19) gives

$\begin{split}P(S_{0,k+1}\backslash\triangle_{s,s,s})=&P(S_{0,k}\backslash\triangle_{s,e,e})+2(q^{1/2}-q^{-1/2})P(S_{0,k}\backslash\triangle_{s,s,e})+\\\
&(q-1+q^{-1})P(S_{0,k}\backslash\triangle_{s,s,s})\leavevmode\nobreak\
.\end{split}$ (3.34)

Using the induction hypothesis we get precisely Equation 3.31 for $k+1$. ∎

###### Proof of 3.10.

Using the gluing formula (3.18) once more we obtain:

$\begin{split}P(S_{0,k})&=P(S_{0,k}\backslash\triangle_{e,e,e})+3P(S_{0,k}\backslash\triangle_{s,s,e})+(q^{1/2}-q^{-1/2})P(S_{0,k}\backslash\triangle_{s,s,s})\\\
&=(1+q)^{k-4}(1+3q+q^{2}-q)+(1+q^{-1})^{k-4}(1+3q^{-1}+q^{-2}-q^{-1})\\\
&=(1+q)^{k-2}+(1+q^{-1})^{k-2}\leavevmode\nobreak\ .\end{split}$ (3.35)

∎

##  4 Graphical calculus

In this section, we present a diagrammatic interpretation of our polynomial.
In particular we give a graphical way to multiply elements in the Hecke
algebra. In this section, we work with a Coxeter system $(W,S)$ associated to
a Weyl group.

We start from the observation that the structure constants in the Hecke
algebra are linked to configurations of flags. From that, we define graphs
with labeled edges which count the contributions to our polynomial.

###  4.1 Structure constants and flag counting

There is a well-known link between the structure constants in the Hecke
algebra and triples of flags (see e.g. [Cur88, Proposition 2.2]). We present a
short way to get this link.

In Section 2.1, we introduced the Hecke algebra using generators and
relations. However the original definition by Iwahori [Iwa64] is geometric.
Consider a finite field $\mathbb{F}_{q}$, a simple Lie group $G$ over
$\mathbb{F}_{q}$ and fix a Borel subgroup $B$ of $G$. The Hecke algebra
$\mathcal{H}^{q}_{G}$ is the algebra of functions on $G$ invariant under left
and right shift by $B$. Multiplication is given by the convolution divided by
the order of the Borel subgroup $\\#B$. Note that the double quotient
$B\backslash G/B$ is in bijection with $W$. This is why the Hecke algebra has
a basis $(T_{w})_{w\in W}$ parameterized by the Weyl group. In this geometric
approach the generators $(T_{s})_{s\in S}$ satisfy the braid relations and the
quadratic relation reads:

$T_{s}^{2}=(q-1)T_{s}+q\leavevmode\nobreak\ .$ (4.1)

The structure constants $\tensor{C}{{}_{xy}^{z}}$ are defined by

$T_{x}T_{y}=\sum_{z\in W}\tensor{C}{{}_{xy}^{z}}(q)T_{z}\leavevmode\nobreak\
.$ (4.2)

From the definition of the convolution product, it follows that

$\tensor{C}{{}_{xy}^{z}}(q)=\\#\\{h\in x\mid h^{-1}g\in
y\\}/\\#B\leavevmode\nobreak\ ,$ (4.3)

where we interpret $x,y$ and $z$ as elements of $W\cong B\backslash G/B$ and
where $g$ is any representative of $z\in B\backslash G/B$.

The quotient $G/B$ is called the flag variety of $G$. For
$G=\operatorname{GL}_{n}(\mathbb{C})$ this is the space of complete flags in
$\mathbb{C}^{n}$. Recall the isomorphism $G\backslash(G/B)^{2}\cong
B\backslash G/B$ given by $h\mapsto(F_{0},hF_{0})$ where $F_{0}\in G/B$ is the
class of the unit element in $G$. Combining this isomorphism with $B\backslash
G/B\cong W$, we see that the relative positions of two flags is described by
the Weyl group. Using this isomorphism we can rewrite the formula for the
structure constants as

$\begin{split}\tensor{C}{{}_{xy}^{z}}(q)&=\\#\\{h\in
G\mid(F_{0},hF_{0})=x,(F_{0},h^{-1}gF_{0})=y\\}/\\#B\\\ &=\\#\\{h\in
G\mid(F_{0},hF_{0})=x,(hF_{0},gF_{0})=y\\}/\\#B\leavevmode\nobreak\
.\end{split}$ (4.4)

Taking into account that $B$ acts freely on the right on the set $\\{h\in
x\mid h^{-1}g\in y\\}$ and denoting $gF_{0}$ by $F_{2}$ and $hF_{0}$ by
$F_{1}$, we finally get:

######  Proposition 4.1.

The structure constants $\tensor{C}{{}_{xy}^{z}}(q)$ count the configuration
of three flags with prescribed relative positions:

$\tensor{C}{{}_{xy}^{z}}(q)=\\#\\{F_{1}\in
G/B\mid(F_{0},F_{1})=x,(F_{1},F_{2})=y\\}\leavevmode\nobreak\ ,$ (4.5)

where $(F_{0},F_{2})=z$.

######  Example 4.2.

Let $G=\operatorname{SL}(2)$. Then $G/B=\mathbb{P}^{1},W=\\{1,s\mid
s^{2}=1\\}$,
$\tensor{C}{{}_{11}^{1}}=1,\tensor{C}{{}_{11}^{s}}=\tensor{C}{{}_{s1}^{1}}=\tensor{C}{{}_{1s}^{1}}=0$
and $\tensor{C}{{}_{1s}^{s}}=\tensor{C}{{}_{s1}^{s}}=1$ obviously. To compute
$\tensor{C}{{}_{ss}^{1}}(q)$, choose $F_{0}=F_{2}=\infty$. Then
$\tensor{C}{{}_{ss}^{1}}(q)=\\#\\{p\in\mathbb{P}^{1}(\mathbb{F}_{q})\mid
p\neq\infty\\}=q$. To compute $\tensor{C}{{}_{ss}^{s}}(q)$, choose $F_{0}=0$
and $F_{2}=\infty$. Then
$\tensor{C}{{}_{ss}^{s}}(q)=\\#\\{p\in\mathbb{P}^{1}(\mathbb{F}_{q})\mid p\neq
0,\infty\\}=q-1$.

###  4.2 Finite higher laminations for triangles

In this subsection, we present a graphical way to compute the product in the
Hecke algebra in the standard basis. Our graphs are very similar to the ones
that appear in the article [EW16] by Elias–Williamson, in the context of the
categorification of the Hecke algebras in terms of Soergel bimodules.

####  4.2.1 Definition

We use the normalized version of the Hecke algebra where the quadratic
relation reads $h_{s}^{2}=1+Qh_{s}$ where $Q=v-v^{-1}=q^{1/2}-q^{-1/2}$.

The idea is to use the interpretation of the structure constants as triples of
flags from the previous subsection. The transition between two flags with
given relative position is decomposed into elementary moves. This corresponds
to a decomposition of a product into elementary ones involving simple elements
only. For example the quadratic relation is graphically given by the following
picture:

Figure 4.1: Graphical multiplication in Hecke algebra

More generally for $x,y\in W$ the product $h_{x}h_{y}$ is represented as a
formal sum of graphs with edges labeled by simple reflections. First one has
to fix reduced expressions of $x$ and $y$, using the simple reflections. Each
simple element $h_{s}$ is represented graphically by an edge labeled by $s\in
S$. Elementary computations in $\mathcal{H}$ are either braid or quadratic
relations, hence the vertices of the graph are of two types:

  * $\bullet$

trivalent with the three edges carrying the same label, called ramification
point,

  * $\bullet$

of braid type at the crossing of edges carrying the labels $s$ and $t$ such
that $(st)^{m}=e$ in $W$. In that case there are $m$ incident edges of type
$s$ and $m$ of type $t$ which alternate (the cases $m=2$ and $m=3$ are drawn
in the middle and on the right of Figure 4.2).

Figure 4.2: Vertex types

######  Definition 4.3.

A finite higher lamination of type $(W,S)$ (in short higher lamination) on a
triangle $t$ is an equivalence class of planar graphs $\Gamma\subset t$ with
edges labeled by elements in $S$, and satisfying the following criteria:

  1. 1.

each vertex is either trivalent or of braid type,

  2. 2.

the edges of $\Gamma$ intersect the boundary of $t$ transversally,

  3. 3.

reading the labels along an edge of $t$ gives a reduced word in $W$,

  4. 4.

the graph is minimal in the sense described below.

The equivalence class is generated by isotopy and the relations (1.1), (2.1),
(2.2), (3.1) and (3.2) described below in Section 4.2.2.

######  Remark 4.4.

We call these graphs finite higher laminations because they generalize
rational bounded measured laminations as described in [FG07]. The rough idea
is that one recovers rational bounded measured laminations from the higher
laminations for the affine Weyl group $\widehat{A}_{1}=\langle s,t\mid
s^{2}=t^{2}=1\rangle$ after removing the singular leaves. Details of this
correspondence and higher laminations corresponding to affine Coxeter systems
will be discussed in details in a forthcoming publication. Aspects of higher
laminations as tropical points of higher Teichmüller spaces and
generalizations of rational measured laminations have been studied in [Xie13]
and [Le16].

##### Minimality.

Let $\Gamma$ be an $S$-labeled graph on an oriented triangle $t$ satisfying
the first three conditions of Definition 4.3, and let $x,y,z$ be the elements
of $W$ corresponding to the reduced words on the sides of $t$. A configuration
of flags on $t\backslash\Gamma$, one flag for each face, is called valid if
the relative position of two adjacent flags is given by the simple element
which appears as the label on the edge between the two corresponding faces.

######  Definition 4.5.

The graph $\Gamma$ is minimal if for all triple of flags, one flag for each
vertex of $t$, there is at most one valid configuration of flags on
$t\backslash\Gamma$ which extends the triple of flags.

An example of a non-minimal graph is given in Figure 4.3: the flag
corresponding to the inner component of $t\backslash\Gamma$ is not uniquely
determined by the flags around.

Figure 4.3: Example of a non-minimal graph

######  Remark 4.6.

A simple criterion to decide whether a graph is minimal or not is still
lacking. In particular, we would like to find a minimality criterion for
general Coxeter systems $(W,S)$ for which flags are not defined.

####  4.2.2 Relations

Given a triple of flags, there might be several minimal labeled graphs
realizing the configuration. In other words, the decomposition into elementary
moves between flags is not unique. For example, given two flags in relative
position $w\in W$ any reduced expression for $w$ is a working decomposition
into elementary moves. Hence we have to quotient out by relations in order to
count each triple of flags only once in 4.1. We describe here a set of
relations that we conjecture to be complete.

There are relations involving one, two and three different labels (also called
“colors”). The one-color relation is given by the following picture:

Figure 4.4: One-color relation

The two-color relations are of two kinds: the first states that one can
simplify two neighbor vertices when they are of the same braid type, as shown
on the left of Figure 4.5 for $m=3$ (for $m=2$ this gives the second
Reidemeister move). The second relation describes how to glide a trivalent
vertex through a vertex of braid type and is shown on the center (resp. right)
of Figure 4.5 for $m=2$ (resp. $m=3$).

Figure 4.5: Two-color relation

Eventually there is a three-color relation for each parabolic subgroup of rank
3 in $W$, displayed in [EW16, Section 5]. Two examples are presented in Figure
4.6: the upper-one is the relation corresponding to a subgroup of type
$A_{1}\times A_{1}\times A_{1}$ (it is the third Reidemeister move) and the
lower-one corresponds to a subgroup of type $A_{3}$.

Figure 4.6: Three-color relation

######  Remark 4.7.

All the relations of [EW16] which do not imply a loose end are relations for
our higher laminations. It seems there is a link between the graphical
calculus in the Hecke algebra we introduced and the graphical calculus of
Elias–Williamson describing morphisms between Bott–Samelson bimodules, even we
do not understand the correspondence yet.

Figure 4.7 shows the equivalence of two seemingly different graphs through
these relations.

Figure 4.7: An example of applying relations

The two-color relation (2.1) and all three-color relations give a complete
list of relations for reduced expressions in $W$: two reduced expressions of
an element of $W$ can be related through a finite number of these relations
[Ron09, Chapter 2, §5]. This is enough to show that the latter are all the
relations we need in the case of higher laminations without ramification
points.

In general, we conjecture the following:

######  Conjecture 4.8.

The relations of above are complete: two minimal labeled graphs corresponding
to the same triple of flags on the vertices of the triangle can be related
through a finite sequence of them.

The conjecture seems reasonable since we only have to look for relations
involving ramification points. Our one-color relation describes how two
ramification points interact and the two-color relation (2.2) describes the
interaction between a ramification point and a vertex of braid type. It does
not seem too presumptuous to expect that these are the only cases one needs to
consider.

####  4.2.3 Existence

We show that we can associate a set of representatives $\Gamma$ of higher
laminations to any multiplication in the Hecke algebra. The non-trivial part
is to show minimality, that is, the existence of a unique configuration of
flags on the connected components of $t\backslash\Gamma$ (henceforth called
the faces of $\Gamma$).

Let $\Gamma$ be a representative of a higher lamination. Recall that an
assignment of flags to the faces of $\Gamma$ is valid if the relative position
of any two adjacent flags is the label of the edge between the corresponding
faces.

We start with some easy results, the proof of which is left as an exercise for
the interested reader.

######  Lemma 4.9.

Let $v,w\in W$ such that $vw$ is not reduced. Then there are reduced
expressions for $v$ and $w$ of the form $v=v^{\prime}s$ and $w=sw^{\prime}$
where $s\in S$ is a simple reflection.

######  Corollary 4.10.

For $s,t\in S$, consider a reduced expression for $w\in W$ of the form
$w=w^{\prime}u$ where $u$ is a word in the letters $s$ and $t$ with maximal
length. Denote by $\bar{u}$ the word obtained from $u$ by exchanging $s$ and
$t$. Then $w^{\prime}\bar{u}$ is also reduced.

The two other lemmas concern the extension of flag configurations.

######  Lemma 4.11.

Given two flags in opposite faces with respect to a vertex a braid type, and
whose relative position is compatible with the local structure around the
vertex, there is a unique way to associate a valid configuration of flags
around the vertex extending the initial data.

In type $A$ this can be verified easily by a computation in $\mathbb{P}^{2}$.

######  Lemma 4.12.

Given two flags $F_{0}$ and $F_{1}$ in relative position $w$ and a reduced
word $w=s_{i_{1}}\cdots s_{i_{k}}$, there is a unique sequence of flags
$F_{0}=F_{i_{1}},F_{i_{2}},...,F_{i_{k+1}}=F_{1}$ such that
$(F_{i_{l}},F_{i_{l+1}})=s_{i_{l}}$ for all $l=1,...,k$.

Any product $h_{x}h_{y}$ in $\mathcal{H}$ of elements of the standard basis
can be decomposed into elementary moves which are either a quadratic relation
or a braid relation, and we have diagrams for these. By juxtaposition we get a
collection of graphs corresponding to the product $h_{x}h_{y}$.

######  Proposition 4.13.

The graphs obtained from expressing $h_{x}h_{y}$ in the standard basis of
$\mathcal{H}$ are representatives of higher laminations.

Figure 4.8: Local structure around $F_{1}$. Left: simple edge. Middle:
ramification point. Right: braid vertex.

###### Proof.

Let $x,y\in W$ and $h_{x},h_{y}$ be the corresponding elements in
$\mathcal{H}$. The latter can be written as a product of the $h_{s_{i}}$
corresponding to the elements in $S$. Let us fix such a decomposition for
$h_{x}$ and $h_{y}$. As one computes the product $h_{x}h_{y}$ in
$\mathcal{H}$, one uses braid relations and quadratic relations until all the
terms one obtains are products of $h_{s_{i}}$ corresponding to reduced words
in $W$. Each quadratic relation increments the number of terms in the
expression by one. Let us consider one of the terminal terms and assume it
corresponds to $h_{z}$ for $z\in W$. Let $\Gamma$ be the corresponding graph
on a triangle $t$ whose sides are respectively labeled by the reduced words
corresponding to $y$, $x$ and $z^{-1}$ as one reads the boundary
counterclockwise.

By 4.1, we can associate a triple of flags $(F_{0},F_{1},F_{2})$ to the
vertices of $t$ such that $(F_{2},F_{1})=y$, $(F_{1},F_{0})=x$ and
$(F_{2},F_{0})=z$ in $G\backslash(G/B)^{2}$.

The only non-trivial fact to check is that $\Gamma$ is minimal, i.e. that we
can extend the triple of flags to a unique valid configuration. We prove this
by induction on the number of faces of $\Gamma$. The initialisation with one
face is trivial since it corresponds to $1\times 1=1$.

In general, by Lemma 4.12 we have flags assigned to all boundary faces. We can
assume that the local configuration around the vertex with flag $F_{1}$ is
given by one of the three cases shown in Figure 4.8. This is because away from
that vertex, we can only apply braid relations (since $x$ and $y$ are reduced)
which amount to choose another reduced expression for $x$ or $y$.

Case 1: There is a simple edge next to $F_{1}$.

The region just this edge $F_{1}$ already has an associated flag
$F_{1}^{\prime}$ since it is a boundary face. The restriction of $\Gamma$ to
the triangle $(F_{0},F_{1}^{\prime},F_{2})$ has strictly less faces and we can
choose the boundary of this new triangle so that it is still a higher
lamination: first one can clearly assume that $\Gamma$ intersects the boundary
of the new triangle transversely and the sides chosen close to the sides of
the original triangle so that the words assigned to them are the words on the
sides of the original triangle with the letter corresponding to the edge
between $F_{1}$ and $F_{1}^{\prime}$ removed. Thus we can apply the induction
hypothesis.

Case 2: There is a ramification point next to $F_{1}$.

As in case 1, the two flags below $F_{1}$ are determined by Lemma 4.12. Let
$F_{1}^{\prime}$ be one of these flags. The restriction of $\Gamma$ to
$(F_{0},F_{1}^{\prime},F_{2})$ has strictly less faces, we can again assume
that it is a higher lamination and hence apply the induction hypothesis.

Case 3: There is a braid vertex next to $F_{1}$.

Since all boundary faces are already assigned a flag, there are two opposite
faces around the braid vertex carrying a a flag and this assignment is
consistent with the local structure of $\Gamma$. By Lemma 4.11, we can
uniquely extend this configuration around all the regions around the braid
vertex. Let $F_{1}^{\prime}$ be the flag below the braid vertex as shown on
the right of Figure 4.8. Let us show that we can apply the induction
hypothesis to the restriction of $\Gamma$ to the triangle formed by
$(F_{0},F_{1}^{\prime},F_{2})$. The only non-trivial fact to check is that the
expressions induced on the edges $(F_{0},F_{1}^{\prime})$ and
$(F_{1}^{\prime},F_{2})$ are reduced. By changing the reduced expression for
$x$ or $y$, we can assume that the number of incoming edges in the braid
vertex is maximal. The fact that the restriction of $\Gamma$ is reduced then
follows from 4.10. ∎

Given a representative $\Gamma$ of a higher lamination one wonders how many
triples of flags modulo $G$ correspond to it. Let $\operatorname{ram}(\Gamma)$
be the number of trivalent vertices of $\Gamma$ and recall that we denoted
$l:W\rightarrow\mathbb{N}$ the Bruhat length on $(W,S)$.

######  Proposition 4.14.

Let $\Gamma$ be a representative of a higher lamination on a triangle $t$
inducing the labels $x,y,z\in W$ on the sides, and $F_{0}$, $F_{2}$ two flags
in relative position $z$. The number of flags $F_{1}$ such that
$(F_{0},F_{1},F_{2})$ extends to a valid configuration on $t\backslash\Gamma$
is:

$Q^{\operatorname{ram}(\Gamma)}q^{1/2(l(x)+l(y)-l(z))}\leavevmode\nobreak\ .$
(4.6)

The proof is similar to the one for 4.13, with a distinction of the same three
cases from Figure 4.8. Note that by 4.13, any contribution from a triple of
flags appears at most once.

###### Proof.

We reason by induction on the number of faces of $\Gamma$. The proposition is
true for the empty graph.

Case 1: There is a simple edge next to $F_{1}$.

The restriction $\Gamma^{\prime}$ of $\Gamma$ to the triangle
$(F_{0},F_{1}^{\prime},F_{2})$ is a representative of a higher lamination with
strictly less faces. We have
$\operatorname{ram}(\Gamma^{\prime})=\operatorname{ram}(\Gamma)$,
$z^{\prime}=z$, $l(x^{\prime})=l(x)-1$ and $l(y^{\prime})=l(y)-1$. The only
restriction on $F_{1}$ is its relative position to $F_{1}^{\prime}$ which is a
simple reflection. If $F_{1}^{\prime}$ is fixed this are $q$ flags satisfying
this constraint. Using the induction hypothesis, the number of possible
$F_{1}$’s is:

$q\times
Q^{\operatorname{ram}(\Gamma^{\prime})}q^{1/2(l(x^{\prime})+l(y^{\prime})-l(z^{\prime}))}=Q^{\operatorname{ram}(\Gamma)}q^{1/2(l(x)+l(y)-l(z))}\leavevmode\nobreak\
.$ (4.7)

Case 2: There is a ramification point next to $F_{1}$.

Let $F_{1}^{\prime}$ be a flag in one of the regions below $F_{1}$ (say on the
boundary of $y$). The restriction $\Gamma^{\prime}$ of $\Gamma$ to the
triangle $(F_{0},F_{1}^{\prime},F_{2})$ is a representative of a higher
lamination with strictly less faces. We have
$\operatorname{ram}(\Gamma^{\prime})=\operatorname{ram}(\Gamma)-1$,
$z^{\prime}=z$, $x^{\prime}=x$ and $l(y^{\prime})=l(y)-1$. If $F_{1}^{\prime}$
is fixed the, by Lemma 4.12 the flag $F_{1}^{\prime\prime}$ next to
$F_{1}^{\prime}$ is uniquely determined. The only restriction on $F_{1}$ is
its relative position to $F_{1}^{\prime}$ and $F_{1}^{\prime\prime}$. Since
these are the same simple reflection, there are $q-1$ possible $F_{1}$’s.
Using the induction hypothesis and the fact that $Q=q^{1/2}-q^{-1/2}$, the
number of possible $F_{1}$’s is:

$(q-1)\times
Q^{\operatorname{ram}(\Gamma^{\prime})}q^{1/2(l(x^{\prime})+l(y^{\prime})-l(z^{\prime}))}=Q^{\operatorname{ram}(\Gamma)}q^{1/2(l(x)+l(y)-l(z))}\leavevmode\nobreak\
.$ (4.8)

Case 3: There is a braid vertex next to $F_{1}$.

Let $F_{1}^{\prime}$ be the flag assigned to the face of $\Gamma$ below
$F_{1}$ as shown in Figure 4.8. The restriction $\Gamma^{\prime}$ of $\Gamma$
to the triangle $(F_{0},F_{1}^{\prime},F_{2})$ is a representative of a higher
lamination with strictly less faces (as follows from the same reasoning as in
the proof of 4.13). Furthermore, all parameters
$\operatorname{ram}(\Gamma^{\prime}),l(x^{\prime}),l(y^{\prime})$ and
$l(z^{\prime})$ are the same as the ones corresponding to $\Gamma$. If we know
the flag $F_{1}^{\prime}$, by Lemma 4.12 this fixes all boundary flags of the
triangle $(F_{0},F_{1}^{\prime},F_{2})$. In particular, this gives two flags
in opposite regions around the braid vertex. By Lemma 4.11, this determines
uniquely $F_{1}$. We conclude by the induction hypothesis. ∎

A consequence of this last proposition is that the number of ramification
points $\operatorname{ram}(\Gamma)$ does not depend on the representative of
the higher lamination. One can easily check that all the relations of Section
4.2.2 preserve the number of ramification points.

####  4.2.4 Product in the Hecke algebra

We now describe the graphical computation of a product $h_{x}h_{y}$ in the
Hecke algebra, for some $x,y\in W$.

Let us choose reduced expressions for $x$ and $y$ and write them on the two
upper sides of the triangle following the counterclockwise orientation of the
boundary. For each triple of flags appearing in the product as described in
4.1 we choose a corresponding graph $\Gamma$. By 4.13, these graphs represent
higher laminations. On the last edge of the triangle we read a reduced
expression for some element $h_{z}(\Gamma)\in\mathcal{H}$.

######  Theorem 4.15.

We have

$h_{x}h_{y}=\sum_{\Gamma}Q^{\operatorname{ram}(\Gamma)}h_{z}(\Gamma)\leavevmode\nobreak\
,$ (4.9)

where the sum runs over all isotopy classes of graphs $\Gamma$ coming from
triples of flags associated to $h_{x}h_{y}$. Assuming 4.8, the sum can be
taken over all higher laminations inducing $x$ and $y$ on the two upper sides
of the triangle.

###### Proof.

Recall the structure constants $\tensor{C}{{}_{xy}^{z}}$ of the Hecke algebra
with quadratic relation $T_{s}^{2}=(q-1)T_{s}+q$. Combining 4.1 with 4.14, we
get

$\begin{split}\tensor{C}{{}_{xy}^{z}}&=\\#\\{F_{1}\in
G/B\mid(F_{0},F_{1})=x,(F_{1},F_{2})=y\\}\\\
&=\sum_{\Gamma_{z}}Q^{\operatorname{ram}(\Gamma_{z})}q^{1/2(l(x)+l(y)-l(z))}\leavevmode\nobreak\
,\end{split}$ (4.10)

where the sum is taken over all graphs $\Gamma_{z}$ which induce $z$ on the
third side of the triangle.

Let us now relate the structure constants of $\mathcal{H}$ in the basis
$(T_{w})_{w\in W}$ to those in the standard basis $(h_{w})_{w\in W}$. The two
basis are linked by $T_{w}=q^{l(w)/2}h_{w}$. Hence:

$\tensor{c}{{}_{xy}^{z}}=q^{-1/2(l(x)+l(y)-l(z))}\tensor{C}{{}_{xy}^{z}}\leavevmode\nobreak\
.$ (4.11)

Therefore, we get:

$h_{x}h_{y}=\sum_{z\in W}\tensor{c}{{}_{xy}^{z}}h_{z}=\sum_{z\in
W}\sum_{\Gamma_{z}}Q^{\operatorname{ram}(\Gamma_{z})}h_{z}=\sum_{\Gamma}Q^{\operatorname{ram}(\Gamma)}h_{z}(\Gamma)\leavevmode\nobreak\
.$ (4.12)

Assuming 4.8 we can uniquely associate a higher lamination to each triple of
flags appearing in $h_{x}h_{y}$. ∎

Let us give a concrete example of the graphical interpretation of a product in
the Hecke algebra $\mathcal{H}_{(\mathfrak{S}_{3},\\{s,t\\})}$.

######  Example 4.16.

Let us multiply $h_{sts}$ with $h_{st}$ in
$\mathcal{H}_{(\mathfrak{S}_{3},\\{s,t\\})}$. The direct computation reads:

$\begin{split}h_{sts}h_{st}&=h_{s}h_{t}h_{s}^{2}h_{t}\\\
&=h_{s}h_{t}^{2}+Qh_{s}h_{t}h_{s}h_{t}\\\
&=h_{s}+Qh_{s}h_{t}+Qh_{s}^{2}h_{t}h_{s}\\\
&=h_{s}+Qh_{st}+Qh_{ts}+Q^{2}h_{sts}\leavevmode\nobreak\ ,\end{split}$ (4.13)

and it corresponds to the graphs of Figure 4.9.

Figure 4.9: The graphical analogue of the product $h_{sts}h_{st}$.

######  Remark 4.17.

To a higher lamination $\Gamma$ on a triangle $t$ with set $R$ of ramification
points, one can associate a monodromy map $\pi_{1}(t\backslash R)\to W$ in the
following way. To any based loop $\gamma\in\pi_{1}(t\backslash R)$ which
intersects $\Gamma$ transversely one associates the product of all labels of
$\gamma\cap\Gamma$ following the orientation of $\gamma$. It is easy to check
that it only depends on the homotopy class of $\gamma$.

For higher lamination in type $A_{n}$, it is then possible to associate to
$\Gamma$ an $n$-sheeted cover with simple ramification points at $R$ and a
trivialization over each connected component of $t\backslash\Gamma$, such that
the transition between two adjacent regions is given by the label on the
separating edge. This is why we call the trivalent vertices of $\Gamma$
ramification points.

###  4.3 Higher laminations for ciliated surfaces

We now define higher laminations on surfaces, which provides a diagrammatic
way to compute our polynomial invariant. This viewpoint gives a direct proof
of the invariance under $q\mapsto q^{-1}$ for a closed surface (this is 4.21
below).

Let us consider a ciliated surface $\Sigma$ where each boundary component is
labeled by an element of the Weyl group, and fix a triangulation $\mathsf{T}$
of $\Sigma$. A higher lamination $\Gamma$ on $\Sigma$ is the gluing of
representatives of higher laminations on all triangles of $\mathsf{T}$ (their
boundary data has of course to coincide).

######  Remark 4.18.

It should be possible to define higher laminations without using a
triangulation as an equivalence class of minimal labeled graphs inducing the
elements of $W$ on the boundary of $\Sigma$, mimicking the definition for
triangles. The non-trivial point is to define minimality.

Let us see how to compute our polynomial using the graphical calculus. Draw
all possible higher laminations on $\Sigma$ compatible with the boundary data.
Recall that $\operatorname{ram}(\Gamma)$ is the number of trivalent vertices
of $\Gamma$. As in 4.15 we fix representatives of the higher laminations
associated to triples of flags. Assuming 4.8, this choice is irrelevant.

######  Theorem 4.19.

The polynomial invariant for a ciliated surface $\Sigma$ and a Weyl group $W$
is given by

$P_{\Sigma,W}(Q)=\sum_{\Gamma}Q^{\operatorname{ram}(\Gamma)}\leavevmode\nobreak\
,$ (4.14)

where the sum runs over all higher laminations of type $W$.

###### Proof.

Let us consider a triangulation $\mathsf{T}$ of $\Sigma$ and a higher
lamination $\Gamma$. The latter allows to associate elements of the standard
basis $h_{w}$ to each edge of $\mathsf{T}$. By 4.15 we know that for a
triangle $t$ the contribution of $\Gamma$ to the structure constant associated
to $t$ is given by $Q^{m_{t}}$ where $m_{t}$ is the number of ramification
points in the triangle. We conclude by the definition of the polynomial
invariant:

$\begin{split}P_{\Sigma,W}(Q)&=\sum_{e}\prod_{t}c_{xyz}(Q)=\sum_{e}\prod_{t}\sum_{\Gamma_{e}}Q^{\operatorname{ram}(\Gamma|_{t})}\\\
&=\sum_{e}\sum_{\Gamma_{e}}Q^{\operatorname{ram}(\Gamma)}\\\
&=\sum_{\Gamma}Q^{\operatorname{ram}(\Gamma)}\leavevmode\nobreak\
,\end{split}$ (4.15)

where $\textstyle\sum_{e}$ is the sum over all possible labels of the edges of
$\mathsf{T}$ by elements of $W$, $\textstyle\prod_{t}$ is the product over all
faces $t$ of $\mathsf{T}$ and $\textstyle\prod_{\Gamma_{e}}$ is the product
over all higher laminations compatible with the labels on the edges of
$\mathsf{T}$. ∎

Let us see how this works in a simple case:

######  Example 4.20.

Let us consider the sphere with three holes $\Sigma_{0,3}$ and the Hecke
algebra associated with the Coxeter group $\mathfrak{S}_{2}$. The surface
$\Sigma_{0,3}$ is drawn as a triangle in the plane (together with a point at
infinity one gets the sphere). We look for all possible higher laminations.

The following pictures are possible:

Note that the higher lamination which goes around two vertices of the triangle
is the same as the circle around the third vertex, since we are on a sphere.

This shows that our polynomial is given by $4+Q^{2}=q+2+q^{-1}$.

######  Proposition 4.21.

For a punctured surface $\Sigma_{g,k}$, the invariant $P_{g,k,W}$ is a
polynomial in $q=v^{-2}$. Furthermore, it is invariant under the
transformation $q\mapsto q^{-1}$.

###### Proof.

Using the graphical calculus, we have seen in 4.19 that

$P_{g,k,W}(Q)=\sum_{\Gamma}Q^{\operatorname{ram}(\Gamma)}\leavevmode\nobreak\
,$ (4.16)

where the sum runs over all higher laminations.

For a punctured surface, the only vertices of odd degree in a higher
lamination are the ramification points. Hence there is an even number of them.
Therefore our polynomial is given by a polynomial expression in
$Q^{2}=q-2+q^{-1}$ which is both a polynomial in $q$ and invariant under
$q\mapsto q^{-1}$. ∎

Using the graphical calculus, we can compute the first example of an invariant
for a Hecke algebra corresponding to a Coxeter system other than
$\mathfrak{S}_{2}$.

######  Example 4.22.

Let us consider $\Sigma_{0,3}$ with $\mathfrak{S}_{3}$.

If we use only one color, then we are reduced to the case $\mathfrak{S}_{2}$,
so the contributions from one color higher laminations to our polynomial are
given by $2(4+Q^{2})-1$ (we have to subtract one in order to count the empty
higher lamination only once).

Using both colors, here are the possible pictures with their contributions:

The contribution of a picture is given by $Q$ to the power the number of
ramification points, times the multiplicity. For example, the fifth picture
has no ramification point and multiplicity 8 since for each of the three
dashed loops there are two possibilities (going around a vertex of the
triangle or not). For the sixth picture the dashed circles can be there or not
with the only requirement that there must be at least one otherwise it is a
one-color diagram, hence there are $8-1=7$ possibilities. Then colors can be
exchanged, but this gives only 3 new possibilities.

Adding up all the terms, we get

$36+27Q^{2}+8Q^{4}+Q^{6}=q^{3}+2q^{2}+10q+10+10q^{-1}+2q^{-2}+q^{-3}.$

You see the weakness of the graphical calculus: there is no easy way to check
whether all the possible higher laminations have been found.

######  Remark 4.23.

In 4.17, we have seen that higher laminations are linked to ramified covers of
special type (some marking and a minimality condition). Our polynomial counts
these ramified covers.

##  5 Hecke topological quantum field theory

In this section we construct the invariants introduced above in a more
intrinsic way. Given a Coxeter system $(W,S)$, we construct a $1$-dimensional
topological quantum field theory (TQFT) which associates a copy of the Hecke
algebra $\mathcal{H}$ or its dual $\mathcal{H}^{*}$ to topological segments,
while ciliated surfaces play the role of cobordisms. Punctured surfaces define
elements of the base ring $\mathbb{Z}[v^{\pm 1}]$ which are nothing else than
the invariants of Section 3.2.

In the general case since the $1$-dimensional manifolds on which the TQFTs are
defined are not closed, the order of the factors in the tensor products of
$\mathcal{H}$ and $\mathcal{H}^{*}$ matters: these TQFTs are non-commutative.

###  5.1 Triangle invariants and gluing revisited

We redefine our construction in more intrinsic terms, showing in particular
its independence from any choice of basis in $\mathcal{H}$ and to allow
boundary labels to be any elements in $\mathcal{H}$ and not only those of the
standard basis.

##### Structure constants.

We have seen in Equation 3.1 that the product in the Hecke algebra corresponds
to a tensor

$c\in\mathcal{H}^{*}\otimes\mathcal{H}^{*}\otimes\mathcal{H}\ ,$ (5.1)

whose elements in the standard basis are exactly these structure constants.
Let us assign this tensor $c$ to the triangle with two sides ($1$ and $2$ on
the left of Figure 5.1) oriented counterclockwise and the last one ($3$),
clockwise. For any assignment $h,h^{\prime}\in\mathcal{H}$ to the first two
sides and $g\in\mathcal{H}^{*}$ to the last one, the tensor $c$ evaluated on
these yields $c(h,h^{\prime},g)\in\mathbb{Z}[v^{\pm 1}]$ as illustrated on the
right of Figure 5.1.

Figure 5.1: Assigning a tensor to triangles.

Since the product in the Hecke algebra is non commutative in general, one sees
that the order of the factor matters: $c(h,h^{\prime},g)\neq
c(h^{\prime},h,g)$ in general.

######  Remark 5.1.

Let $x,y,z\in W$. Then $c(h_{x},h_{y},h^{z})=\tensor{c}{{}_{xy}^{z}}$ as
defined in Equation 3.1.

We can associate a tensor to any triangle with oriented sides: assign a copy
of $\mathcal{H}$ to each side oriented counterclockwise and a copy of
$\mathcal{H}^{*}$ to each side oriented clockwise, as in Figure 5.1. The
nature of the tensor associated with the triangle must be such that it can be
evaluated on vectors in $\mathcal{H}$ or $\mathcal{H}^{*}$ for each side,
depending on its orientation. Hence one always obtains a scalar in
$\mathbb{Z}[v^{\pm 1}]$ as one evaluates the tensor corresponding to a
triangle on the vectors assigned to its side. For example, then tensor
corresponding to a triangle whose side are all oriented clockwise is in
$\mathcal{H}\otimes\mathcal{H}\otimes\mathcal{H}$ so that one can pair it
naturally with the three vectors in $\mathcal{H}^{*}$ one may assign to the
sides.

The tensors corresponding to triangles whose boundaries are not oriented as in
Figure 5.1 are derived from the tensor of Equation 5.1 corresponding to the
product, using the isomorphism between $\mathcal{H}$ and $\mathcal{H}^{*}$
given by the trace.

##### The standard trace on $\mathcal{H}$ again.

The choice of an isomorphism between $\mathcal{H}$ and its dual
$\mathcal{H}^{*}$ corresponds to the choice of a non-degenerate pairing:

$g:\mathcal{H}\otimes\mathcal{H}\rightarrow\mathbb{Z}[v^{\pm 1}]\ .$ (5.2)

We saw in Equation 3.3 that the pairing we are after is symmetric and defined
by

$g(h_{x},h_{y})=\delta_{x,y^{-1}}$ (5.3)

for $x,y\in W$. We know from Equation 3.5 that it is the standard trace in
$\mathcal{H}$.

######  Remark 5.2.

Another way to see that the choice of pairing of Equation 5.3 is meaningful to
us comes from the graphical calculus. Consider an edge with insertions of a
higher lamination corresponding to a reduced expression for an $x\in W$ in
terms of the generators in $S$ if one reads the edge following its
orientation. Now if one reverts the orientation, the edge corresponds to a
reduced expression for $x^{-1}\in W$ instead of $x$.

The choice of pairing of Equation 5.3 induces the isomorphism
$\mathcal{H}\to\mathcal{H}^{*}$ given by:

$\displaystyle\sum_{w\in W}c_{w}h_{w}\mapsto\displaystyle\sum_{w\in
W}c_{w}h^{(w^{-1})}$ (5.4)

The main interest of this re-definition of the triangle invariants is that the
latter do not depend on the choice of a particular basis of $\mathcal{H}$
anymore, since the pairing of Equation 5.3 can be expressed in any basis. We
will stick to the standard convention in tensor calculus that lower-indices
refer to the coordinates of elements in $\mathcal{H}^{*}$ or elements of a
base in $\mathcal{H}$, while upper-indices correspond to coordinates of
elements in $\mathcal{H}$ or elements of a base in $\mathcal{H}^{*}$. As usual
one lowers indices using the trace $g$ and one raises indices using its
inverse:

$g^{-1}:\mathcal{H}^{*}\otimes\mathcal{H}^{*}\rightarrow\mathbb{Z}[v^{\pm 1}]$
(5.5)

defined via the isomorphism of Equation 5.4.

Let $g^{ij}$ refer to the elements of the inverse trace in some fixed basis
$(C_{i})_{i\in I}$ of $\mathcal{H}$ indexed by some finite set $I$ whose
cardinal is $|W|$. Let $(C^{i})_{i\in I}$ be the dual basis of
$\mathcal{H}^{*}$ determined by the trace. Let $c_{ijk}$ be the coordinates of
the tensor $c\in\left(\mathcal{H}^{*}\right)^{\otimes 3}$ corresponding to a
triangle with sides oriented counterclockwise:

$c=c_{ijk}C^{i}C^{j}C^{k}\leavevmode\nobreak\ ,$ (5.6)

where here and in what follows the summation over repeated indices is
understood. The tensor $c$ corresponds to a tensor in
$\left(\mathcal{H}\right)^{\otimes 3}$ via the trace. Its coordinates are
given by:

$c^{ijk}=g^{il}g^{jm}g^{kn}c_{lmn}\leavevmode\nobreak\ .$ (5.7)

The invariance under cyclic permutation of indices of the $c_{ijk}$ proved in
3.1 transposes into the same property for the $c^{ijk}$.

##### Gluing edges.

The natural pairing

$\mathcal{H}^{*}\otimes\mathcal{H}\rightarrow\mathbb{Z}[v^{\pm 1}]$ (5.8)

provides a way to glue triangles along edges.

Consider the two triangles drawn on the left of Figure 5.2. The upper-one
corresponds to the tensor
$c_{u}=c\in\mathcal{H}^{*}_{1}\otimes\mathcal{H}^{*}_{2}\otimes\mathcal{H}^{*}_{3}$
which is the one of Equation 3.3, and the lower-one, to the corresponding
tensor
$c_{d}\in\mathcal{H}_{4}\otimes\mathcal{H}^{*}_{5}\otimes\mathcal{H}^{*}_{6}$.
The indices tell to which edge the copies of $\mathcal{H}$ or
$\mathcal{H}^{*}$ correspond. The edge $3$ is positively oriented with respect
to the upper triangle hence is associated a copy of $\mathcal{H}$ while the
edge $4$ is negatively oriented with respect to the lower one and is
associated a copy of $\mathcal{H}^{*}$. Hence we can glue the edges $3$ and
$4$ together.

Figure 5.2: Gluing two triangles along an edge.

Let $c_{ijk}$ be the coefficients of $c_{u}$ in the basis we fixed and
$\tensor{(c_{d})}{{}^{l}_{mn}}$ those of $c_{d}$ for $i,j,k,l,m,n\in I$. We
have seen that:

$\tensor{(c_{d})}{{}^{l}_{mn}}=g^{lp}c_{pmn\ .}$ (5.9)

Gluing the two triangles on the left of Figure 5.2 yields the quadrilateral on
the right, to which corresponds a tensor
$c_{q}\in\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\mathcal{H}_{5}\otimes\mathcal{H}_{6}$
whose coordinates are:

$(c_{q})_{ijkl}=c_{ijm}\tensor{(c_{d})}{{}^{m}_{kl}}=g^{mn}c_{ijm}c_{nkl}\leavevmode\nobreak\
.$ (5.10)

######  Proposition 5.3.

The tensor $c_{q}$ does not depend on the orientation of the edge along which
one glues the two triangles.

###### Proof.

Since:

$(c_{q})_{ijkl}=c_{ijm}(c_{d})^{m}_{\ \
kl}=g^{mn}c_{ijm}c_{nkl}=\tensor{c}{{}_{ij}^{m}}\tensor{(c_{d})}{{}_{mkl}}\ ,$
(5.11)

the proposition follows. ∎

######  Proposition 5.4.

The tensor $c_{q}$ does not depend on the diagonal of the quadrilateral along
which one does the gluing.

###### Proof.

The associativity of the product in the Hecke algebra and more precisely
Equation 3.16 implies that:

$(c_{q})_{ijkl}=g^{mn}c_{ijm}c_{nkl}=g^{mn}c_{jkm}c_{nli}\ ,$ (5.12)

which amounts to say that the tensor corresponding to the quadrilateral on the
right of Figure 5.2 can equivalently be obtained by gluing the two triangles
defined by the choice of one diagonal or the other. ∎

One can glue arbitrary ciliated surfaces together along edges in the same way,
with obvious generalizations of 5.3 and of 5.4. Since any ciliated surface of
our interest admits a triangulation, it is assigned a tensor invariant by
gluing the invariants corresponding to the faces of its triangulation. This
tensor does not depend on the triangulation since any two triangulations can
be related by a finite sequence of flips and thanks to 5.4, hence it is a
topological invariant of the ciliated surface.

###  5.2 Hecke topological quantum field theories

First we describe our category $\mathcal{C}$ of cobordisms: the objects are
disjoint unions of oriented segments, and the morphisms from an object $A$ to
another object $B$ are the ciliated surfaces whose boundary is the disjoint
union of $B$ and $A$ with the orientation reversed.

Let us consider the functor

$F:\mathcal{C}\rightarrow\leavevmode\nobreak\ \mathbb{Z}[v^{\pm
1}]-\mathrm{Mod}$ (5.13)

which associates a copy of $\mathcal{H}$ to the positively oriented segment, a
copy of $\mathcal{H}^{*}$ to the negatively oriented segment, and the proper
tensor product of copies of $\mathcal{H}$ and $\mathcal{H}^{*}$ to a disjoint
union of oriented segments. For $A$ and $B$ two objects in $\mathcal{C}$ and a
ciliated surface $\Sigma\in\mathrm{Hom}(A,B)$, the morphism $F(\Sigma)$ is the
tensor invariant constructed in Section 5.1 which is indeed in
$F(\partial\Sigma)$. By 5.4, we know that this tensor only depends on $\Sigma$
and not on a triangulation.

######  Theorem 5.5.

For any finite Coxeter system $(W,S)$ the functor $F$ satisfies the axioms of
a topological quantum field theory listed in [Ati88].

###### Proof.

Our construction makes clear that $F$ is invariant under orientation
preserving diffeomorphisms of ciliated surfaces and their boundary, that it is
involutory, and that it is multiplicative. ∎

Let $\Sigma$ be a ciliated surface without cilia and hence without boundary.
Then $F(\Sigma)\in\mathbb{Z}[v^{\pm 1}]$, that is, it a Laurent polynomial in
$v$. It clearly coincides with the invariant for punctured surfaces defined in
Section 3.

###  5.3 Traces in the Hecke algebra and invariants of $n$-gons

Let again $(C_{i})_{i\in I}$ be a basis of $\mathcal{H}$ and $(C^{i})_{i\in
I}$ the trace-dual basis in $\mathcal{H}$:

$\operatorname{tr}C_{i}C^{j}=\delta_{i}^{j}\leavevmode\nobreak\ .$ (5.14)

The basis $(C^{i})_{i\in I}$ of $\mathcal{H}$ is obtained from the basis in
$\mathcal{H}^{*}$ dual to $(C_{i})_{i\in I}$ via the isomorphism of Equation
5.4. Similarly to 3.1, the structure constants are given by:

$c_{ijk}=\operatorname{tr}C_{i}C_{j}C_{k}\leavevmode\nobreak\ .$ (5.15)

More generally, products of $C_{i}$’s and $C^{j}$’s can be expressed with the
trace. For example:

$C_{i}C_{j}C^{k}C_{l}=\operatorname{tr}(C_{i}C_{j}C^{k}C_{l}C^{m})C_{m}\leavevmode\nobreak\
,$ (5.16)

(Einstein summation convention is again understood) as can be seen by
multiplying both sides by $C^{n}$ and taking the trace. Let us define:

$\tensor{c}{{}_{ij}^{k}{}_{l}^{m}}:=\operatorname{tr}(C_{i}C_{j}C^{k}C_{l}C^{m})$
(5.17)

and similarly for the trace of other products.

Let us come back to the invariant corresponding to a quadrilateral obtained in
Equation 5.10. We have:

$(c_{q})_{ijkl}=\operatorname{tr}(C_{i}C_{j}C_{m})\operatorname{tr}(C^{m}C_{k}C_{l})=\operatorname{tr}(C^{m}C_{n})\operatorname{tr}(C_{i}C_{j}C_{m})\operatorname{tr}(C^{n}C_{k}C_{l})\leavevmode\nobreak\
,$ (5.18)

hence the gluing of the sides labeled by $C_{m}$ and $C^{n}$ corresponds to
the multiplication by
$\operatorname{tr}C^{m}C_{n}=\tensor{c}{{}^{m}_{n}}=\delta^{m}_{n}$ since we
used the trace to identify the dual $\mathcal{H}^{*}$ with the Hecke algebra.

The following property is the translation of the contraction of indices for
tensors in terms of trace.

######  Proposition 5.6.

In a product of two traces repeated indices can be contracted. For example:

$\tensor{c}{{}_{ijk}^{lm}}\tensor{c}{{}_{mn}^{pq}}=\tensor{c}{{}_{ijk}^{l}{}_{n}^{pq}}\leavevmode\nobreak\
.$ (5.19)

###### Proof.

We illustrate the proof for the example given in the proposition. We have seen
that traces appear as coefficients in product expansions, hence:

$C_{i}C_{j}C_{k}C^{l}=\operatorname{tr}(C_{i}C_{j}C_{k}C^{l}C^{m})C_{m}=\tensor{c}{{}_{ijk}^{lm}}C_{m}\leavevmode\nobreak\
.$ (5.20)

By linearity of the trace:

$\tensor{c}{{}_{ijk}^{lm}}\tensor{c}{{}_{mn}^{pq}}=\operatorname{tr}(\tensor{c}{{}_{ijk}^{lm}}C_{m}C_{n}C^{p}C^{q})=\operatorname{tr}(C_{i}C_{j}C_{k}C^{l}C_{n}C^{p}C^{q})=\tensor{c}{{}_{ijk}^{l}{}_{n}^{pq}}\leavevmode\nobreak\
.$ (5.21)

∎

######  Remark 5.7.

Note that you can not contract two indices at once. For example we have
$c_{ji}c^{ij}\neq 1$. Rather
$c_{ji}c^{ij}=\tensor{c}{{}_{j}^{j}}=\delta_{j}^{j}=|W|$ again with Einstein’s
summation convention.

Coming back to Equation 5.18, we see that:

$(c_{q})_{ijkl}=\operatorname{tr}(C_{i}C_{j}C_{k}C_{l})=c_{ijkl}\leavevmode\nobreak\
.$ (5.22)

This is a general fact: the invariant corresponding to an $n$-gon with labels
$h_{1},...,h_{n}$ in either $\mathcal{H}$ or $\mathcal{H}^{*}$ on the sides
when read in a counterclockwise order is $\operatorname{tr}(h_{1}...h_{n})$.
Even more generally it is always possible to express our polynomial as a
single trace. Let us consider for example the sphere with three punctures: in
Example 5.8 we saw that $P_{0,3,W}=c_{xyz}c^{xzy}$. Contracting indices we
obtain $P_{0,3,W}=\tensor{c}{{}_{xy}^{yx}}$. This generalizes easily to:

$P_{0,n,W}=\tensor{c}{{}_{w_{1}w_{2}...w_{n-1}}^{w_{n-1}w_{n-2}...w_{1}}}\leavevmode\nobreak\
.$ (5.23)

Thus we can extend the definition of our polynomial to $\Sigma_{0,2}$ and even
$\Sigma_{0,1}$: $P_{0,2,W}=|W|$ and $P_{0,1,W}=1$.

An $n$-gon can be interpreted as a morphism from the trivial object of the
category $\mathcal{C}$ of the previous section to the $n$-gon with positively
oriented sides, that is, the disjoint union of $n$ positively oriented
segments. The TQFT corresponding to a system $(W,S)$ maps it to the morphism
$\mathbb{Z}[v^{\pm 1}]\rightarrow\mathcal{H}^{\otimes n}$ given by

$1\mapsto\operatorname{tr}(C^{w_{1}}\cdots
C^{w_{n}})C_{w_{1}}\otimes\cdots\otimes C_{w_{n}}\leavevmode\nobreak\ .$
(5.24)

Interpreting now an $n$-gon with negatively oriented sides as a cobordism from
the disjoint union of $n$ negatively oriented segments to the trivial object
in $\mathcal{C}$ yields a map

$\left.\begin{array}[]{rcl}(\mathcal{H}^{*})^{\otimes
n}&\rightarrow&\mathbb{Z}[v^{\pm 1}]\\\ C^{w_{1}}\otimes
C^{w_{2}}\otimes\cdots\otimes
C^{w_{n}}&\mapsto&\operatorname{tr}(C^{w_{1}}C^{w_{2}}\cdots
C^{w_{n}})\end{array}\right.\leavevmode\nobreak\ .$ (5.25)

######  Example 5.8.

Consider the sphere with three punctures, represented by the gluing of two
triangles as in Figure 5.3. Reading the picture yields a map
$\mathbb{Z}[v^{\pm 1}]\rightarrow\mathbb{Z}[v^{\pm 1}]$ given by

$1\mapsto\operatorname{tr}(C^{x}C^{y}C^{z})\operatorname{tr}(C_{x}C_{z}C_{y})=c^{xyz}c_{xzy}\leavevmode\nobreak\
.$ (5.26)

Figure 5.3: Reading off the polynomial

Given a $n$-gon with some of its sides identified to form a ciliated surface
$\Sigma$, we can now easily describe the invariant $P_{\Sigma,W}$.

######  Proposition 5.9.

Associate an element $h_{i}$ in $\mathcal{H}$ or $\mathcal{H}^{*}$ to each of
the sides of the $n$-gon in such a way that the choice of $\mathcal{H}$ or
$\mathcal{H}^{*}$ is consistent with the orientation and that identified edges
carry the same variable. The polynomial $P_{\Sigma,W}$ is the trace of the
product of the $h_{i}$, following the border of the polygon in a
counterclockwise way. On the left of Figure 5.4 for example the polygon is
given by

$\operatorname{tr}(C_{x_{1}}C_{x_{2}}C^{x_{3}}C^{x_{2}}C_{x_{3}}C_{x_{4}}C^{x_{1}}C^{x_{4}}).$
(5.27)

Figure 5.4: Trace expression for the polynomial

###### Proof.

Consider the triangulation shown on the right of Figure 5.4. Our polynomial is
obtained through the gluing of the structure constants corresponding to the
faces of the triangulation, which can be expressed as traces. In the example
we get:

$P_{\Sigma,W}=\tensor{c}{{}_{x_{1}x_{2}}^{y_{1}}}\tensor{c}{{}_{y_{1}}^{x_{3}y_{2}}}\tensor{c}{{}_{y_{2}}^{x_{2}y_{3}}}\tensor{c}{{}_{y_{3}x_{3}}^{y_{4}}}\tensor{c}{{}_{y_{4}x_{4}}^{y_{5}}}\tensor{c}{{}_{y_{5}}^{x_{1}x_{4}}}.$
(5.28)

Contracting the repeated $y_{i}$ indices gives the desired expression. ∎

Choosing a polygon such as the one on the left of Figure 3.5 we get a simple
expression for the invariant corresponding to punctured surfaces:

$P_{g,k,W}=\operatorname{tr}\;(C_{w}C^{w})^{k-1}(C_{x}C_{y}C^{x}C^{y})^{g}.$
(5.29)

######  Remark 5.10.

Any expression as the one of Equation 5.27 is equivalent to another one
normalized as in Equation 5.29, since the invariant is independent from the
triangulation.

##  6 Explicit expression and positivity

In this section, we explicitly compute the polynomial invariant for ciliated
surfaces and determine in which cases positivity hold. The key observation is
that for a punctured surface, the polynomial is the trace of a central element
in the Hecke algebra $\mathcal{H}$. The structure of the center can be
understood using the Wedderburn decomposition of $\mathcal{H}$. The polynomial
is a sum of powers of Schur elements multiplied by irreducible characters.

###  6.1 Schur elements and Wedderburn decomposition

The Hecke algebra with its standard trace is a specific example of a symmetric
algebra, that is, an algebra with a non-degenerate trace function. There is a
general theory of these algebras which we present briefly here. In particular
we expose some general facts about Schur elements and the Wedderburn
decomposition for Hecke algebras. Main references are [GP00], Chapter 7 and 8
of [Chl16] and the article [Neu06].

##### Symmetric algebras.

Fix a commutative integral domain $A$. Let $H$ be a free and finitely
generated algebra over $A$. A trace on $H$ is a linear map $H\to A$ vanishing
on the commutator subalgebra $[H,H]$. Denote by $T(H)\simeq(H/[H,H])^{*}$ the
space of traces on $H$. An algebra $H$ is called symmetric if we fix a trace
$\tau$ on it such that the symmetric bilinear form

$B_{\tau}(h_{1}\otimes h_{2})=\tau(h_{1}h_{2})$ (6.1)

is nondegenerate. We further assume $H$ semisimple. Note that if
$A=\mathbb{R}$ and $B$ is positive definite, the algebra $H$ is automatically
semisimple since an orthogonal complement to a left ideal of $H$ is a right
ideal and visa versa.

Denote by $\mathrm{Irr}(H)$ the set of irreducible representations of $H$ and
for $\lambda\in\mathrm{Irr}(H)$ denote by the same letter the map
$\lambda:H\to\mathrm{End(V_{\lambda})}$. Then the semisimplicity of $H$
implies that

$H=\bigoplus_{\lambda\in\mathrm{Irr}(H)}\mathrm{End(V_{\lambda})}\leavevmode\nobreak\
.$ (6.2)

This is the Artin–Wedderburn decomposition, or a version of the Peter–Weyl
theorem in this setting.

Observe that for any $z$ in the center $Z(H)$ of $H$ one can construct another
trace $h\mapsto\tau(zh)$. Conversely, given a trace $t$ one can construct an
element $(1\otimes t)B_{\tau}^{-1}$ of $H$, where we consider $B_{\tau}^{-1}$
as an element of $H\otimes H$. These two maps are obviously inverse to each
other and give therefore an isomorphism

$Z(H)\simeq T(H)\leavevmode\nobreak\ .$ (6.3)

For any representation $\lambda$ of $H$ one can associate a character denoted
by $\chi_{\lambda}\in T(H)$, defined by
$\chi_{\lambda}(h)=\mathrm{tr}\lambda(h)$ and a corresponding element
$Z_{\lambda}$ of the center $Z(H)$. The sets $\\{\chi_{\lambda}\\}$ and
$\\{Z_{\lambda}\\}$ for $\lambda\in\mathrm{Irr}(H)$ form orthogonal bases in
the spaces $T(H)$ and $Z(H)$, respectively. Denote by $s_{\lambda}$ the
inverses the coefficients of the decomposition of unity in $H$ with respect to
the base of the $Z_{\lambda}$:

$\textstyle\sum_{\lambda}\frac{1}{s_{\lambda}}Z_{\lambda}=1\leavevmode\nobreak\
.$ (6.4)

The $s_{\lambda}$ are called Schur elements. From Equation 6.4 we immediately
get

$\tau=\textstyle\sum_{\lambda}\frac{1}{s_{\lambda}}\chi_{\lambda}\leavevmode\nobreak\
.$ (6.5)

Further, by Equation 6.2, we see that $Z_{\lambda}$ acts on $V_{\lambda}$ as
$s_{\lambda}\operatorname{id}$ (which is the usual definition of the Schur
elements). The elements $Z_{\lambda}$ can be computed as follows:

######  Proposition 6.1.

Let $(C_{w})_{w\in W}$ be a basis of $H$ and let $(C^{w})$ be its trace-dual
basis in $H$. The central element $Z_{\lambda}$ can be computed by

$Z_{\lambda}=\sum_{w\in W}\chi_{\lambda}(C_{w})C^{w}\leavevmode\nobreak\ .$
(6.6)

###### Proof.

For $h\in H$, we compute (using Einstein summation convention):

$\tau(Z_{\lambda}h)=\chi_{\lambda}(h)=\chi_{\lambda}(\tau(C^{w}h)C_{w})=\tau(C^{w}h)\chi_{\lambda}(C_{w})=\tau(\chi_{\lambda}(C_{w})C^{w}h)$
(6.7)

where we used that $h=\tau(C^{w}h)C_{w}$ by definition of $C^{w}$. Hence the
proposition follows by the non-degeneracy of $\tau$. ∎

There is another definition of Schur elements which will be useful in the
sequel. Let $\varphi:V\rightarrow V^{\prime}$ be an $A$-morphism between right
$H$-modules. Let $I(\varphi):V\rightarrow V^{\prime}$ be defined by:

$I(\varphi)\cdot v=\sum_{w}\varphi(vC_{w})C^{w}\leavevmode\nobreak\ .$ (6.8)

The morphism $I(\varphi)$ does not depend on the choice of basis $(C_{w})$ and
it is a morphism of $H$-modules [GP00, Lemma 7.1.10].

For $V=V^{\prime}=V_{\lambda}$ an irreducible representation, we have [GP00,
Theorem 7.2.1]:

######  Proposition 6.2.

Let $\varphi\in\operatorname{End}(V_{\lambda})$. Then:

$I(\varphi)=s_{\lambda}\operatorname{tr}(\varphi)\operatorname{id}\leavevmode\nobreak\
.$ (6.9)

##### Hecke algebras.

Now, we specialize to the Hecke algebra $(\mathcal{H},\operatorname{tr})$ with
its standard trace. We consider $K\mathcal{H}=\mathcal{H}\otimes_{A}K$ where
$K$ is the field of fractions of $A=\mathbb{Z}[v^{\pm 1}]$. The
Artin–Wedderburn theorem implies that

$K\mathcal{H}\simeq\bigoplus_{\lambda\in\operatorname{Irr}(K\mathcal{H})}\operatorname{End}(V_{\lambda})\leavevmode\nobreak\
.$ (6.10)

######  Remark 6.3.

For Hecke algebras, the irreducible representations are all inside so-called
left cell representations. For type $A$, the left cell representations are all
irreducible. The article [Neu06] by Neunhöffer describes explicitly the
adapted basis for type $A$, i.e. the matrix elements for each factor
$\operatorname{End}(V_{\lambda})$. We will use these cell representations only
for 6.12 below.

The decomposition of Equation 6.10 implies that the center of the Hecke
algebra is given by diagonal matrices.

######  Proposition 6.4.

The elements $(Z_{\lambda})_{\lambda\in\operatorname{Irr}(\mathcal{H})}$ form
a basis of the center $Z(\mathcal{H})$ such that:

$Z_{\lambda}Z_{\mu}=\delta_{\lambda,\mu}s_{\lambda}Z_{\lambda}\;\;\forall\;\lambda,\mu\in\operatorname{Irr}(K\mathcal{H})\leavevmode\nobreak\
.$ (6.11)

###### Proof.

Since the characters form a basis of the space of trace functions, the
$Z_{\lambda}$’s form a basis of the center $Z(H)$. The Wedderburn
decomposition implies that $Z_{\lambda}Z_{\mu}=0$ for $\lambda\neq\mu$. Now,
for all $h\in H$:

$\operatorname{tr}(Z_{\lambda}^{2}h)=\chi_{\lambda}(Z_{\lambda}h)=\chi_{\lambda}(s_{\lambda}h)=\operatorname{tr}(s_{\lambda}Z_{\lambda}h)\leavevmode\nobreak\
.$ (6.12)

Hence $Z_{\lambda}^{2}=s_{\lambda}Z_{\lambda}$. ∎

There is a special symmetry in the Schur elements of Hecke algebras. Let
$\gamma$ be the $\mathbb{Z}[v^{\pm 1}]$-algebra homomorphism on $\mathcal{H}$
given by $\gamma(h_{s})=-qh_{s}^{-1}$. For
$\lambda\in\operatorname{Irr}(\mathcal{H})$ let $\lambda^{*}$ be the
composition $\lambda\circ\gamma$ called the dual representation. Proposition
9.4.3 in [GP00] states that:

######  Proposition 6.5.

For $\lambda\in\operatorname{Irr}(\mathcal{H})$, we have

$s_{\lambda^{*}}(q)=s_{\lambda}(q^{-1}).$

###  6.2 Central elements

Recall the expression of the invariant for punctured surfaces in terms of the
standard trace of Equation 5.29:

$P_{g,k,W}=\operatorname{tr}\;(C_{w}C^{w})^{k-1}(C_{x}C_{y}C^{x}C^{y})^{g}.$

For surfaces with boundary labeled by element in $\mathcal{H}$, terms of the
form $C_{w}hC^{w}$ arise.

######  Theorem 6.6.

The elements of the form $(C_{w}C^{w})^{k-1}(C_{x}C_{y}C^{x}C^{y})^{g}$ and
$C_{w}hC^{w}$ (for $h\in\mathcal{H}$) are in the center of the Hecke algebra.

###### Proof.

For the first statement, let $s=(C_{w}C^{w})^{k-1}(C_{x}C_{y}C^{x}C^{y})^{g}$.
By the non-degeneracy of the trace, it is sufficient to show that

$\operatorname{tr}(sh_{1}h_{2})=\operatorname{tr}(h_{1}sh_{2})\;\forall\;h_{1},h_{2}\in\mathcal{H}.$
(6.13)

The expression $\operatorname{tr}(sh_{1}h_{2})$ is the invariant of a surface
with two boundary components labeled by $h_{1}$ and $h_{2}$. The same holds
for $\operatorname{tr}(h_{1}sh_{2})=\operatorname{tr}(sh_{2}h_{1})$ with
$h_{1}$ and $h_{2}$ exchanged.

A $\pi$-rotation of the surface as shown in Figure 6.1 exchanges $h_{1}$ and
$h_{2}$. Since it does not change the topology, the invariants of the two
surfaces coincide. This implies Equation 6.13 and thus the theorem.

The second statement is analogous. The surface corresponding to the invariant

$\operatorname{tr}C_{w}hC^{w}h_{1}h_{2}$ (6.14)

is a cylinder with one boundary labeled by $h$ and another boundary with two
cilia and labels $h_{1}$ and $h_{2}$. Again a $\pi$-rotation exchanges $h_{1}$
and $h_{2}$. ∎

Figure 6.1: Exchanging $h_{1}$ and $h_{2}$.

Note that the central elements $C_{w}C^{w}$ and $C_{x}C_{y}C^{x}C^{y}$ can be
seen as Casimir elements of order 2 and 4 for the Hecke algebra.

From 6.4 we know that the Schur elements
$(Z_{\lambda})_{\lambda\in\operatorname{Irr}(\mathcal{H})}$ form a basis of
the center $Z(\mathcal{H})$ of the Hecke algebra. We can determine the
decomposition of the two building blocks $C_{w}hC^{w}$ and
$C_{x}C_{y}C^{x}C^{y}$ of expressions in the form of Equation 5.29 in this
basis:

######  Proposition 6.7.

For $h\in\mathcal{H}$, we have:

$C_{w}hC^{w}=\textstyle\sum_{\lambda}\chi_{\lambda}(h)Z_{\lambda}\leavevmode\nobreak\
,$ (6.15)

and

$C_{x}C_{y}C^{x}C^{y}=\textstyle\sum_{\lambda}s_{\lambda}Z_{\lambda}\leavevmode\nobreak\
.$ (6.16)

Note that in particular (for $h=1$), we get

$C_{w}C^{w}=\textstyle\sum_{\lambda}\dim(V_{\lambda})Z_{\lambda}\leavevmode\nobreak\
.$ (6.17)

###### Proof.

For the first assertion, take $V=V^{\prime}=V_{\lambda}$ and
$\varphi_{h}\in\operatorname{End}(V_{\lambda})$, the action induced by right
multiplication by $h$. Let $h^{\prime}\in V_{\lambda}$ and write
$C_{x}hC^{x}=\sum_{\mu}e_{\mu}Z_{\mu}$. On the one hand we have

$h^{\prime}C_{x}hC^{x}=I(\varphi_{h}).h^{\prime}=s_{\lambda}\operatorname{tr}(\varphi_{h})h^{\prime}=s_{\lambda}\chi_{\lambda}(h)h^{\prime}$
(6.18)

thanks to 6.2, and on the other

$h^{\prime}C_{x}hC^{x}=\textstyle\sum_{\mu}e_{\mu}Z_{\mu}h^{\prime}=e_{\lambda}s_{\lambda}h^{\prime}$
(6.19)

since $Z_{\lambda}h^{\prime}=s_{\lambda}h^{\prime}$ and 0 for other values of
$\mu$. Comparing coefficients we get $e_{\lambda}=\chi_{\lambda}(h)$.

For the second assertion, consider the map
$\varphi_{\lambda,x}:V_{\lambda}\rightarrow V_{\lambda}$ given by

$\varphi_{\lambda,x}(v)=vC^{x}\leavevmode\nobreak\ .$ (6.20)

Thus:

$I(\varphi_{\lambda,x}).v=\sum_{y}\varphi_{\lambda,x}(vC_{y})C^{y}=\sum_{y}vC_{y}C^{x}C^{y}\leavevmode\nobreak\
,$ (6.21)

and by 6.2:

$I(\varphi_{\lambda,x}).v=s_{\lambda}\operatorname{tr}(\varphi_{\lambda,x})v=s_{\lambda}\,\chi_{\lambda}(C^{x})v$
(6.22)

where we have used the definition of the character $\chi_{\lambda}$. Hence:

$\sum_{y}vC_{y}C^{x}C^{y}=s_{\lambda}\,\chi_{\lambda}(C^{x})\,v\text{ for
}v\in V_{\lambda}.$ (6.23)

Recall the Wedderburn decomposition

$\mathcal{H}\cong\bigoplus_{\lambda}\operatorname{End}(V_{\lambda})\leavevmode\nobreak\
.$ (6.24)

Let $(B_{x})$ be an adapted basis of $\mathcal{H}$ given by the matrix
coefficients in the factors of the decomposition, and $(B^{x})$ its trace-dual
basis in $\mathcal{H}$. Let $\lambda(x)$ be the unique $\lambda$ such that
$B_{x}\in V_{\lambda}$.

Since $B_{x}B_{y}=0$ whenever $\lambda(x)\neq\lambda(y)$, we know that
$B^{x}\in\operatorname{Span}(B_{y}\mid\lambda(y)=\lambda(x))$, and hence
$B^{x}B_{y}=0$ for $\lambda(x)\neq\lambda(y)$. Hence $\chi_{\lambda}(A^{x})=0$
if $\lambda\neq\lambda(x)$.

Equation 6.23 implies that:

$\sum_{y}B_{x}B_{y}B^{x}B^{y}=s_{\lambda(x)}\chi_{\lambda(x)}(B^{x})B_{x}\leavevmode\nobreak\
,$ (6.25)

and eventually:

$\begin{split}\sum_{x,y}B_{x}B_{y}B^{x}B^{y}&=\sum_{x}s_{\lambda(x)}\chi_{\lambda(x)}(B^{x})B_{x}\\\
&=\sum_{\lambda}s_{\lambda}\sum_{x\,\mid\,\lambda(x)=\lambda}\chi_{\lambda}(B^{x})B_{x}\\\
&=\sum_{\lambda}s_{\lambda}\sum_{x}\chi_{\lambda}(B^{x})B_{x}\\\
&=\sum_{\lambda}s_{\lambda}Z_{\lambda}\end{split}$ (6.26)

where we used that $\chi_{\lambda}(B^{x})=0$ if $\lambda\neq\lambda(x)$ and
that $Z_{\lambda}=\sum_{x}\chi_{\lambda}(B^{x})B_{x}$ from 6.1. This concludes
the proof. ∎

###  6.3 Explicit expression

We are now ready to compute the invariants for ciliated surfaces. We start
with punctured surfaces.

######  Theorem 6.8.

The polynomial invariant corresponding to a punctured surface is given by

$P_{g,k,W}(q)=\sum_{\lambda}(\dim
V_{\lambda})^{k}s_{\lambda}(q)^{2g-2+k}\leavevmode\nobreak\ .$ (6.27)

###### Proof.

We have that:

$\begin{split}P_{g,k,W}(q)&=\operatorname{tr}\left(\left(C_{w}C^{w}\right)^{k-1}\left(C_{x}C_{y}C^{x}C^{y}\right)^{g}\right)\\\
&=\operatorname{tr}\left(\textstyle\sum_{\lambda}\dim
V_{\lambda}Z_{\lambda}\right)^{k-1}\left(\textstyle\sum_{\lambda}s_{\lambda}Z_{\lambda}\right)^{g}\\\
&=\operatorname{tr}\sum_{\lambda}(\dim
V_{\lambda})^{k-1}s_{\lambda}^{2g-2+k}Z_{\lambda}\\\ &=\sum_{\lambda}(\dim
V_{\lambda})^{k}s_{\lambda}^{2g-2+k}\leavevmode\nobreak\ ,\end{split}$ (6.28)

where the first equality is Equation 5.29, the second one comes from 6.7, the
third one from 6.4, and in the last one we used
$\operatorname{tr}Z_{\lambda}=\chi_{\lambda}(1)=\dim V_{\lambda}$. ∎

For example, let us consider the case $W=\mathfrak{S}_{2}$. There are two
Schur elements in $\mathcal{H}_{\mathfrak{S}_{2}}$, respectively $s_{1}=1+q$
and $s_{2}=1+q^{-1}$. Hence the invariant corresponding to a genus $g$ surface
with $k$ punctures is given by

$P_{g,k,\mathfrak{S}_{2}}=(1+q)^{2g-2+k}+(1+q^{-1})^{2g-2+k}\leavevmode\nobreak\
.$ (6.29)

This results agrees with 3.10.

Looking at the explicit expression in 6.8, we observe several phenomena:

  * •

We can evaluate the expression at $k=0$, although we have no definition of our
invariant for closed surfaces. In view of 4.17 this is not surprising since
our polynomial seems to count special ramified coverings over $\Sigma$.

  * •

Specializing to $q=1$ and using $s_{\lambda}(1)=|W|(\dim V_{\lambda})^{-1}$
(in type $A$ this is the hook length formula), we recover the result of
Equation 3.21.

  * •

The invariance under $q\mapsto q^{-1}$ of our polynomial can be obtained from
the explicit expression of 6.8 and the duality property of Schur elements of
6.5. Interestingly, the coefficients of each Schur element are symmetric as
can be observed by a case-by-case study (see B.2). It would be nice to recover
this property using the TQFTs developed in this paper.

Let us turn to the case of a ciliated surface
$\Sigma_{g,k,\\{p_{1},...,p_{n}\\}}$ of genus $g$, $k\geq 1$ punctures and
boundary components with $p_{i}\geq 1$ cilia. Let $h_{i}\in\mathcal{H}$ be the
product of the elements along the $i$-th boundary component following the
orientation of $\Sigma$ ($h_{i}$ per se is not well-defined but its conjugacy
class is and it is enough to write 6.9 unambiguously since characters are
class functions).

######  Theorem 6.9.

For a ciliated surface $\Sigma_{g,k,\\{p_{1},...,p_{n}\\}}$ with labels
$(h_{1},...,h_{n})\in\mathcal{H}^{n}$, the polynomial invariant is:

$P_{\Sigma,W}(q)=\sum_{\lambda}(\dim
V_{\lambda})^{k}s_{\lambda}(q)^{2g-2+k+n}\chi_{\lambda}(h_{1})\cdots\chi_{\lambda}(h_{n})\leavevmode\nobreak\
.$ (6.30)

###### Proof.

We can suppose $n\geq 1$ since the punctures case was already treated in 6.8.
There is a polygonal gluing yielding the ciliated surface $\Sigma$ in the
following way: start with a disc with one cilia on the boundary. Then glue $g$
handles to it, giving a term $(C_{x}C_{y}C^{x}C^{y})^{g}$, then add $k$
punctures, giving a term $(C_{w}C^{w})^{k}$, add other boundary components
with labels $h_{2}$ to $h_{n}$, giving a term
$\prod_{i=2}^{n}C_{w}h_{i}C^{w}$, and finally add the label $h_{1}$ to the
initial boundary circle.

Hence, we get:

$\begin{split}P_{\Sigma,W}(q)&=\operatorname{tr}\;(C_{x}C_{y}C^{x}C^{y})^{g}(C_{w}C^{w})^{k}\textstyle\prod_{i=2}^{n}(C_{w}h_{i}C^{w})\;h_{1}\\\
&=\operatorname{tr}\;(\textstyle\sum_{\lambda}s_{\lambda}^{2g-1}Z_{\lambda})(\textstyle\sum_{\mu}(\dim
V_{\mu})^{k}s_{\mu}^{k-1}Z_{\mu})\textstyle\prod_{i=2}^{n}(\sum_{\lambda_{i}}\chi_{\lambda_{i}}(h_{i})Z_{\lambda_{i}})\;h_{1}\\\
&=\sum_{\lambda}(\dim
V_{\lambda})^{k}s_{\lambda}^{2g-2+k+n}\textstyle\prod_{i=2}^{n}\chi_{\lambda}(h_{i})\;\operatorname{tr}(Z_{\lambda}h_{1})\\\
&=\sum_{\lambda}(\dim
V_{\lambda})^{k}s_{\lambda}^{2g-2+k+n}\textstyle\prod_{i=1}^{n}\chi_{\lambda}(h_{i})\leavevmode\nobreak\
,\end{split}$ (6.31)

where we used Equation 5.29, 6.4 and the equality
$\operatorname{tr}(Z_{\lambda}h_{1})=\chi_{\lambda}(h_{1})$. ∎

Two remarks on the explicit expression:

  * •

A puncture is equivalent to a boundary component labeled by $1\in\mathcal{H}$,
since $\dim(V_{\lambda})=\chi_{\lambda}(1)$.

  * •

The compatibility of 6.9 with the gluing property of Equation 3.17 is ensured
by the orthogonality of the irreducible characters (see [GP00, Corollary
7.2.4]):

$\chi_{\lambda}(C_{w})\chi_{\mu}(C^{w})=\delta_{\lambda,\mu}s_{\lambda}\dim(V_{\lambda})\leavevmode\nobreak\
.$ (6.32)

###  6.4 Positivity properties

We now turn to the positivity properties of our invariants. For punctured
surfaces this reduces to the study of positivity properties of the Schur
elements. For a ciliated surface, the characters of elements in the
Kazhdan–Lusztig basis appear.

In Appendix B we carry out the study of positivity properties of Schur
elements, using a formula of [Chl16]. We obtain the following theorem:

######  Theorem 6.10.

The following positivity properties hold for punctures surfaces:

  1. (i)

The polynomial invariant $P_{g,k,W}(q)$ has positive coefficients for all
classical $W$ and for the exceptional types $H_{3},E_{6}$ and $E_{7}$.

  2. (ii)

For all other types, i.e. $I_{2}(m)$ for $m\geq 5$, $H_{4},F_{4}$ and $E_{8}$,
the invariants may have negative coefficients.

The first part (apart from $H_{3}$) is a direct consequence of the positivity
of Schur elements, the second is a case-by-case study. We are thankful to
Sebastian Manecke for his contribution to the case $H_{3}$.

###### Proof.

By B.1, the Schur elements have positive coefficients in all classical types
and for $E_{6}$ and $E_{7}$. Hence 6.8 implies the positivity of the
coefficients of the invariants for punctured surfaces in those cases.

Let us now study the case $H_{3}$. An explicit computation shows that there
are only two Schur elements with negative coefficients which only differ by a
shift by $q^{5}$. In other words, we can write these two elements as some
Laurent polynomial $P$ and $q^{5}P$. The corresponding irreducible
representations of $\mathcal{H}_{H_{3}}$ are $3$-dimensional. There are two
other Schur elements whose representation is $3$-dimensional. These are of the
form $Q$ and $q^{5}Q$ for some other Laurent polynomial $Q$. More explicitly
let $a=\cos(\tfrac{2\pi}{5})$ and $b=\cos(\tfrac{4\pi}{5})$. Then:

$\begin{split}P(q)=&\;(2-2b)q+(6-6b)+(7-2b)q^{-1}-10aq^{-2}\\\
&\;-10aq^{-3}+(7-2b)q^{-4}+(6-6b)q^{-5}+(2-2b)q^{-6}\\\
Q(q)=&\;(2-2a)q+(6-6a)+(7-2a)q^{-1}-10bq^{-2}\\\
&\;-10bq^{-3}+(7-2a)q^{-4}+(6-6a)q^{-5}+(2-2a)q^{-6}\leavevmode\nobreak\
.\end{split}$ (6.33)

We are going to show that _$P^{l}+Q^{l}$ has positive coefficients for all
$l$_. This implies the positivity of the coefficients of the invariants
corresponding to $H_{3}$ by the explicit formula of 6.8.

We note that the coefficients of $Q$ are positive, and so are those of $Q^{l}$
for all $l\geq 1$. We prove by induction $l\mapsto l+5$ that $P^{l}$ has
positive coefficients for $l\geq 5$. This is checked by direct computation for
$P^{5}$ to $P^{9}$. Then $P^{l+5}=P^{5}P^{l}$ gives the induction heredity.
For $l\in\\{1,2,3,4\\}$ one checks explicitly that $P^{l}+Q^{l}$ is positive.

To prove the second part, we note by explicit computation that in types
$G_{2},F_{4},H_{4}$ and $E_{8}$, the polynomial has negative coefficients for
$g=0$ and $k=3$.

Consider now type $I_{2}(m)$ with $m=2l+1$ odd. Then by [GP00, Thm 8.3.4], the
Schur elements are given by

$\begin{split}s_{0}&=1+2q+2q^{2}+...+2q^{m-1}+q^{m}\\\
s_{l+1}&=1+2q^{-1}+...+2q^{1-m}+q^{-m}\\\ s_{j}&=\frac{m}{2-2\cos(\frac{2\pi
j}{m})}(q-2\cos(\tfrac{2\pi j}{m})+q^{-1})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{for}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq j\leq l\end{split}$ (6.34)

where $s_{0}$ and $s_{l+1}$ correspond to 1-dimensional representations and
all others to representations which are of dimension 2. Hence the constant
term of $P_{0,3,W}(q)$ is given by

$2-8m\sum_{j=1}^{l}\frac{\cos(\frac{2\pi j}{m})}{1-\cos(\frac{2\pi
j}{m})}=2-8m\left(-l+\frac{1}{2}\sum_{j=1}^{l}\frac{1}{\sin^{2}(\frac{\pi
j}{m})}\right)\leavevmode\nobreak\ .$ (6.35)

For $l\geq 5$ we have $\frac{\pi^{2}}{m^{2}}<\frac{2}{5l}$ (by analysis of the
roots of a quadratic polynomial in $l$). Using $\sin(x)\leq x$ we obtain:

$-l+\frac{1}{2}\sum_{j=1}^{l}\frac{1}{\sin^{2}(\frac{\pi
j}{m})}>-l+\frac{1}{2}\frac{1}{\sin^{2}(\frac{\pi}{m})}>\frac{l}{4}\leavevmode\nobreak\
.$ (6.36)

Thus, the constant term is smaller than $2-2ml$ which is strictly negative for
$l\geq 5$. For $l=3$ and $l=4$, explicit computations also show negative
constant terms. For $l=2$, one checks that for $g=0,k=4$ there are negative
coefficients in the polynomial.

Consider type $I_{2}(m)$ with $m=2l$ even. Then again by [GP00] Theorem 8.3.4,
in addition to the Schur elements given above (for $1\leq j\leq l-1$ for the
2-dimensional representations), there are two others whose corresponding
representation is 1-dimensional given by:

$s_{\varepsilon_{1}}=s_{\varepsilon_{2}}=l(q+2+q^{-1})\leavevmode\nobreak\ .$
(6.37)

Hence the constant term of $P_{0,3,W}(q)$ is given by

$2+2m-8m\left(-(l-1)+\sum_{j=1}^{l-1}\frac{1}{2\sin^{2}(\frac{\pi
j}{m})}\right)\leavevmode\nobreak\ .$ (6.38)

The same techniques as for $m$ odd apply and give that this constant term is
negative for $l\geq 5$. For $l=3,4$, explicit computations show that the
constant term is still negative. This concludes for the cases $I_{2}(m)$ with
$m\geq 5$. ∎

######  Remark 6.11.

Recall that the invariants are traces of elements of the form

$(C_{w}C^{w})^{k-1}(C_{x}C_{y}C^{x}C^{y})^{g}\leavevmode\nobreak\ .$ (6.39)

These expressions are independent of the choice of basis $(C_{w})$, and hence
we may choose the Kazhdan–Lusztig basis $(b_{w})$ to do the computations.
Despite all the known positivity results concerning this very special basis of
the Hecke algebra, we are unable to deduce from that the positivity of our
invariants because of the appearance of the dual basis $(b^{w})$. 6.10 can be
seen as a family of positivity properties in the center of the Hecke algebra
indexed by punctured surfaces.

Let us analyze the case of a general ciliated surface with boundary components
labeled by elements of the Kazhdan–Lusztig basis (KL basis for short in the
sequel).

######  Proposition 6.12.

In type $A$, all irreducible characters evaluated at elements of the
Kazhdan–Lusztig basis are positive.

###### Proof.

Fix an irreducible representation with character $\chi_{\lambda}$. In type
$A$, we know that this representation is a cell representation for some left
cell $\Lambda$. Further by [Neu06, Eq. 4] , we have

$\chi_{\lambda}(b_{w})=\sum_{x\in\Lambda}\operatorname{tr}b_{w}b_{x}b^{x}\leavevmode\nobreak\
.$ (6.40)

Since the structure constants in the KL basis are positive, all
$\operatorname{tr}b_{x}b^{x}b_{w}$ are positive. ∎

Let $\mathcal{H}_{\geq 0}$ denote the set of elements in the Hecke algebra
which have non-negative coordinates in the KL basis.

######  Corollary 6.13.

In type $A$, the invariant corresponding to a ciliated surface with boundary
labels in $\mathcal{H}_{\geq 0}$, has positive coefficients.

###### Proof.

Let $\alpha$ be the product of the labels on the boundary following the latter
according to the orientation induced by the surface. Since $\mathcal{H}_{\geq
0}$ is stable under product, we get $\alpha\in\mathcal{H}_{\geq 0}$. In type
$A$ both the Schur elements and the characters $\chi_{\lambda}(b_{w})$ are
positive, and we conclude with the explicit expression of 6.9. ∎

For all other types the characters $\chi_{\lambda}(b_{w})$ can have negative
coefficients and hence the invariant can have negative coefficients as well.
It is for example the case for $g=2,k=1,W=B_{2}$ and $\alpha=b_{rsr}$.

Using the link between the standard trace and the expansion of an element in a
given basis, we get:

######  Corollary 6.14.

In type $A$, any expression of the form
$(C_{w}C^{w})^{k}(C_{x}C_{y}C^{x}C^{y})^{g}$ has positive coefficients in the
dual Kazhdan–Lusztig basis $(b^{w})$.

###### Proof.

The coefficient of $g=(C_{w}C^{w})^{h}(C_{x}C_{y}C^{x}C^{y})^{g}$ along
$b^{w}$ when expressed in the KL basis is $\operatorname{tr}b_{w}g$ which is
positive by the previous corollary. ∎

######  Example 6.15.

For $W=\mathfrak{S}_{2}$, the Hecke algebra is commutative and the invariant
is given by $\operatorname{tr}((C_{1}C^{1}+C_{s}C^{s})^{m})$ in any basis
$(C_{1},C_{s})$. The dual KL basis is
$(b^{1},b^{s})=(h_{1}-q^{-1/2}h_{s},h_{s})$. Using induction, one easily
checks that

$\operatorname{tr}((b_{1}b^{1}+b_{s}b^{s})^{m})=((1+q)^{m-1}+(1+q^{-1})^{m-1})b^{1}+q^{(m-1)/2}(q^{1/2}+q^{-1/2})^{m}b^{s}\leavevmode\nobreak\
.$ (6.41)

Each coefficient is indeed positive. Note that the first coefficient is
$P_{0,m+1,\mathfrak{S}_{2}}(q)$.

If we use labels with positive coefficients in the basis adapted to the
Wedderburn decomposition, we get:

######  Proposition 6.16.

Consider a ciliated surface with labeled boundary, such that the product of
the labels on each boundary component has positive coefficients in the
Wedderburn-adapted basis. Then the polynomial invariant is positive.

###### Proof.

This follows from the explicit expression in 6.9 and the fact that the
characters evaluated at the Wedderburn adapted basis are non-negative. ∎

##  7 Perspectives

Our construction can be generalized in different seemingly promising ways,
that we list and comment briefly.

First, it is clear that our construction of a $1$-dimensional quantum field
theory for which cobordisms are ciliated surfaces does not only apply to Hecke
algebras corresponding to finite Coxeter systems, but works for any symmetric
finitely generated algebra over nice rings. For example, there should be
similar TQFTs for cyclotomic Hecke algebras and Yokonuma–Hecke algebras
[Chl16]. We believe that the explicit expressions of 6.8 and 6.9 stay true,
but properties like the invariance under $q\mapsto q^{-1}$, positivity or the
interpretation via counting of higher laminations might not hold in general.

A further generalization would be to aim to similar TQFTs associated with
symmetric algebras which are not finitely generated, such as affine Hecke
algebras. As already underlined in the introduction and in 4.4, our original
motivation for the present work was to study the space of functions over
character varieties corresponding to non-closed surfaces and its canonical
basis: higher laminations. The Satake correspondence is expected to play a
role in this story. Since it identifies the spherical affine Hecke algebra
corresponding to an algebraic reductive Lie group $G$ with the space of
representations of the Langlands dual $G^{\vee}$, the generalization of the
TQFTs constructed above to spherical affine Hecke algebras should be related
to the corresponding character varieties. Since those symmetric algebras are
not finitely generated anymore, the construction of the TQFTs must involve
some regularization of the infinite sums which then appear in the gluing
process. Higher laminations, which would generalize finite higher laminations
to this new setup at least in the case of affine Hecke algebras, would be very
similar to the spectral networks of [GMN13], calling for a physical
interpretation of our construction and possibly an understanding in terms of
BPS states counting in $4d$ $\mathcal{N}=2$ theories of class $S$.

Hecke algebras are famously categorified into Soergel bimodules [Soe07]. Can
our construction also be expressed in term of these bimodules? Whereas it was
tempting to look for such a categorified version of our TQFTs to explain the
positivity properties of the invariants we observed at the very beginning of
our study, we have seen that the positivity does not hold in general and hence
it cannot be directly related to Soergel bimodules. The latter indeed exist
for all the Hecke algebras corresponding to finite Coxeter systems. Thus we
can wonder whether there exists another categorification - maybe only for type
$A$, which would explain the positivity properties of the Schur elements of
the Hecke algebras as described in Appendix B, or the positivity properties of
the invariants corresponsing to punctured surfaces.

##  Appendix A Computation using Sage and CHEVIE

In this appendix, we describe two ways to compute our invariants by computer,
the first using Sage [S+21] and the second using CHEVIE, a package of Gap3
(see [Mic15] and [GHL+96]).

###  A.1 Sage

We recommend the online platform SageMathCell where you can perform
computations using Sage without any installation. To compute the polynomial
invariant, we use Equation (5.29). For example, a possible code for computing
the polynomial $P_{1,3,A_{3}}$ is:

    Ψk = 3
    Ψg = 1
    ΨR.<v> = LaurentPolynomialRing(QQ)
    ΨH = IwahoriHeckeAlgebra(’A3’, v, -1/v)
    ΨW = H.coxeter_group()
    ΨT = H.T();
    ΨS = (sum(T(i)*T(i.inverse()) for i in W))**(k-1)*(sum
    Ψ(sum(T(i)*T(j)*T(i.inverse())*T(j.inverse()) for j in W)
    Ψfor i in W))**g
    Ψ

Using the parameters $v$ and $-1/v$ for H corresponds to the normalized
version of the Hecke algebra in which $(h_{s}+v)(h_{s}-1/v)=0$. An H.T(w) for
w in the Coxeter group is the standard basis element of H corresponding to w,
while H.Cp(w) is the element in the KL basis associated with w.

This code yields the sum $(h_{w}h^{w})^{k-1}(h_{x}h_{y}h^{x}h^{y})^{g}$.
However we are interested in its trace, hence the $P_{g,k,W}$ is the
coefficient of $1$ in the result, in which one can replace $v^{-2}$ by $q$.
Here are some examples of computations:

  * •

$P_{0,3,\mathfrak{S}_{3}}(q)=q^{3}+2q^{2}+10q+10+10q^{-1}+2q^{-2}+q^{-3}.$

  * •

$P_{1,1,\mathfrak{S}_{3}}(q)=q^{3}+2q^{2}+4q+4+4q^{-1}+2q^{-2}+q^{-3}.$

  * •

$P_{0,4,\mathfrak{S}_{3}}(q)=q^{6}+4q^{5}+8q^{4}+10q^{3}+24q^{2}+36q+50+...$

  * •

$P_{0,3,\mathfrak{S}_{4}}(q)=q^{6}+3q^{5}+5q^{4}+33q^{3}+67q^{2}+108q+142+...$

  * •

$P_{0,3,G_{2}}(q)=q^{6}+2q^{5}+2q^{4}+2q^{3}+2q^{2}+72q-18+...$

For a surface with one boundary component labeled by $h$ one needs to multiply
the big sum by $h$, and expand the result in the standard basis (to read of
the constant term). Here is an example for type $A_{3}$ and $h$ an element in
the KL-basis:

    ΨCp = H.Cp()
    Ψr,s,t = W.simple_reflections()
    ΨT(Cp(r*s*t*r)*S)
    Ψ

###  A.2 CHEVIE

CHEVIE is a package of Gap3 (not included in Gap4). We recommend the
installation from the webpage of Jean Michel. The advantage of CHEVIE is that
it knows the Schur elements and characters. So we can use the explicit
expression from 6.8 to compute our polynomial.

Here is a code computing $P_{0,3,E_{8}}$:

    Ψg:=0;;
    Ψk:=3;;
    ΨW:= CoxeterGroup("E",8);;
    Ψv:=X(Cyclotomics);; v.name:="v";;
    ΨH:=Hecke(W,[[v,-v^-1]]);;
    ΨT:=Basis(H,"T");;
    ΨCp:=Basis(H,"C’");;
    Ψschur:=SchurElements(H);;
    Ψlist:=[1..Length(schur)];;
    Ψdim:=HeckeCharValues(T());;
    ΨSum(list,i->dim[i]^k*schur[i]^(2*g-2+k));
    Ψ

Using the explicit formula from 6.8, we get the following general formula for
type $A_{2}$ where $m=2g-2+k$:

$P_{g,k,\mathfrak{S}_{3}}(q)=(1+2q+2q^{2}+q^{3})^{m}+(1+2q^{-1}+2q^{-2}+q^{-3})^{m}+2^{k}(q+1+q^{-1})^{m}.$

For type $G_{2}$:

$\displaystyle P_{g,k,G_{2}}(q)=$
$\displaystyle\;(1+2q+...+2q^{5}+q^{6})^{m}+(1+2q^{-1}+...+2q^{-5}+q^{-6})^{m}$
$\displaystyle\;+2(3q^{-1}+6+3q)^{m}+2^{k}(6q-6+6q^{-1})^{m}+2^{k}(2q+2+2q^{-1})^{m}.$

For a surface with one boundary component, we can compute the polynomial using
6.9: one only need to change the last row of the code above to:

    Ψh:=Cp(1);;
    ΨSum(list,i->dim[i]^(k-1)*schur[i]^(2*g-2+k)*HeckeCharValues(h)[i]);
    Ψ

One can of course change the value of $h$ at will.

##  Appendix B Positivity for Schur elements

We study in detail the coefficients of Schur elements associated to a
Iwahori–Hecke algebra and their positivity. The main tool is an explicit
formula for the Schur elements using generalized hook lengths from [Chl16]
(see in particular Example 2.5 for more references).

######  Theorem B.1.

The Schur elements $s_{\lambda}(q)$ have positive coefficients for all Coxeter
groups of classical type and for the exceptional types $E_{6}$ and $E_{7}$.

###### Proof.

For the exceptional types $E_{6}$ and $E_{7}$ an explicit computation using
CHEVIE shows the positivity of the Schur elements, while in all other
exceptional types there are Schur elements with negative coefficients.

For the classical types we use the explicit formula from Theorem 4.3 in
[Chl16] which was first published in [CJ12]. We need to introduce some
notations to state the formula. For classical types, an irreducible
representation $\lambda$ is described by a set of Young diagrams
$(\lambda^{(0)},...,\lambda^{(l-1)})$ also called a multipartition (for Weyl
groups we have $l=1$ or $l=2$). The generalized hook length at
$(i,j)\in\lambda$ with respect to two Young diagrams $\lambda$ and $\mu$ is

$h_{i,j}^{\lambda,\mu}=\lambda_{i}-i+\mu_{j}^{\prime}-j+1\leavevmode\nobreak\
,$ (B.1)

where $\lambda_{i}$ denotes the length of the $i$-th row in $\lambda$ and
$\mu^{\prime}$ is the conjugated Young diagram. For $\lambda=\mu$ this gives
the usual hook length.

We can now state the explicit formula. The Schur element indexed by a
multipartition $\lambda=(\lambda^{(0)},...,\lambda^{(l-1)})$ of the integer
$n$ is given by

$s_{\lambda}(q)=(-1)^{n(l-1)}q^{-N(\bar{\lambda})}\prod_{0\leq s\leq
l-1}\prod_{(i,j)\in\lambda^{(s)}}\left([h_{i,j}^{\lambda^{(s)},\lambda^{(s)}}]_{q}\prod_{0\leq
t\leq l-1,t\neq
s}(q^{h_{i,j}^{\lambda^{(s)},\lambda^{(t)}}}Q_{s}Q_{t}^{-1}-1)\right)\leavevmode\nobreak\
.$ (B.2)

Note that we have not introduced the notation $N(\bar{\lambda})$ because we
can ignore this part of the formula for our purposes.

Type $A$ corresponds to $l=1$ hence the formula simplifies to:

$s_{\lambda}(q)=q^{-N(\bar{\lambda})}\prod_{i,j\in\lambda}[h_{i,j}^{\lambda}]_{q}\leavevmode\nobreak\
.$ (B.3)

The Schur element is thus a product of quantized hook lengths (modulo a
shift). These quantum integers are all positive, and hence is the Schur
element $s_{\lambda}(q)$.

Type $B$ corresponds to $l=2,Q_{0}=q$ and $Q_{1}=-1$. An irreducible
representation is parameterized by a pair of Young diagrams $(\lambda,\mu)$,
which yields:

$s_{\lambda,\mu}(q)=q^{-N(\lambda\cup\mu)}\prod_{(i,j)\in\lambda}[h_{i,j}^{\lambda,\lambda}]_{q}(q^{2+\lambda_{i}-i+\mu_{j}^{\prime}-j}+1)\times\prod_{(i,j)\in\mu}[h_{i,j}^{\mu,\mu}]_{q}(q^{\mu_{i}-i+\lambda_{j}^{\prime}-j}+1)\leavevmode\nobreak\
.$ (B.4)

Since the quantum integers are positive, the Schur element again has positive
coefficients.

To obtain the results for type $D$, we have to use a link to type $B$ which is
established in [Chl09, Part 2.3] using Clifford theory. The result is that we
can reduce type $D$ to $l=2$ with parameters $Q_{0}=q$ and $Q_{1}=-q$. An
irreducible representation is given by an unordered pair of Young diagrams
$(\lambda,\mu)$. If $\lambda\neq\mu$ we get:

$s_{\lambda,\mu}(q)=\frac{1}{2}q^{-N(\lambda\cup\mu)}\prod_{(i,j)\in\lambda}[h_{i,j}^{\lambda,\lambda}]_{q}(q^{\lambda_{i}-i+\mu_{j}^{\prime}-j+1}+1)\times\prod_{(i,j)\in\mu}[h_{i,j}^{\mu,\mu}]_{q}(q^{\mu_{i}-i+\lambda_{j}^{\prime}-j+1}+1)\leavevmode\nobreak\
.$ (B.5)

For $\lambda=\mu$ we get

$s_{\lambda,\lambda}(q)=q^{-N(\lambda\cup\lambda)}\prod_{(i,j)\in\lambda}[h_{i,j}^{\lambda,\lambda}]_{q}^{2}(q^{\lambda_{i}-i+\mu_{j}^{\prime}-j+1}+1)^{2}\leavevmode\nobreak\
.$ (B.6)

In both cases the expressions show the positivity of the Schur elements. ∎

Apart from the positivity, the coefficients of the Schur elements for Coxeter
groups satisfy other interesting properties. Call a sequence of integers
$(a_{1},a_{2},...,a_{n})$ symmetric if $a_{i}=a_{n-i}$ and call it log-concave
if $a_{i}^{2}\geq a_{i-1}a_{i+1}$ (taking the logarithm gives precisely the
condition for a concave function).

######  Proposition B.2.

For any Coxeter group, the coefficients of the Schur elements are symmetric.
In type $A_{n}$, the coefficients are in addition log-concave.

The proof of the first part relies on a general formula for Schur elements in
terms of cyclotomic polynomials which are symmetric. The second part relies on
Chlouveraki–Jacon’s formula and the observation that log-concavity is
multiplicative.

###### Proof.

In [Chl16], Formula (2.2) gives the following expression of Schur elements:

$s_{\lambda}=\xi_{\lambda}q^{-a_{\lambda}}\prod_{\Phi\in\operatorname{Cyc}_{\lambda}}\Phi(q^{n_{\lambda,\Phi}})\leavevmode\nobreak\
,$ (B.7)

where $\xi_{\lambda}\in\mathbb{R}$, $n_{\lambda,\Phi}\in\mathbb{Z}_{>0}$ and
$\operatorname{Cyc}_{\lambda}$ is a set of cyclotomic polynomials. This
formula comes from a case-by-case study.

Since we are interested in the symmetry of the sequence of coefficients, we
can ignore the prefactor $\xi_{\lambda}q^{-a_{\lambda}}$. Furthermore,
cyclotomic polynomials are known to be symmetric. Finally a product of
symmetric polynomials is still symmetric. This proves the first part.

In type $A_{n}$, we have seen above in Equation (B.3) that

$s_{\lambda}(q)=q^{-N(\bar{\lambda})}\prod_{i,j\in\lambda}[h_{i,j}^{\lambda}]_{q}\leavevmode\nobreak\
.$ (B.8)

The sequence of coefficients in a quantum integer is $(1,1,...,1)$ which is
positive and log-concave. By Proposition 2 in [Sta89], the product of positive
log-concave sequences preserves these properties, thus the coefficients of the
Schur elements are log-concave. ∎

## References

  * [Ati88] Michael F. Atiyah.  Topological quantum field theory.  Publications Mathématiques de l’IHÉS, 68:175–186, 1988\.
  * [Chl09] Maria Chlouveraki.  Blocks and families for cyclotomic Hecke algebras.  Springer, 2009.
  * [Chl16] Maria Chlouveraki.  Hecke algebras, generalisations and representation theory.  https://hal.archives-ouvertes.fr/tel-01411063/document, 2016.  Habilitation thesis.
  * [CJ12] Maria Chlouveraki and Nicolas Jacon.  Schur elements for the Ariki–Koike algebra and applications.  Journal of Algebraic Combinatorics, 35(2):291–311, 2012.
  * [Cur88] Charles W. Curtis.  Representations of Hecke algebras.  Astérisque, 168:13–60, 1988.
  * [EW16] Ben Elias and Geordie Williamson.  Soergel calculus.  Representation Theory of the American Mathematical Society, 20(12):295–374, 2016.  arXiv:math/1309.0865.
  * [FG07] Vladimir Fock and Alexander Goncharov.  Dual Teichmüller and lamination spaces.  Handbook of Teichmüller theory, 1(11):647–684, 2007.  arXiv:math/0510312.
  * [GHL+96] Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer.  Chevie—a system for computing and processing generic character tables.  Applicable Algebra in Engineering, Communication and Computing, 7(3):175–210, 1996.
  * [GMN13] Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke.  Wall-crossing, Hitchin systems, and the WKB approximation.  Advances in Mathematics, 234:239–403, 2013.  arXiv:hep-th/0907.3987.
  * [GP00] Meinolf Geck and Götz Pfeiffer.  Characters of finite Coxeter groups and Iwahori-Hecke algebras.  Number 21. Oxford University Press, 2000.
  * [Iwa64] Nagayoshi Iwahori.  On the structure of a Hecke ring of a Chevalley group over a finite field.  Journal of the Faculty of Science, University of Tokyo. Sect. 1, Mathematics, Astronomy, Physics, Chemistry, 10(2):215–236, 1964.
  * [KL79] David Kazhdan and George Lusztig.  Representations of Coxeter groups and Hecke algebras.  Inventiones mathematicae, 53(2):165–184, 1979.
  * [Le16] Ian Le.  Higher laminations and affine buildings.  Geometry & Topology, 20(3):1673–1735, 2016.  arXiv:math/1209.0812.
  * [Lib19] Nicolas Libedinsky.  Gentle introduction to Soergel bimodules I: the basics.  Sao Paulo Journal of Mathematical Sciences, 13(2):499–538, 2019\.  arXiv:math/1702.00039.
  * [LZ13] Sergei Lando and Alexander Zvonkin.  Graphs on surfaces and their applications, volume 141.  Springer Science & Business Media, 2013.
  * [Mat64] Hideya Matsumoto.  Générateurs et relations des groupes de Weyl généralisés.  Comptes rendus hebdomadaires de l’académie des sciences, 258(13):3419, 1964.
  * [Mic15] Jean Michel.  The development version of the chevie package of gap3.  Journal of Algebra, 435:308–336, 2015.
  * [Neu06] Max Neunhöffer.  Kazhdan–Lusztig basis, Wedderburn decomposition, and Lusztig’s homomorphism for Iwahori–Hecke algebras.  Journal of Algebra, 303(1):430–446, 2006.
  * [Ron09] Mark Ronan.  Lectures on buildings: updated and revised.  University of Chicago Press, 2009.
  * [S+21] W. A. Stein et al.  Sage Mathematics Software (Version 10.15.7).  The Sage Development Team, 2021.  http://www.sagemath.org.
  * [Soe07] Wolfgang Soergel.  Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen.  Journal of the Institute of Mathematics of Jussieu, 6(3):501–525, 2007.
  * [Sta89] Richard P. Stanley.  Log-concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry.  Ann. New York Acad. Sci, 576(1):500–535, 1989.
  * [Wil16] Geordie Williamson.  The Hodge theory of the Hecke category.  arXiv preprint arXiv:1610.06246, 2016.  arXiv:math/1610.06246.