We describe a reduced representative for the homotopy class of any positive (or negative)
Rouquier complex.
It is obtained via Gaussian elimination from the ordinary representative, given by a tensor
product of standard complexes. The underlying graded object is still a direct sum of
Bott-Samelson objects, but over subwords rather than subexpressions.

§ INTRODUCTION

§.§ The 2-braid group

Rouquier complexes were introduced in <cit.> to study actions of the (generalized)
braid group $B_W$ on categories.

These actions appeared in representation theory already in the works of Carlin
<cit.> or Rickard <cit.>, and
a precise definition was made by Deligne <cit.>.
They describe higher symmetries of the categories acted upon and usually give rich information
about them. A classical example is the
braid action on $\mathcal{D}^{\text{b}}(\mathcal{O})$, the bounded derived category of
the category $\mathcal{O}$ of representations of a complex semisimplie Lie algebra $\mathfrak{g}$.
A geometric counterpart is the action on $\mathcal{D}^{\text{b}}(G/B)$, the bounded derived category of
sheaves over the flag variety $G/B$ associated with a reductive algebraic group $G$ and
a Borel subgroup $B$.

Rouquier pointed out the interest of the category of self-equivalences
induced by such an action. So he introduced the
2-braid group $\mathscr{B}_W$, that upgrades $B_W$ to a category and serves as a
model for this purpose.

It is a subcategory of $\Kb(\D)$, the homotopy category of the Hecke category $\D$, and its
objects, called Rouquier complexes, are all possible tensor products between certain
standard and costandard complexes $F_s$ and $F_s^{-1}$, for $s\in S$.
Such products satisfy, up to homotopy, the relations of the braid group
so their isomorphism classes
form at least a quotient of $B_W$. See Remark <ref> below for more detail.

This category also plays an important role in algebraic topology.
Khovanov and Rozansky <cit.> began the construction of link invariants from
Rouquier complexes, that was made precise by Khovanov <cit.> as
a triply graded link homology. The idea is that the homology
of the complex obtained computing Hochschild (co)homology of the Rouquier complex $F_\omega$
associated to a braid $\omega$, is an invariant, up to an overall shift, of the link $\overline{\omega}$ obtained by closing $\omega$
(Rouquier complexes come with a cohomological and a polynomial degree, and Hochschild homology
provides a third grading).

When working with Rouquier complexes, it is useful to consider appropriate representatives
in the category $\Cb(\D)$ of complexes. The natural ones are just the tensor products between
the standard and costandard complexes.

When the category $\D$ is Krull-Schmidt (see Theorem <ref>), any complex
admits a minimal subcomplex: a homotopy equivalent summand with no
contractible direct summand.
In this	paper we find an intermediary reduction for positive (or negative) Rouquier complexes
(i.e. with only $F_s$'s, or resp. $F_s^{-1}$'s) which is
available with no restriction on the coefficients. The minimal subcomplex is very hard to
find in general, but one can see our result as a first step of such a reduction.

§.§ Diagrammatic Hecke category

As we mentioned, Rouquier complexes are certain objects of the homotopy
category of the Hecke category. The original definition was in terms of Soergel bimodules.
In this paper we will use the diagrammatic presentation of the category due to Elias
and Williamson <cit.>.

The generating objects, denoted $B_s$, correspond to simple reflections, and are
represented as colored points (one color for each $s\in S$).
Products between these generators are called Bott-Samelson objects. They correspond
to words and are represented by sequences of points in a line.
The morphisms between
Bott-Samelson objects are described by certain diagrams inside the strip
$\mathbb{R}\times [0,1]$. Namely, these are
planar graphs obtained from some generating vertices, that connect the sequences of
colored points corresponding to the source and the target,

identified by some relations.
This diagrammatic Hecke category is available for any Coxeter system, equipped with a realization $\mathfrak{h}$ (i.e. a
finite rank representation over the coefficient ring satisfying certain properties).
For more detail see  <ref> below. Under certain conditions this category is equivalent to
that of Soergel bimodules and has a geometric counterpart, when $W$ is a Weyl group, given by
equivariant parity sheaves on the corresponding flag variety.

Let us pass to the homotopy category. For any $s\in S$, the standard and costandard
complexes $F_s$ and $F_{s}^{-1}$ are defined as

\[
\begin{tikzcd}[row sep=small,%
execute at end picture={%
\def\hdott{.32cm}
\pic[red,yshift=.3em] at (A) {dot={\hdott}};
\pic[red,yshift=1.5em,yscale=-1] at (B) {dot={\hdott}};
F_{\re{s}}=\,\cdots\ar[r]		& 0 \ar[r]		& 0	\ar[r]								& B_{\re{s}}\ar[r,""{coordinate,name=A}]	& \un(1)\ar[r]	& 0\ar[r]		& \cdots \\
& \cohdeg{-2}	& \cohdeg{-1}							& \cohdeg{0}											& \cohdeg{1}	& \cohdeg{2}	& \\
F_{\re{s}}^{-1}=\,\cdots\ar[r]	& 0\ar[r]		& \un(-1)\ar[r,""{coordinate,name=B}]	& B_{\re{s}}\ar[r]							& 0	\ar[r]		& 0\ar[r]		& \cdots \\
\end{tikzcd}
\]

where the numbers in the middle row denote the
cohomological degree in $\Kb(\D)$.
Then Rouquier complexes are are all possible products between the $F_{s}^{\pm 1}$'s.

§.§ Results

We consider a positive Rouquier complex, of the form:

\[
F_{\uw}^\bullet=F_{s_1}F_{s_2}\dots F_{s_n}
\]

for $s_i\in S$. As a graded object, this is the direct sum of Bott-Samelson
objects indexed by the $2^n$ subexpressions of $\uw=s_1s_2\dots s_n$.
We find a summand $F_{\uw}$, in the category $\Cb(\D)$ which,
as a graded object, is a direct sum indexed over subwords (each of which can correspond
to several subexpressions). Then we show the following (see Theorem <ref> below).

The inclusion of the complex $F_{\uw}$ in $\Roudg{\uw}$ is a homotopy equivalence.

Let us see an example of our simplification.

Let $\uw=ssttss$. The original complex $\Roudg{\uw}$ is drawn in Figure <ref>.
It is a cube of dimension $6$, whose vertices are labeled by the $2^6=64$ subexpressions of
$ssttss$. The arrows (the edges of the cube) describe the differential map: each subexpression
has a map towards those subexpressions that can be obtained by canceling a symbol from it.

(v000000) at (0.3,0);
(v000001) at (1,5);(v000010) at (1,3);(v000100) at (1,1);(v001000) at (1,-1);(v010000) at (1,-3);(v100000) at (1,-5);
(v000011) at (2,14);(v000101) at (2,12);(v000110) at (2,10);(v001001) at (2,8);(v001010) at (2,6);(v001100) at (2,4);(v010001) at (2,2);(v010010) at (2,0);(v010100) at (2,-2);(v011000) at (2,-4);(v100001) at (2,-6);(v100010) at (2,-8);(v100100) at (2,-10);(v101000) at (2,-12);(v110000) at (2,-14);
(v000111) at (3,19);(v001011) at (3,17);(v001101) at (3,15);(v001110) at (3,13);(v010011) at (3,11);(v010101) at (3,9);(v010110) at (3,7);(v011001) at (3,5);(v011010) at (3,3);(v011100) at (3,1);(v100011) at (3,-1);(v100101) at (3,-3);(v100110) at (3,-5);(v101001) at (3,-7);(v101010) at (3,-9);(v101100) at (3,-11);(v110001) at (3,-13);(v110010) at (3,-15);(v110100) at (3,-17);(v111000) at (3,-19);
(v001111) at (4,14);(v010111) at (4,12);(v011011) at (4,10);(v011101) at (4,8);(v011110) at (4,6);(v100111) at (4,4);(v101011) at (4,2);(v101101) at (4,0);(v101110) at (4,-2);(v110011) at (4,-4);(v110101) at (4,-6);(v110110) at (4,-8);(v111001) at (4,-10);(v111010) at (4,-12);(v111100) at (4,-14);
(v011111) at (5,5);(v101111) at (5,3);(v110111) at (5,1);(v111011) at (5,-1);(v111101) at (5,-3);(v111110) at (5,-5);
(v111111) at (5.7,0);
in 0,1
i̱n 0,1
i̧n 0,1
ịn 0,1
in 0,1
in 0,1
(v̧̱) circle (1pt);
(ņ̱) at (v̧̱) ;

i̱n 0,1
i̧n 0,1
ịn 0,1
in 0,1
in 0,1
[-latex,gray] (n0̧̱)–(n1̧̱);

in 0,1
i̧n 0,1
ịn 0,1
in 0,1
in 0,1
[-latex,gray] (n0̧̣)–(n1̧̣);

in 0,1
i̱n 0,1
ịn 0,1
in 0,1
in 0,1
[-latex,gray] (n0̱)–(n1̱);

in 0,1
i̱n 0,1
i̧n 0,1
in 0,1
in 0,1
[-latex,gray] (ņ̱0)–(ņ̱1);

in 0,1
i̱n 0,1
i̧n 0,1
ịn 0,1
in 0,1
[-latex,gray] (ņ̱0̣)–(ņ̱1̣);

in 0,1
i̱n 0,1
i̧n 0,1
ịn 0,1
in 0,1
[-latex,gray] (ņ̱0)–(ņ̱1);

A picture of the complex $\Roudg{\underline{ssttss}}$.

The complex $F_{\uw}$ instead is showed in Figure <ref>. The vertices this time are labeled by the
23 subwords and the arrows (representing the new differential) connect a certain word with all its subwords (the symbol $\emptyset$ denotes the empty word).

(v) at (5.6,0);
(s) at (5,1);(t) at (5,-1);
(st) at (4,3);(ss) at (4,1);(tt) at (4,-1);(ts) at (4,-3);
(sst) at (3,5);(stt) at (3,3);(sss) at (3,1);(sts) at (3,-1);(tss) at (3,-3);(tts) at (3,-5);
(sstt) at (2,5);(ssts) at (2,3);(ssss) at (2,1);(stts) at (2,-1);(stss) at (2,-3);(ttss) at (2,-5);
(sstss) at (1,0);(sstts) at (1,3);(sttss) at (1,-3);
(ssttss) at (0,0);
(nv) at (v) $\emptyset$;
ıin s,t,ss,st,ts,tt,sss,sst,sts,stt,tss,tts,ssss,ssts,sstt,stss,stts,ttss,sstts,sstss,sttss,ssttss

(nı) at (ı) $\i$;

[-latex,gray] (nssttss)–(nsstts);[-latex,gray] (nssttss)–(nsstss);[-latex,gray] (nssttss)–(nsttss);
[-latex,gray] (nsstts)–(nsstt);[-latex,gray] (nsstts)–(nssts);[-latex,gray] (nsstts)–(nstts);
[-latex,gray] (nsstss)–(nssts);[-latex,gray] (nsstss)–(nssss);[-latex,gray] (nsstss)–(nstss);
[-latex,gray] (nsttss)–(nstts);[-latex,gray] (nsttss)–(nstss);[-latex,gray] (nsttss)–(nttss);
[-latex,gray] (nssss)–(nsss);
[-latex,gray] (nssts)–(nsst);[-latex,gray] (nssts)–(nsss);[-latex,gray] (nssts)–(nsts);
[-latex,gray] (nsstt)–(nsst);[-latex,gray] (nsstt)–(nstt);
[-latex,gray] (nstss)–(nsts);[-latex,gray] (nstss)–(nsss);[-latex,gray] (nstss)–(ntss);
[-latex,gray] (nstts)–(nstt);[-latex,gray] (nstts)–(nsts);[-latex,gray] (nstts)–(ntts);
[-latex,gray] (nttss)–(ntts);[-latex,gray] (nttss)–(ntss);
[-latex,gray] (nsss)–(nss);
[-latex,gray] (nsst)–(nss);[-latex,gray] (nsst)–(nst);
[-latex,gray] (nsts)–(nst);[-latex,gray] (nsts)–(nss);[-latex,gray] (nsts)–(nts);
[-latex,gray] (nstt)–(nst);[-latex,gray] (nstt)–(ntt);
[-latex,gray] (ntss)–(nts);[-latex,gray] (ntss)–(nss);
[-latex,gray] (ntts)–(ntt);[-latex,gray] (ntts)–(nts);
[-latex,gray] (nss)–(ns);
[-latex,gray] (nst)–(ns);[-latex,gray] (nst)–(nt);
[-latex,gray] (nts)–(nt);[-latex,gray] (nts)–(ns);
[-latex,gray] (ntt)–(nt);
[-latex,gray] (ns)–(nv);[-latex,gray] (nt)–(nv);

The complex $F_{\underline{ssttss}}$.

This reduction is obtained via a strong use of Gaussian elimination for complexes.
This is a tool from homological algebra which allows to reduce a given complex when some component
of the differential is an isomorphism. In  <ref>, we develop a
“large scale” version of this technique.

§ THE DIAGRAMMATIC HECKE CATEGORY

We now recall the construction,
by Elias and Williamson <cit.>, of the diagrammatic Hecke category associated with
(a realization of) an arbitrary Coxeter system.	We begin with some background about Coxeter
systems, braid groups and Hecke algebras.

§.§ Notation for Coxeter systems

Let $(W,S)$ be a Coxeter system. For each $s,t\in S$ let $m_{st}$ be the order of $st$ in $W$ (in particular $m_{ss}=1$),
so that

\[
W=\langle s\in S\mid (st)^{m_{st}}=1, \text{ if } m_{st}<\infty \rangle.
\]

We will restrict to the case $|S|<\infty$ of finitely generated Coxeter systems.

Let $\cword$ be the free monoid generated by $S$. Its elements are called Coxeter words
and will be denoted by underlined letters.
If $\uw$ is a Coxeter word, we say that it expresses
the element $w\in W$ if $w$ is its image via the natural morphism $\cword\rightarrow W$ (this morphism
is surjective because each $s$ is an involution).
We say that a word is monotonous if it contains only (repetitions of) one letter.
Let $\ell(\uw)$ denote the length of $\uw$, i.e. the number of its letters.
The word $\uw$ is said to be reduced if there is no shorter word expressing $w$.

For $\uw, \ux \in \cword$, we say that $\ux$ is a subword of $\uw$, and we
write $\ux\preceq \uw$,
if $\ux$ is obtained from $\uw$ by erasing some letters. In other words,
if $\uw=s_1s_2\cdots s_n$ with $s_i\in S$, then
$\ux$ is of the form $s_{i_1}s_{i_2}\cdots s_{i_r}$,
with $1\le i_1 \le i_2 \le \dots \le i_r\le n$. A subexpression is instead the datum of a subword
together with the information of the precise positions of the letters in the original word. This
can be encoded in a 01-sequence $\boe\in\{0,1\}^n$ where the ones correspond to the letters
that we pick and the zeros to those we do not pick.

Let $\uw=s_4s_2s_5s_1s_1s_3$. Then $\ux=s_4s_2s_1$ is a subword, obtained by erasing $s_5$, $s_3$ and one of the two
$s_1$'s. It corresponds to both subexpressions $110010$ and $110100$.

§.§ Braid groups

Let $(W,S)$ be a Coxeter system. We associate to it the corresponding
Artin-Tits group (or generalized braid group):

\[
B_W=\langle \sigma_s, s\in S\mid \underbrace{\sigma_s\sigma_t \dots}_{m_{st}}=\underbrace{\sigma_t\sigma_s \dots}_{m_{st}} \rangle.
\]

Hence we have a natural surjection $B_W\twoheadrightarrow W$ sending both $\sigma_s$ and
$\sigma_s^{-1}$ to $s$. Let $\Sigma^+=\{\sigma_s\}_{s\in S}$,
$\Sigma^-=\{\sigma_s^{-1}\}_{s\in S}$ and $\Sigma=\Sigma^+ \sqcup \Sigma^-$.
Let $\bword$ be the free monoid generated by $\Sigma$. Its elements are called
braid words. We call the elements of $\Sigma^+$ (and $\Sigma^-$) positive letters
(respectively negative letters).
A braid word is called positive (or negative) if all its letters are.
Subwords and subexpressions are defined in $\bword$ in the same way as in $\cword$.

We call Coxeter projection the map $\bword\rightarrow\cword$, sending both
$\sigma_s$ and $\sigma_s^{-1}$ to $s$. This has two distinguished sections: any Coxeter word $\uw=s_1s_2\dots s_k$
has a positive word lift $\sigma_{s_1}\sigma_{s_2}\dots \sigma_{s_k}$
and a negative word lift $\sigma_{s_1}^{-1}\sigma_{s_2}^{-1}\dots \sigma_{s_k}^{-1}$.

Given a non-empty $\uom\in\bword$, we can write it as

\begin{equation}\label{eq_posnegdec}
\uom_1\uom_2\dots \uom_k
\end{equation}

where $\uom_i$ are non-empty words such that: each $\uom_i$ is either positive or negative;
$\uom_i$ is positive if and only if $\uom_{i+1}$ is negative.
We call (<ref>) the alternating decomposition of $\uom$.

Finally let $\varpi:\bword\rightarrow (\ZZ,+)$ be the morphism of monoids with
$\varpi(\sigma_s)=1$ and $\varpi(\sigma_s^{-1})=-1$.
Hence $\varpi$ counts the difference between the number of positive and negative letters.

§.§ The Hecke algebra

Let $(W,S)$ be a Coxeter system and consider the ring $\Laur$ of Laurent polynomial in one
variable with integer coefficients.

The Hecke algebra $\HH$ associated with $(W,S)$ is the $\Laur$-algebra generated by $\{\delta_s\}_{s\in S}$
with relations

\[
\begin{cases}
(\delta_s+v)(\delta_s-v^{-1})=0& 	\text{for all $s\in S$}\\
\underbrace{\delta_s\delta_t\dots}_{m_{st}}=\underbrace{\delta_t\delta_s\dots}_{m_{st}}& 		\text{if $m_{st}<\infty$}
\end{cases}
\]

Notice that $\HH$ can be seen as the quotient of the group algebra of the braid group
$B_W$ over $\Laur$, by the above $v$-deformed version of the
involution relation in $W$.

For each $s\in S$ let $b_s:=\delta_s+v$. By the first relation, one gets

\begin{equation}\label{eq_bsbsalgebra}
\end{equation}

Recall that $\HH$ is a free $\Laur$-module with a
standard basis $\{\delta_w\}_{w\in W}$, where
$\delta_w:=\delta_{s_1}\delta_{s_2}\cdots \delta_{s_k}$ for any reduced
expression $\uw=s_1 s_2 \cdots s_k$ for $w$ (one shows that it does not depend on the chosen expression). One can actually start from this free module and
define the algebra structure in terms of the standard basis, see <cit.>,
<cit.>, or <cit.>.

The category that we are going to describe is a categorification
of the Hecke algebra. We need to introduce the notion
of realization of a Coxeter system.

§.§ Realizations of Coxeter systems
Let $(W,S)$ be a Coxeter system
and $\kk$ a commutative ring. A realization
of $W$, in the sense of <cit.>, consists in a free,
finite rank, $\kk$-module $\mathfrak{h}$, with distinguished elements
$\{\alpha_s^\vee\}_{s\in S} \subset \mathfrak{h}$ and
$\{ \alpha_s \}_{s\in S} \subset \mathfrak{h}^*=\Hom(\mathfrak{h},k)$,
that we call respectively simple coroots and simple roots,
satisfying the following conditions.

* If $\langle \cdot\,,\cdot\rangle:\mathfrak{h}^*\times \mathfrak{h}\rightarrow \kk$ is the natural
pairing, then $\langle \alpha_s,\alpha_s^\vee\rangle =2$, for each $s\in S$.

*  The $\kk$-module $\mathfrak{h}$ is a representation of $W$ via

\[
s(v)=v-\langle \alpha_s, v\rangle \alpha_s^{\vee},\quad\forall v\in \mathfrak{h},\,\forall s \in S.
\]

Then $W$ acts on $\mathfrak{h}^*$ by the contragredient representation, described by similar formulas:

\begin{equation}\label{eq_realiz2}
s(\lambda)=\lambda-\langle \lambda, \alpha_s^\vee \rangle \alpha_s,\quad\quad\forall \lambda\in \mathfrak{h}^*,\,\forall s \in S.
\end{equation}

* The technical condition <cit.> is satisfied.

The action of the realization over simple roots and coroots is encoded in the coefficients[For
the notation, we follow Bourbaki <cit.>: notice that the pairing considered here is the transposed
of that of <cit.>, and <cit.>.]
$a_{st}:=\langle \alpha_s,\alpha_t^\vee\rangle$, that we store in the
Cartan matrix $( a_{st} )_{s,t\in S}$.

Let $R=S(\mathfrak{h}^*)$, with $\mathfrak{h}^*$ in degree 2. The action of $W$ on
$\mathfrak{h}^*$ defined by (<ref>) extends naturally to $R$.
We define, for each $s\in S$, the Demazure operator $\partial_s : R \rightarrow R$, via:

\[
f \mapsto \frac{f-s(f)}{\alpha_s}.
\]

Then, in particular, we have[
Notice that $\partial_s(\alpha_t)$ is the same as in <cit.>, so this notation avoids
transposition issues.]
$\partial_s(\alpha_t)=\langle \alpha_t,\alpha_s^\vee\rangle$.

In this paper we will only consider balanced realizations (see <cit.>).
We also assume Demazure surjectivity (see <cit.>), namely we suppose
that, for each $s\in S$, the maps:

\begin{align*}
&\langle \alpha_s,\cdot \rangle: \mathfrak{h}\rightarrow \kk&
&\langle \cdot\,,\alpha_s^\vee \rangle: \mathfrak{h}^*\rightarrow \kk
\end{align*}

are surjective. In this case we can (and do) choose some $\delta_s\in\mathfrak{h}^*$ such that
$\langle \delta_s,\alpha_s^\vee\rangle=1$.
Notice that if $2$ is invertible in $\kk$ then this assumption always holds (take for instance
For more details, see <cit.>.

Here are some interesting examples of balanced realizations
(cf. <cit.> and <cit.>).

*  For any Coxeter group $(W,S)$ one can consider the geometric representation
$\hh:=\bigoplus_{s\in S}\mathbb{R}\alpha_s^\vee$ with
$\alpha_s\in \hh^*$ defined via $\langle \alpha_s,\alpha_t^\vee\rangle=-2\cos(\pi/m_{st})$,
with $m_{ss}:=1$ as usual, and $\pi/\infty:=0$.
Then the simple roots are linearly independent if and only if $W$ is finite.

*  If $W_\mathrm{f}$ is the Weyl group associated with a root datum
$(X,\roo,X^\vee,\roo^\vee)$, as defined in <cit.>, then one has a natural realization coming from the root datum.
One takes $\mathfrak{h}^*=\kk \otimes X$ (so $\mathfrak{h}=\kk \otimes X^\vee$), and
chooses sets $\{\alpha_s\}$ and $\{\alpha_s^\vee\}$ of simple roots and coroots respectively (in the usual sense).
One can also take $\mathfrak{h}$ to be $\kk\otimes \roo^\vee$ (then a natural
assumption to make is that $\kk\otimes \roo \times \kk\otimes \roo^\vee\rightarrow \kk$
is a perfect pairing so as to identify $\mathfrak{h}^*$ with the base-changed root lattice).

*  For the affine case $W=W_\mathrm{f}\ltimes \roo^\vee$, if $s\in S\setminus \Sf$ is an
affine simple reflection, consider the image of $s$ under the projection $W\rightarrow \Wf$. This will be a
reflection associated to some positive coroot $\tilde{\alpha}^\vee$. Then we put
$\alpha_{s}^\vee:=-\tilde{\alpha}^\vee$ and $\alpha_{s}:=-\tilde{\alpha}$.
In this case neither $\{\alpha_s\}$, nor $\{\alpha_s^\vee\}$ are linearly independent.
This is the realization considered in
<cit.>, up to a different choice of notation (exchanging roots and coroots).
Notice that the action of $W$ here factors through that of $\Wf$.

*  One can start with a generalized Cartan matrix
$(a_{st})_{s,t\in S}$ (i.e. a matrix satisfying $a_{ss}=2$, $a_{st}\in\ZZ_{\le 0}$ and
$a_{st}=0\Leftrightarrow a_{ts}=0$)
and build a realization as described in <cit.>.
One considers the Kac-Moody Lie algebra $\mathfrak{g}$, defined by $(a_{st})$,
and takes $\hh$ to be any $\ZZ$-lattice of the Cartan subalgebra of $\mathfrak{g}$,
containing the coroot lattice and such that the dual lattice $\hh^*$ contains the root
lattice. In this way one obtains the Kac-Moody realization of the Weyl group
$W$ associated with $\mathfrak{g}$.
Here both the simple roots and the simple coroots
are linearly independent.

*  More generally,
one can consider any Kac-Moody root datum associated to
a generalized Cartan matrix as above. This is a free $\ZZ$-module of finite rank $\Lambda$
with distinguished elements $\{\alpha_s\}\subset \Lambda$ and $\{\alpha_s^\vee\}\subset \Hom(\Lambda,\ZZ)$,
such that $\alpha_s^\vee(\alpha_t)=a_{ts}$.	Then one can take $\mathfrak{h}^*=\kk\otimes \Lambda$ and
$\mathfrak{h}=\kk\otimes \Hom(\Lambda,\ZZ)$.
From the same data one can construct a Kac-Moody ind-group scheme $\mathscr{G}$ over $\ZZ$, with
canonical Borel subgroup and a maximal torus, such that the character group of the
latter identifies with $\Lambda$. Then $\hh$ is a realization of the Weyl group associated to $\mathscr{G}$.
Hence this example is a generalization of <ref>.

§.§ Definition of the category

Given a realization $\mathfrak{h}$ over $\kk$ of a Coxeter system $(W,S)$,
one constructs the corresponding diagrammatic Hecke category $\D=\D(\mathfrak{h},\kk)$.
This is a $\kk$-linear monoidal category enriched in graded $R$-bimodules.
First one defines the Bott-Samelson category $\D_{\mathrm{BS}}$ by generators and relations,
then one gets $\D$ as the Karoubi envelope of the closure of $\D_{\mathrm{BS}}$ by direct
sums and shifts.

* The objects of $\DBS$ are generated by tensor product from objects $B_s$ for $s\in S$. So a general object
corresponds to a Coxeter word: if
$\uw=s_1\dots s_n$, let $B_{\uw}$ denote the object
$B_{s_1}\otimes \cdots \otimes B_{s_n}$. Let also $\un$ denote the monoidal unit, corresponding to the
empty word.

* Morphisms in $\Hom_{\DBS}(B_{\uw_1},B_{\uw_2})$
are $\kk$-linear combinations of Soergel graphs, which are defined as follows.

* We associate a color to each simple reflection.

* A Soergel graph is then a colored, decorated planar graph
contained in the planar strip $\mathbb{R}\times [0,1]$, with boundary in

* The bottom (and top) boundary is the arrangement
of boundary points colored according to the letters of the source word $\uw_1$
(and target word $\uw_2$, respectively).

* The edges of the graph are colored in such a way that those connected with the
boundary have colors consistent with the boundary points.

* The other vertices of the graph are either:

* univalent (called dots), which are declared of degree 1, or;

* trivalent	with three edges of the same color, of degree $-1$, or;

* $2m_{st}$-valent with edges of alternating colors corresponding to $s$ and $t$,
if $m_{st}<\infty$, of degree 0.
By poetic licence, we also call them $(s,t)$-ars.

[gray,dashed] (0,0) circle (1cm);
[red] (0,-1)–(0,0); [red] (0,0) circle (2pt);
[anchor=north,inner sep=8pt] at (0,-1) (a) Dot;
[gray,dashed] (0,0) circle (1cm);
[red] (0,1)–(0,0);
[red] (-30:1cm)–(0,0)–(-150:1cm);
[anchor=north,inner sep=8pt] at (0,-1) (b) Trivalent;

[gray,dashed] (0,0) circle (1cm);
[red] (0,1)–(0,0);
[red] (-30:1cm)–(150:1cm);
[blue,rotate=30] (0,1)–(0,0);
[blue,rotate=30] (-30:1cm)–(150:1cm);
ıin -60,-75,-90
(ı:.5cm) circle (.5pt);

[anchor=north,xshift=1.4*,inner sep=8pt] at (0,-1) (c) $(s,t)$-ar;

* Decorations are boxes labeled by homogeneous elements in
$R$ that can appear in
any region (i.e. connected component of the complement of the graph):
we will usually omit the boxes and just write the polynomials.

Then, composition of morphism is given by gluing diagrams vertically, whereas tensor product
is given by gluing them horizontally. The identity morphism of the object $B_{\uw}$ is
the diagram with parallel vertical strands colored according to the word $\uw$.

* These diagrams are identified via some relations:

Polynomial relations. The first such relation says that whenever two polynomials are in the same region, they multiply.
This gives morphism spaces the structure of $R$-bimodules, by acting on the leftmost or the
rightmost region.

The other polynomial relations are:

\begin{align*}
& \text{\emph{Barbell} relation:}  &
\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,.5)--(0,-.5); \fill[red] (0,.5) circle (2pt);\fill[red] (0,-.5) circle (2pt);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
\draw[gray,dashed] (0,0) circle (1cm); \node[draw] at (0,0) {$\alpha_{\re{s}}$};
\end{tikzpicture}			\\
& \text{\emph{Sliding} relation:} &
\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,1)--(0,-1);
\node[draw] at (.5,0) {$f$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,1)--(0,-1);
\node[draw, inner sep =.08cm] at (-.5,0) {$\re{s}(f)$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,1)--(0,.5); \fill[red] (0,.5) circle (2pt);
\draw[red] (0,-1)--(0,-.5); \fill[red] (0,-.5) circle (2pt);
\node[draw] at (0,0) {$\partial_{\re{s}}(f)$};
\end{tikzpicture}
\end{align*}

One color relations. These are the following:

\begin{align}
& \text{\emph{Frobenius associativity}:} &
\begin{tikzpicture}[baseline=-0.1cm,color=red,scale=sqrt(2)/3,scale=.7, transform shape]\label{associativity}
\draw (-1.5,-1.5)--(-0.6,0)--(0.6,0)--(1.5,-1.5);
\draw (-1.5,1.5)--(-0.6,0); \draw(0.6,0)--(1.5,1.5);
\draw[dashed,gray] (0,0) circle ({3/sqrt(2)});
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.1cm,color=red,scale=sqrt(2)/3,rotate=90,scale=.7, transform shape]
\draw (-1.5,-1.5)--(-0.6,0)--(0.6,0)--(1.5,-1.5);
\draw (-1.5,1.5)--(-0.6,0); \draw(0.6,0)--(1.5,1.5);
\draw[dashed,gray] (0,0) circle ({3/sqrt(2)});
\end{tikzpicture} \\
& \text{\emph{Frobenius unit}:} &
\begin{tikzpicture}[color=red,baseline=-0.1cm,scale=.7, transform shape]\label{dotline}
\draw[gray,dashed] (0,0) circle (1cm);
\draw (0,-1) -- (0,1); \draw (0,0)--(.3,0);\fill (0.3,0) circle (2pt);
\end{tikzpicture}
\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
\draw[gray,dashed] (0,0) circle (1cm);
\draw[red] (0,-1) -- (0,1);
\end{tikzpicture}
\\
&\text{\emph{Needle} relation:} &
\begin{tikzpicture}[baseline=0,scale=.7, transform shape]\label{eq_needle}
\draw[gray,dashed] (0,0) circle (1cm);
\draw[red] (0,-1)--(0,-0.3) arc (-90:270:0.3cm);
\end{tikzpicture}
&= 0 \notag
\end{align}

Relations (<ref>) and (<ref>) could be phrased by saying that
$B_s$ is a Frobenius algebra object in the category $\DBS$, which explains the terminology.
This was pointed out in
<cit.>. See <cit.>.

Two color relations. These allow to move dots,
or trivalent vertices, across $(s,t)$-ars.
We give the two versions, according to the parity of $m_{st}$:

\[
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.3,0); \coordinate (b) at (0.5,0);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
\draw[red] ({180-2*360/11}:1cm)--(a); \draw[blue] ({180-5*360/11}:1cm)--(b);
\draw[red] ({180-4*360/11}:1cm)--(a);% \draw[blue] ({180-6*360/11}:1cm)--(b);
\draw[blue] ({180+360/11}:1cm)--(a);
\draw[red] ({180+2*360/11}:1cm)--(a); \draw[blue] ({180+5*360/11}:1cm)--(b);
\draw[red] ({180+4*360/11}:1cm)--(a);% \draw[blue] ({180+6*360/11}:1cm)--(b);
\draw[blue](a)--(b);
\node at (-0.05,.5) {...};\node at (-0.05,-.5) {...};
\end{tikzpicture}
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.5,0); \coordinate (b) at (0.15,0.35); \coordinate (c) at (0.15,-0.35);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(b);
\draw[red] ({180-2*360/11}:1cm)--(b); \draw[blue] ({180-5*360/11}:1cm)--(b);
\draw[red] ({180-4*360/11}:1cm)--(b);% \draw[blue] ({180-6*360/11}:1cm)--(b);
\draw[blue] ({180+360/11}:1cm)--(c);
\draw[red] ({180+2*360/11}:1cm)--(c); \draw[blue] ({180+5*360/11}:1cm)--(c);
\draw[red] ({180+4*360/11}:1cm)--(c);% \draw[blue] ({180+6*360/11}:1cm)--(c);
\draw[blue](c)..controls (-0.15,0)..(b); \draw[red] (b) -- (c);\draw[red] (b)--(a)--(c);
\draw[red] (b) ..controls (0.7,0).. (c);
\node at (.2,.6) {...};\node at (.2,-.6) {...};\node at (.35,0) {...};
\end{tikzpicture}\quad\text{or}\quad
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.3,0); \coordinate (b) at (0.5,0);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
\draw[red] ({180-2*360/11}:1cm)--(a); \draw[red] ({180-5*360/11}:1cm)--(b);
\draw[blue] ({180-4*360/11}:1cm)--(a);% \draw[blue] ({180-6*360/11}:1cm)--(b);
\draw[blue] ({180+360/11}:1cm)--(a);
\draw[red] ({180+2*360/11}:1cm)--(a); \draw[red] ({180+5*360/11}:1cm)--(b);
\draw[blue] ({180+4*360/11}:1cm)--(a);% \draw[blue] ({180+6*360/11}:1cm)--(b);
\draw[red](a)--(b);
\node at (-0.05,.5) {...};\node at (-0.05,-.5) {...};
\end{tikzpicture}
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.5,0); \coordinate (b) at (0.15,0.35); \coordinate (c) at (0.15,-0.35);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(b);
\draw[red] ({180-2*360/11}:1cm)--(b); \draw[red] ({180-5*360/11}:1cm)--(b);
\draw[blue] ({180-4*360/11}:1cm)--(b);% \draw[blue] ({180-6*360/11}:1cm)--(b);
\draw[blue] ({180+360/11}:1cm)--(c);
\draw[red] ({180+2*360/11}:1cm)--(c); \draw[red] ({180+5*360/11}:1cm)--(c);
\draw[blue] ({180+4*360/11}:1cm)--(c);% \draw[blue] ({180+6*360/11}:1cm)--(c);
\draw[blue](c)..controls (-0.15,0)..(b); \draw[red] (b) -- (c);\draw[red] (b)--(a)--(c);
\draw[blue] (b) ..controls (0.7,0).. (c);
\node at (.2,.6) {...};\node at (.2,-.6) {...};\node at (.35,0) {...};
\end{tikzpicture}
\]

\[
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
\draw[blue] ({180-3*360/11}:1cm)--(a);
\draw[red] ({180-4*360/11}:1cm)--(a);% \draw[blue] ({180-6*360/11}:1cm)--(b);
\draw[blue] ({180+360/11}:1cm)--(a);
\draw[blue] ({180+3*360/11}:1cm)--(a); \fill[blue] (b) circle (2pt);
\draw[red] ({180+4*360/11}:1cm)--(a);% \draw[blue] ({180+6*360/11}:1cm)--(b);
\draw[blue](a)--(b);
\node at (-0.4,.5) {...};\node at (-0.4,-.5) {...};
\end{tikzpicture}
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
\draw[blue] ({180-3*360/11}:1cm)--(a);
\draw[blue] ({180+360/11}:1cm)--(a);
\draw[blue] ({180+3*360/11}:1cm)--(a);
\draw[red] ({180+4*360/11}:1cm) -- (b) -- ({180-4*360/11}:1cm); \draw[red](a)--(b);% \draw[blue] ({180+6*360/11}:1cm)--(b);
\node at (-0.4,.6) {...};\node at (-0.4,-.6) {...};
\node[draw,circle, inner sep=.1cm,fill=white] at (a) {$\JW$};
\end{tikzpicture}
\quad\text{or}\quad
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
\draw[red] ({180-3*360/11}:1cm)--(a);
\draw[blue] ({180-4*360/11}:1cm)--(a);% \draw[blue] ({180-6*360/11}:1cm)--(b);
\draw[blue] ({180+360/11}:1cm)--(a);
\draw[red] ({180+3*360/11}:1cm)--(a); \fill[red] (b) circle (2pt);
\draw[blue] ({180+4*360/11}:1cm)--(a);% \draw[blue] ({180+6*360/11}:1cm)--(b);
\draw[red](a)--(b);
\node at (-0.4,.5) {...};\node at (-0.4,-.5) {...};
\end{tikzpicture}
\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
\draw[dashed,gray] (0,0) circle (1cm);
\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
\draw[red] ({180-3*360/11}:1cm)--(a);
\draw[blue] ({180+360/11}:1cm)--(a);
\draw[red] ({180+3*360/11}:1cm)--(a);
\draw[blue] ({180+4*360/11}:1cm) -- (b) -- ({180-4*360/11}:1cm); \draw[blue](a)--(b);% \draw[blue] ({180+6*360/11}:1cm)--(b);
\node at (-0.4,.6) {...};\node at (-0.4,-.6) {...};
\node[draw,circle, inner sep=.1cm,fill=white] at (a) {$\JW$};
\end{tikzpicture}
\]

where the circles labeled $\JW$ are the Jones-Wenzel morphisms.
These are certain $\kk$-linear combinations of diagrams (with circular boundary
and $2m_{st}-2$ boundary points around it)
that can be described in terms of the 2-colored Temperley-Lieb category.

We will not need these relations in this paper. For further detail, we refer the reader to
<cit.>, <cit.> or <cit.>.

Three color relations. For each finite parabolic subgroup $W_I$
of rank 3, there is a relation ensuring compatibility between the three
corresponding $2m$-valent vertices. This is an analog of the
Zamolodchikov tetrahedron equation for braided monoidal
2-categories, generalized to all types (not just $A$).
We will not need them either, so for more detail see <cit.>.

Notice that all the relations are homogeneous, so the morphism spaces are graded

This completes the definition of $\DBS$.

The diagrammatic Hecke category $\D$ is the Karoubi envelope of the closure of $\DBS$
by shifts, denoted by $(\cdot)$, and direct sums.

§.§ Main properties of the Hecke category

The split Grothendieck group $[\D]_{\oplus}$ is naturally a $\Laur$-algebra: the ring structure is induced by the tensor product
and the action of $v$ corresponds to the shift (more precisely $v[B]:=[B(1)]$).
Hence we can state the following result (see <cit.>).

If $\kk$ is a complete local ring, then $\D$ is Krull-Schmidt and

there is a unique isomorphism of $\Laur$-algebras, called character,

\[
\mathrm{ch}:[\D]\overset{\sim}{\rightarrow}\HH
\]

sending the class of $B_s$ to $b_s$;

* Under the hypotheses of Theorem <ref> one can also classify the
indecomposable objects: they are parametrized, up to isomorphism and shift, by $W$.
Moreover morphism spaces are free as left $R$-modules and one can express
their graded rank in term of their classes in $\HH$.

* If $\chr(\kk)\neq 2$ and that the realization
$\mathfrak{h}$ is reflection faithful in the sense of <cit.>,

then $\D$ is equivalent to the category of Soergel bimodules, see <cit.>.

This assumption can be dropped by using the variant of the category of
Soergel bimodules introduced by Abe <cit.>.

*  Let $G$ be a reductive algebraic group over the complex numbers with
a Borel subgroup $B$ and a maximal torus $T$.

Then one can consider the realization of the Weyl group of Example <ref> <ref>.
Riche and Williamson <cit.> showed that, if $\kk$ is a complete local ring (with 2 invertible in
some cases) the category $\D$ is equivalent to that of $B$-equivariant parity sheaves
over the flag variety $G/B$ with coefficients in $\kk$, in the sense of <cit.>,
endowed with its convolution operation.
One also has a version for the affine Weyl group by considering the affine flag variety
$G(\!(t)\!)/I$ and the realization from Example <ref> <ref> (see <cit.>),
and more generally for any cristallographic group using the Kac-Moody flag variety and the realization in Example
<ref> <ref> (see <cit.>).
If $\chr(\kk)=0$, then, thanks to the Decomposition Theorem, this category conincides
with that of semisimple complexes.

The basic example of a diagrammatic computation in $\D$ is the following (see <cit.>).
If $s\in S$, then

\begin{equation}\label{eq_BsBscat}
B_s\otimes B_s\cong B_s(-1)\oplus B_s(1)
\end{equation}

In fact consider the following maps (where we omit the bottom and top lines):

\[
\def\d{1cm}
\iota_1=	\begin{tikzpicture}[yscale=-1,baseline=-.5*\h]
%							\draw (.5*\d,0)--(2.5*\d,0);
%							\draw (.5*\d,\h)--(2.5*\d,\h);
%							\draw[red] (\d,0) ..controls (\d,.5*\h).. (1.5*\d,.5*\h) ..controls (2*\d,.5*\h).. (2*\d,0);
\draw[red] (\d,0) -- (1.5*\d,.5*\h) -- (2*\d,0);
\draw[red] (1.5*\d,.5*\h)--(1.5*\d,\h);
\node at (1.5*\d,0) {$\delta_s$};
\end{tikzpicture},
\qquad
\pi_1=	\begin{tikzpicture}[baseline=.5*\h]
%							\draw (.5*\d,0)--(2.5*\d,0);
%							\draw (.5*\d,\h)--(2.5*\d,\h);
\draw[red] (\d,0)--(1.5*\d,.5*\h)--(2*\d,0);
\draw[red] (1.5*\d,.5*\h)--(1.5*\d,\h);
\end{tikzpicture},
\qquad
\iota_2=	\begin{tikzpicture}[baseline=-.5*\h,yscale=-1]
%							\draw (.5*\d,0)--(2.5*\d,0);
%							\draw (.5*\d,\h)--(2.5*\d,\h);
\draw[red] (\d,0)--(1.5*\d,.5*\h)--(2*\d,0);
\draw[red] (1.5*\d,.5*\h)--(1.5*\d,\h);
\end{tikzpicture},
\qquad
\pi_2=	-\begin{tikzpicture}[baseline=.5*\h]
%							\draw (.5*\d,0)--(2.5*\d,0);
%							\draw (.5*\d,\h)--(2.5*\d,\h);
%							\draw[red] (\d,0) ..controls (\d,.5*\h).. (1.5*\d,.5*\h) ..controls (2*\d,.5*\h).. (2*\d,0);
\draw[red] (\d,0) -- (1.5*\d,.5*\h) -- (2*\d,0);
\draw[red] (1.5*\d,.5*\h)--(1.5*\d,\h);
\node[yshift=-1pt] at (1.5*\d,0) {$s(\delta_s)$};
\end{tikzpicture}.
\def\d{.5cm}
\]

Using the relations, one can check that $\iota_1\pi_1+\iota_2\pi_2=\id_{B_sB_s}$, as well as

\begin{align*}
&\pi_1\iota_1=\id_{B_s(-1)},&	 &\pi_2\iota_2=\id_{B_s(1)},&	&\pi_i\iota_j=0\text{ if }i\neq j.
\end{align*}

The decomposition (<ref>) lifts the equality (<ref>) in the Hecke algebra.

The Hecke category plays an important role in representation theory. Let us mention some examples.

*  Let $\mathfrak{g}$ be a complex semisimple Lie algebra and fix a Cartan subalgebra $\mathfrak{h}$.
Let $W$ be the corresponding Weyl group and $S$ the set of simple reflections corresponding to a
choice of simple roots. Consider dot action of $W$ on $\mathfrak{h}^*$ given
by $w\cdot \lambda=w(\lambda+\rho)-\rho$ where $\rho$ is half the sum of positive
roots. Consider the category $\mathcal{O}$ of Bernstein-Gel'fand-Gel'fand and its
decomposition into blocks $\mathcal{O}=\bigoplus\mathcal{O}_\lambda$. For each
$s\in S$ let $\mu\in \mathfrak{h}^*$ be a weight with stabilizer $\{1,s\}$ under
the dot action. Recall that one has translation functors
$T^s:=T_0^{\mu}:\mathcal{O}_0\rightarrow\mathcal{O}_{\mu}$ and
$T_s:=T_\mu^0:\mathcal{O}_\mu\rightarrow\mathcal{O}_{0}$ which are left and right adjoint to each other.
Consider the endofunctor $\Theta_s=T_sT^s$ of $\mathcal{O}_0$.
Let $\D=\D(W,\mathfrak{h})$ be the Hecke category
associated with $(W,S)$ and the realization $\mathfrak{h}$. Then one has a functor
$\D\rightarrow \End(\Proj\mathcal{O}_0)$ sending $B_s$ to $\Theta_s$. This was shown
by Soergel <cit.> in terms of his bimodules, by embedding the subcategory of projectives
inside the category of modules over the coinvariant ring $C=R/R\cdot R^W_+$.

Via the isomorphism of the theorem, the indecomposable objects correspond to a basis of $\HH$
over $\Laur$. Consider realizations as in Example <ref> <ref> and <ref>.
Then one can show (see for example <cit.>) that this basis only depends on
the type of the root datum and on the characteristic $p$ of the residue field of
$\kk$. This is the $p$-canonical basis of $\HH$ appearing in
character formulas for tilting and simple representations for the group $G$:
see the work of Achar, Makisumi, Riche and
Williamson <cit.> and of Riche and Williamson <cit.>.

*  Soergel <cit.> conjectured that, if $\kk$ is a
characteristic zero field, then the latter is the Kazhdan-Lusztig basis.
For realizations over $\mathbb{R}$, this was proved by Elias and Williamson <cit.>.
This was usef to show, for a general Coxeter system, the positivity of the coefficients of
Kazhdan-Lusztig polynomials. Moreover Soergel had showed
that his conjecture implies Kazhdan-Lusztig's conjecture when $W$ is a Weyl group.

§ ROUQUIER COMPLEXES

In this section we introduce Rouquier complexes in the homotopy category of the
Hecke category.

§.§ Definition

Consider the homotopy category $\Kb(\D)$.
Recall that the monoidal structure of $\D$ extends to $\Kb(\D)$ as follows.
Let $\Ao,\Bo\in \Kb(\D)$, then:

\[
(\Ao\otimes \Bo)^i=\bigoplus_{p+q=i} A^p\otimes B^q,
\]

and the differential map, restricted to $A^p\otimes B^q$ is:

\begin{equation}\label{eq_signconv}
d\otimes \id + (-1)^p \id \otimes d.
\end{equation}

The unit is $\un[0]$ (the object $\un$ as a complex concentrated in degree 0).

Consider the standard and costandard complexes $F_s$ and $F_s^{-1}$
from the introduction.

If $\sigma=\sigma_{s}^{\pm 1}\in \Sigma$, then let
$F_\sigma$ denote $F_s^{\pm 1}$.
Then, for any braid word $\uom=\sigma_1 \sigma_2 \dots \sigma_n$, with $\sigma_i\in \Sigma$,
we put

\begin{equation}\label{eq_defrouq}
F_{\uom}^\bullet:=F_{\sigma_1}\otimes F_{\sigma_2}\otimes \dots \otimes F_{\sigma_n}.
\end{equation}

In the sequel we will often omit the symbol $\otimes$.
These objects are called

Rouquier complexes and the subcategory $\mathscr{B}_W$ that they form is called 2-braid group.

These complexes were introduced in <cit.> in terms of Soergel bimodules
to study natural transformations between the endofunctors induced by braid group actions on categories.
As an example, in the setting of Remark <ref> <ref>,
consider the complexes

\begin{align*}
\tilde{F}_s=\dots\rightarrow 0\rightarrow 0\rightarrow &\,\Theta_s\rightarrow \un\rightarrow 0\rightarrow \dots\\
\tilde{F}_s^{-1}=\dots\rightarrow 0\rightarrow\un \rightarrow &\,\Theta_s\rightarrow 0\rightarrow 0\rightarrow\dots
\end{align*}

where the morphisms are
given by the counit and unit of the two adjunctions between $T_s$ and $T^s$,
and where $\Theta_s$ is	in degree 0 for both. One can show that they are mutually inverse
endofunctors for the category $\Kb(\Proj\mathcal{O}_0)=\mathcal{D}^{\mathrm{b}}(\mathcal{O}_0)$.
and the assignment $\sigma_s\mapsto\tilde{F}_s$ defines an action of $B_W$ on this category <cit.>. Furhtermore
one can upgrade this to a functor from $\mathscr{B}_W$ to the category of self-equivalences
of $\mathcal{D}^{\mathrm{b}}(\mathcal{O}_0)$ <cit.>. Hence morphisms between
Rouquier complexes give natural transformations between these endofunctors.

In the geometric setting (see Remark <ref> <ref>), the
homotopy category $\Kb(\D)$ was considered by Achar and Riche
<cit.> as a modular version of the equivariant mixed derived category on
the (affine) flag variety. The standard and costandard complexes $F_s$ and $F_s^{-1}$ correspond to
the standard and costandard sheaves $\stdmix_{s}$ and $\cosmix_{s}$ in that setting.

§.§ Basic properties of Rouquier complexes

These properties were first proved by Rouquier <cit.> in the
language of Soergel bimodules. A sketch of proof is also given in <cit.>. One can find a
diagrammatic argument in <cit.>.

One has the following.

*  Let $s\in S$, then $F_s F_s^{-1}\cong F_s^{-1}F_s\cong \un[0]$.

*  Let $s,t\in S$ with $m_{st}<\infty$, then

\[
\underbrace{F_s F_tF_sF_t \cdots}_{m_{st} \text{ times}} \cong %
\underbrace{F_t F_sF_tF_s \cdots}_{m_{st} \text{ times}}
\]

Hence, for each pair of braid words $\uom_1$, $\uom_2$ expressing
the same element $\omega\in B_W$, there is an isomorphism $F_{\uom_1}^\bullet\cong F_{\uom_2}^\bullet$.

* (Rouquier Canonicity)
for each $\uom_1$ and $\uom_2$ as above, we have

\[
\Hom(F_{\uom_1}^\bullet,F_{\uom_2}^\bullet)\cong R,
\]

and one can chose
$\gamma_{\uom_1}^{\uom_2}$ such that the system
$\{\gamma_{\uom_1}^{\uom_2}\}_{\uom_1,\uom_2}$ is transitive.

Thanks to these properties,	the Rouquier complexes $F_{\omega}$'s associated to $\omega\in B_W$ are
well defined up to a canonical isomorphism and they form a quotient
of the braid group.	Rouquier conjectured that this quotient is just $B_W$ itself.
This faithfulness of the 2-braid group was shown in type $A$ by Khovanov and Seidel <cit.>, in simply laced
finite type by Brav and Thomas <cit.>, and in all finite types by Jensen <cit.>.

We want to investigate representatives for $F_{\omega}$ in the category of complexes $\Cb(\D)$.
First let us introduce some notation.

§.§ Notation for graded objects and complexes

Given a $\Bbbk$-linear additive category $\mathscr{C}$, a graded object is a family $\{A^q\}_{q\in \ZZ}$,
where $A^q$ is an object of $\mathscr{C}$ for all $q\in \ZZ$. If $A$ is an object of $\mathscr{C}$, then
$A[-q]$ denotes the graded object whose only nonzero entry is $A$ in degree $q$.
The direct sum of graded objects is defined in a natural way, so that we can also write

\[
\{A^q\}_{q\in\ZZ}=\bigoplus_{q\in \ZZ} A^q[-q].
\]

A (degree 0) morphism of graded objects $\Ao=\{A^q\}$ and $\Bo=\{B^q\}$ is a family of morphisms

\[
\{f^q:A^q\rightarrow B^{q}\}_{q\in \ZZ}.
\]

This defines a category $\mathcal{G}_0(\mathscr{C})$
of graded objects. The shift functor $[1]$ sends the graded object $\{A^q\}_{q\in \ZZ}$ to
$\{A^{q+1}\}_{q\in \ZZ}$. Then one can define the graded category $\mathcal{G}(\mathscr{C})$ with the
same objects and morphism spaces

\[
\Hom_{\mathcal{G}(\mathscr{C})}(\Ao,\Bo):=\bigoplus_{p\in \ZZ} \Hom_{\mathcal{G}_0(\mathscr{C})}(\Ao,\Bo[p]).
\]

The $p$-th graded piece is denoted by $\Hom^p(\Ao,\Bo)$ and its elements are said
homogeneous of degree $p$.
A complex in $\C(\mathscr{C})$ is then determined by a graded object $\Ao$ endowed with an
endomorphism $d:\Ao\rightarrow \Ao$ of degree 1, such that $d^2=0$.
Viceversa, let $\grad{(\cdot)}:\C(\mathscr{C})\rightarrow\mathcal{G}(\mathscr{C})$ denote
the forgetful functor sending $(\Ao,d)$ to $\Ao$.

§.§ Representatives for Rouquier complexes

The most immediate representatives for Rouquier complexes are just the tensor products (<ref>).
Let us describe them more precisely.

For a given subexpression $\boi$ of a braid word $\uom$, let $\uom_{\boi}$ denote the
subword corresponding to it. Then let $B_{\boi}$ be the Bott-Samelson object
corresponding to the Coxeter projection of $\uom_{\boi}$. For example, given

\[
\uom=\sigma_s\sigma_t\sigma_t\sigma_u^{-1},
\]

and $\boi=0101$, we have
$\uom_{\boi}=\sigma_t\sigma_u^{-1}$ and then $B_{\boi}=B_{\underline{tu}}=B_tB_u$.

By definition of tensor product of complexes, we have

\[
\grad{(\Roudg{\uom})}=\bigoplus_{\boi\in\{0,1\}^{\ell(\uom)}}B_{\boi}\Tate{q_{\boi}},
\]

where $q_{\boi}=\varpi(\uom)-\varpi(\uom_{\boi})$ and $\Tate{1}:=(1)[-1]$ denotes the Tate twist.
One can then compute the components of the differential map, according to
(<ref>). The only nonzero components $B_{\boi}\rightarrow B_{\boi'}$ are those
for which $\boi'$ is obtained from $\boi$ by changing one symbol in such a way that
In this case the component is either

\begin{align*}
\begin{tikzpicture}[x=0.3cm,y=0.4cm,baseline=0.36cm,every node/.style={font=\scriptsize}]
\draw[gray] (0.5,0)--(9.5,0);\draw[gray] (0.5,2)--(9.5,2);
\foreach \i in {1,2,4,6,8,9}{%
\draw[violet] (\i,0)--(\i,2);
\pic[violet] at (5,0) {dot={1cm}};
\node[font=\normalsize] at (3,1) {\dots}; \node[font=\normalsize] at (7,1) {\dots};
%						\node[anchor=north] at (1,0) {$0$};
%						\node[anchor=north] at (2,0) {$1$};
%						\node[anchor=north] at (5,0) {$i$};
\end{tikzpicture}&
\begin{tikzpicture}[x=0.3cm,y=0.4cm,baseline=0.36cm,every node/.style={font=\scriptsize}]
\draw[gray] (0.5,0)--(9.5,0);\draw[gray] (0.5,2)--(9.5,2);
\foreach \i in {1,2,4,6,8,9}{%
\draw[violet] (\i,0)--(\i,2);
\pic[violet,yscale=-1] at (5,2) {dot={1cm}};
\node[font=\normalsize] at (3,1) {\dots}; \node[font=\normalsize] at (7,1) {\dots};
%							\node[anchor=south] at (1,2) {$0$};
%							\node[anchor=south] at (2,2) {$1$};
%							\node[anchor=south] at (5,2) {$i$};
\end{tikzpicture}.&
\end{align*}

where $k$ is the number of 0's in $\boi$ preceding the changed symbol.

Then the complex $F_{\uom}^\bullet$ has the form
of a cube of dimension equal the number of letters in $\uom$: the vertices correspond to the
Bott-Samelson objects obtained from all possible subexpressions of $\uom$ and the edges are the arrows
describing the component of the differential map.

Let $\re{s},\bl{t}\in S$ and
$\uom=\sigma_s\sigma_s\sigma_t^{-1}$. Then the complex $\Roudg{\uom}=F_sF_sF_t^{-1}$
is the following:

[x=2.7cm,y=0.9cm,every pic/.style=x=1cm,scale=0.5,font=,-]

(s1t1) at (2,2) $B_sB_t(1)$;
(s2t1) at (2,0) $B_sB_t(1)$;
(n1) at (2,-2) $\un(1)$;
[-latex] (ss9) to
pic[midway,yshift=.5cm]bottomdot=sst3+ (sst);
[-latex] (ss9) to
pic[midway,yshift=.3cm]topdot=ss2+ (s1);
[-latex] (ss9) to
pic[midway,yshift=-.5cm]topdot=ss1+ (s2);
[-latex] (sst) to
pic[midway,yshift=.3cm]topdot=sst2+ (s1t1);
[-latex] (sst) to
pic[pos=.6,xshift=.8cm]topdot=sst1+ (s2t1);
[-latex] (s1) to
pic[pos=.4,xshift=-.6cm]bottomdot=st2- (s1t1);
[-latex] (s1) to
pic[near start,xshift=-.3cm]topdot=s1+ (n1);
[-latex] (s2) to
pic[pos=.6,xshift=.8cm]bottomdot=st2- (s2t1);
[-latex] (s2) to
pic[midway,yshift=-.3cm]topdot=s1- (n1);
[-latex] (s1t1) to
pic[midway,yshift=.5cm]topdot=st1+ (t2);
[-latex] (s2t1) to
pic[midway,yshift=.3cm]topdot=st1- (t2);
[-latex] (n1) to
pic[midway,yshift=-.5cm]bottomdot=t1+ (t2);
[green,font=] at (0,-3)-1; [green,font=] at (1,-3)0;
[green,font=] at (2,-3)1; [green,font=] at (3,-3) 2;
at (1,1) $\oplus$; at (1,-1) $\oplus$;
at (2,1) $\oplus$; at (2,-1) $\oplus$;

When the category $\D$ is Krull-Schmidt, one can
get rid of a maximal null-homotopic summand of each $F_{\uom}^\bullet$ and obtain
its minimal subcomplex representing $F_{\omega}$.
This is a summand which does not admit null-homotopic factor and one can show that it is
unique up to isomorphism (and not just homotopy equivalence): see <cit.>.

Minimal subcomplexes of Rouquier complexes are hard to find in general and for arbitrary
coefficients, this notion is not well defined.
For $\uom$ positive (or negative) we will now describe a representative $F_{\uom}$
for $F_{\omega}$ which is simpler than $F_{\uom}^\bullet$, and available
without restrictions on $\kk$. This can be consider an intermediate step towards a minimal subcomplex.
This is obtained by getting rid of a large
null-homotopic summand, via Gaussian elimination for complexes.
The graded object of the new complex is a direct sum over subwords, rather than subexpressions.

§ REDUCED REPRESENTATIVES FOR ROUQUIER COMPLEXES

Let us start with a simple example to illustrate our result.

Let $\uom=\sigma_{\re{s}}\sigma_{\re{s}}\sigma_{\re{s}}$. Then $F_{\uom}^\bullet$ is

\[
\begin{tikzcd}[row sep=tiny]
& B_{\underline{ss}}(1) \ar[r]\ar[rdd]		& B_s(2)\ar[rdd]	& \\
& \oplus									& \oplus			& \\
B_{\underline{sss}} \ar[ruu]\ar[r]\ar[rdd]	& B_{\underline{ss}}(1) \ar[ruu]\ar[rdd]		& B_s(2)\ar[r]	& \un(3)\\
& \oplus									& \oplus			& \\
& B_{\underline{ss}}(1) \ar[ruu]\ar[r]		& B_s(2)\ar[ruu]	& \\
\end{tikzcd}
\]

According to Example <ref>, the Bott-Samelson objects appearing above decompose as

\begin{align*}
&B_{\underline{ss}}(1)=B_s\oplus B_s(2), &
&B_{\underline{sss}}=B_s(-2)\oplus B_s^{\oplus 2} \oplus B_s(2).
\end{align*}

Hence the complex can be written

\begin{equation}\label{eq_exadecomcol}
\begin{tikzcd}[row sep=tiny]
& \bl{B_s}\oplus \gr{B_s(2)} \ar[r]\ar[rdd]		& \gr{B_s(2)}\ar[rdd]	& \\
& \oplus										& \oplus				& \\
\textcolor{violet}{B_s(-2)}\oplus \bl{B_s^{\oplus 2}} \oplus \gr{B_s(2)} \ar[ruu]\ar[r]\ar[rdd]	& \bl{B_s}\oplus \gr{B_s(2)} \ar[ruu]\ar[rdd]	& \gr{B_s(2)}\ar[r]		& \un(3)\\
& \oplus										& \oplus				& \\
& \bl{B_s}\oplus \gr{B_s(2)} \ar[ruu]\ar[r]		& \gr{B_s(2)}\ar[ruu]	& \\
\end{tikzcd}
\end{equation}

and can be filtered by the subquotients:

\begin{align*}
&\textcolor{violet}{G_3},& &\bl{G_2},& &\gr{G_1},& &G_0,&
\end{align*}

corresponding to the colors in the above picture (for a precise definition of subquotient in this setting,
see below).
One can now check that the components of the differential map in each $G_i$ are just identities. For instance, the blue and
the green pieces are

\begin{align*}
&\begin{tikzcd}[ampersand replacement=\&,row sep=tiny]
\& \bl{B_s} \\
\bl{B_s}\ar[ru]\ar[rd]	\& \oplus	\\
\oplus					\& \bl{B_s} \\
\bl{B_s}\ar[ru]\ar[rd]	\& \oplus \\
\& \bl{B_s}
\end{tikzcd}&
&\begin{tikzcd}[ampersand replacement=\&,row sep=tiny]
\& \gr{B_s(2)}\ar[r]\ar[rdd]		\& \gr{B_s(2)} \\
\& \oplus						\& \oplus		\\
\gr{B_s(2)}\ar[ruu]\ar[r]\ar[rdd]	\& \gr{B_s(2)}\ar[ruu]\ar[rdd]	\& \gr{B_s(2)} \\
\& \oplus						\& \oplus	 \\
\& \gr{B_s(2)}\ar[ruu]\ar[r]		\& \gr{B_s(2)}\\
\end{tikzcd}
\end{align*}

Notice that we can describe these complexes by simplicial sets whose faces correspond to some
$B_s(i)$, and where face relationship corresponds to the arrows. More precisely,
consider the following simplicial sets.

\begin{align*}
& \begin{tikzpicture}
\draw[blue] (1,0)--(2,0)--(3,0);
\foreach \i in {1,2,3} {%
\fill[blue] (\i,0) circle (1.5pt);
\draw (\i,0) circle (1.5pt);
\end{tikzpicture}&
& \begin{tikzpicture}[baseline=0]
\fill[green,opacity=.4] (-30:1)--(90:1)--(210:1)--cycle;
\draw[green] (-30:1)--(90:1)--(210:1)--cycle;
\foreach \i in {-30,90,210} {%
\fill[green] (\i:1) circle (1.5pt);
\draw (\i:1) circle (1.5pt);
\end{tikzpicture}&
\end{align*}

Namely, in the blue picture, the edges correspond to the two $B_s$'s in lower cohomological degree and the
three vertices to the others, and similarly in the green picture.

One can then use Gaussian elimination (see below) to reduce these complexes: roughly speaking,
one can eliminate arrows which are isomorphisms and obtain a summand of the original complex
which is homotopy equivalent to it. This corresponds, in the simplicial sets, to an elementary collapse:
the removal of a simplex together with one of its faces (see the precise definition below).
For example, for the green part, we can perform three
of these collapses, as shown in the following picture (the removed parts are drawn in gray):

\[
\begin{tikzpicture}[baseline=0]
\fill[green,opacity=.4] (-30:1)--(90:1)--(210:1)--cycle;
\draw[green] (-30:1)--(90:1)--(210:1)--cycle;
\foreach \i in {-30,90,210} {%
\fill[green] (\i:1) circle (1.5pt);
\draw (\i:1) circle (1.5pt);
\end{tikzpicture}
\quad\rightsquigarrow\quad
\begin{tikzpicture}[baseline=0]
\fill[gray,opacity=.1] (-30:1)--(90:1)--(210:1)--cycle;
\draw[green] (-30:1)--(90:1)--(210:1);
\draw[dashed,gray] (-30:1)--(210:1);
\foreach \i in {-30,90,210} {%
\fill[green] (\i:1) circle (1.5pt);
\draw (\i:1) circle (1.5pt);
\end{tikzpicture}
\quad\rightsquigarrow\quad
\begin{tikzpicture}[baseline=0]
\fill[gray,opacity=.1] (-30:1)--(90:1)--(210:1)--cycle;
\draw[green] (-30:1)--(90:1);
\draw[gray,dashed] (90:1)--(210:1)--(-30:1);
\foreach \i in {-30,90} {%
\fill[green] (\i:1) circle (1.5pt);
\draw (\i:1) circle (1.5pt);
\fill[white] (210:1) circle (1.5pt);
\draw[gray] (210:1) circle (1.5pt);
\end{tikzpicture}
\quad\rightsquigarrow\quad
\begin{tikzpicture}[baseline=0]
\fill[gray,opacity=.1] (-30:1)--(90:1)--(210:1)--cycle;
\draw[gray,dashed] (90:1)--(210:1)--(-30:1)--cycle;
\foreach \i in {-30} {%
\fill[green] (\i:1) circle (1.5pt);
\draw (\i:1) circle (1.5pt);
\foreach \i in {90,210}{
\fill[white] (\i:1) circle (1.5pt);
\draw[gray] (\i:1) circle (1.5pt);
\end{tikzpicture}
\]

This transforms the green complex into $B_s(2)[-2]$.
One can then show that all these eliminations can be performed together: each simplicial set can be collapsed to a point,
and the original complex can be reduced to

\[
\begin{tikzcd}
\textcolor{violet}{B_s(-2)}\ar[r] & \bl{B_s}\ar[r] & \gr{B_s(2)}\ar[r] & \un(3)
\end{tikzcd}
\]

Furthermore the procedure gives formulas for the projection corresponding to this summand
and allows to compute the new differential map. In this case this is given by
the following three morphisms:

\begin{align*}
& \delta_s\, \begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
\draw[red](0,-1)--(0,1);
\end{tikzpicture}
\hphantom{\delta_s}
\hphantom{\delta_s}
\begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
\draw[red](0,-1)--(0,1);
\end{tikzpicture}
\,s(\delta_s)&
& \delta_s\, \begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
\draw[red](0,-1)--(0,1);
\end{tikzpicture}
\hphantom{\delta_s}
\hphantom{\delta_s}
\begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
\draw[red](0,-1)--(0,1);
\end{tikzpicture}
\,\delta_s&
& \begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
\draw[red](0,-1)--(0,0);
\fill[red] (0,0) circle (1.5pt);
\end{tikzpicture}
\end{align*}

Finally, notice that each of the colored part that we defined above corresponds to
a subword of $sss$, and reduces to a single object in the final complex.

We would now like to make this procedure more precise and generalize it to any positive
Rouquier complex.

§.§ Large scale Gaussian elimination for complexes

Our main tool is Gaussian elimination for
complexes. Let $\CC$ be an additive category.
A summand of a complex in the category
$\Cb(\CC)$ will be called Gaussian if the projection and the inclusion morphisms
are mutually inverse homotopy equivalences.

The following was first pointed out by Bar-Natan <cit.>.

Consider the following complex in $\Cb(\CC)$

\[
\begin{tikzcd}[ampersand replacement=\&,column sep=large,baseline=.15cm]
A \arrow{r}{ \begin{psmallmatrix}
\epsilon \\ \zeta
\end{psmallmatrix} }		\& B\oplus E \arrow{r}{ \begin{psmallmatrix}
\alpha	& \beta \\
\gamma	& \phi
\end{psmallmatrix} } \& C\oplus E' \arrow{r}{ 	\begin{psmallmatrix}
\eta	& \theta \\
\end{psmallmatrix} } 	\& D
\end{tikzcd}
\]

and suppose that $\phi: E\rightarrow E'$ is an isomorphism in $\CC$.
Then the following

\[
\begin{tikzcd}[column sep=large]
A\ar[r,"\epsilon"] & B\ar[r,"\alpha-\beta\phi^{-1}\gamma"] & C\ar[r,"\eta"] & D
\end{tikzcd}
\]

is a Gaussian summand.

Consider the projection morphism $\pi$ given by:

\begin{equation}\label{eq_projmorph}
\begin{tikzcd}[ampersand replacement=\&,column sep=large,baseline=.15cm]
A \arrow{r}{ \begin{psmallmatrix}
\epsilon \\ \zeta
\end{psmallmatrix} }\ar[d,equal]		\& B\oplus E \arrow{r}{ \begin{psmallmatrix}
\alpha	& \beta \\
\gamma	& \phi
\end{psmallmatrix} }
\arrow{d}{\begin{psmallmatrix}
\id_B & 0
\end{psmallmatrix}}
\& C\oplus E' \arrow{r}{ 	\begin{psmallmatrix}
\eta	& \theta \\
\end{psmallmatrix} }
\arrow{d}{\begin{psmallmatrix}
\id_C & -\beta\phi^{-1}
\end{psmallmatrix}} 	\& D\ar[d,equal]\\
A \ar[r,swap,"\epsilon"]		\& B \ar[r,swap,"\alpha-\beta\phi^{-1} \gamma"]	\& C \ar[r,swap,"\eta"]	\& D
\end{tikzcd}
\end{equation}

and the inclusion morphism $\iota$ described by:

\[
\begin{tikzcd}[ampersand replacement=\&,column sep=large,baseline=.15cm]
A \arrow{r}{ \begin{psmallmatrix}
\epsilon \\ \zeta
\end{psmallmatrix} }		\& B\oplus E \arrow{r}{ \begin{psmallmatrix}
\alpha	& \beta \\
\gamma	& \phi
\end{psmallmatrix} } \& C\oplus E' \arrow{r}{ 	\begin{psmallmatrix}
\eta	& \theta \\
\end{psmallmatrix} }\& D \\
A \ar[r,swap,"\epsilon"]\ar[u,equal]		\& B \ar[r,swap,"\alpha-\beta\phi^{-1} \gamma"]	\arrow{u}{\begin{psmallmatrix}
\id_B \\ -\phi^{-1}\gamma
\end{psmallmatrix}}	\& C \ar[r,swap,"\eta"]	\arrow{u}{\begin{psmallmatrix}
\id_C \\ 0
\end{psmallmatrix}} 	\& D \ar[u,equal]
\end{tikzcd}
\]

It is easy to check that $\pi\iota=\id$. The
idempotent $\iota\pi$ is homotopy equivalent to the identity because

\[
\begin{tikzcd}[ampersand replacement=\&,row sep=huge,column sep=huge]
A \ar[r] \& B\oplus E \ar[r]\ar[ld,swap,"0"] \& C\oplus E' \ar[r]\arrow{ld}[swap,pos=0.6]{\begin{psmallmatrix}
0 & 0 \\
0 & \phi^{-1}
\end{psmallmatrix}}		\& D \ar[ld,swap,"0"] \\
A \ar[r] \& B\oplus E \ar[r] 			\& C\oplus E' \ar[r]												\& D
\end{tikzcd}
\]

is a homotopy from the complementary idempotent to the zero morphism.

We want now to describe a way to perform many Gaussian eliminations on complexes where the
isomorphisms that fit together in a nice way, as in in Example <ref>.

Recall that a polytopal set[The standard terminology (see for example <cit.>)
is polytopal complex but in our context the word “complex” would be
overloaded.] of dimension $n$ is a finite collection $\Pi$ of convex polytopes in
$\mathbb{R}^d$ of dimensions up to $n$ such that:

* a face of a polytope $\sigma\in \Pi$ belongs to $\Pi$;

* if $\sigma,\sigma'\in \Pi$ then the intersection $\sigma\cap\sigma'$ is either
empty or a common face of $\sigma$ and $\sigma'$ (so in particular it belongs to $\Pi$).

Let $\Pi^i$ denote the subset of $i$-dimensional polytopes of $\Pi$.
Here 0-dimensional polytopes are just points.
This generalizes the notion of simplicial complex.

We can now make the above idea more precise.

Let $\Pi$ be a polytopal set of dimension $n$, and let $A$ be a complex in
$\Cb(\CC)$. We say that $A$ is left-described by $\Pi$ if:

* as a graded object, the complex $A$ decomposes as

\[
\bigoplus_{\sigma\in \Pi} A_\sigma[\dim \sigma]
\]

with $A_\sigma$ in $\CC$.
So lower dimensional polytopes correspond to objects in higher cohomological degrees.

* the component $\phi_{\tau,\sigma}:A_{\sigma}\rightarrow A_{\tau}$ of the
differential map is non-zero if and only if $\tau$ is a face of $\sigma$.

* all non-zero components of the differential map are isomorphisms.

One can think of the objects $A_\sigma$ as “living over” the polytopes of $\Pi$.
The dual notion of right-description can be obtained by reversing all the arrows.
The face relationship is also reversed: in a right-description higher dimensional polytopes correspond
to objects in higher cohomological degrees.

Consider the 1-skeleton $\Pi^0\cup \Pi^1$ of a polytopal set $\Pi$ which left-describes
a complex $A$. If $e$ is an oriented edge
from the vertex $p$ to the vertex $q$, then we have isomorphisms $\phi_{p,e}$ and $\phi_{q,e}$.
We define the isomorphism associated to the oriented edge $e$ to be

\[
\Phi(e) = - \phi_{q,e}(\phi_{p,e})^{-1}.
\]

We put a minus sign for a technical reason that will become clear later.
Notice that if $-e$ is the same edge with opposite orientation, then
Given an oriented path $P$ through the oriented edges $e_1,\dots e_r$ along the
1-skeleton of $\Pi$,
the isomorphism associated to $P$ is the composition

\[
\Phi(P):=\Phi(e_r) \cdots \Phi(e_1).
\]

Clearly, if $P'$ is another path starting from the endpoint of $P$, and
$P''$ is the concatenation of $P$ and $P'$, then

\[
\Phi(P'')=\Phi(P')\Phi(P).
\]

If two paths $P$ and $P'$ connecting the point $q$ to $p$ are homotopic in $\Pi$
then $\Phi(P)=\Phi(P')$.

It is sufficient to
prove that the isomorphism associated to a contractible loop $P$

is the identity. Such a loop is the
boundary of the union of a finite number of polygons.
By splitting the isomorphism as shown below,

\[
\begin{tikzpicture}[baseline=1cm,every node/.style={inner sep=.1cm}]
\coordinate (p0) at (0,0); \node[anchor=north east] (pp0) at (p0) {};
\coordinate (p1) at (-1,1); \node[anchor=north east] (pp1) at (p1) {};
\coordinate (p2) at (-1,2); \node[anchor=south east] (pp2) at (p2) {};
\coordinate (p3) at (0,3); \node[anchor=south east] (pp3) at (p3) {};\node[anchor=south west] (pp31) at (p3) {};
\coordinate (p4) at (2,2); \node[anchor=south west] (pp4) at (p4) {};
\coordinate (p5) at (3,1); \node[anchor=south west] (pp5) at (p5) {};\node[anchor=north west] (pp51) at (p5) {};
\coordinate (p6) at (2,0); \node[anchor=north west] (pp6) at (p6) {};
\fill[green!40](p0)--(p1)--(p2)--(p3)--(p4)--(p5)--(p6)--cycle;
\draw[green](p0)--(p1)--(p2)--(p3)--(p4)--(p5)--(p6)--cycle;
\draw[green] (p0)--(p3); \draw[green] (p0)--(p4);
\draw[->,thick] (pp0)--(pp1);
\draw[->,thick] (pp1)--(pp2);
\draw[->,thick] (pp2)--(pp3);
\draw[->,thick] (pp31)--(pp4);
\draw[->,thick] (pp4)--(pp5);
\draw[->,thick] (pp51)--(pp6);
\draw[->,thick] (pp6)--(pp0);
\end{tikzpicture}
\quad\rightsquigarrow\quad
\begin{tikzpicture}[baseline=1cm,every node/.style={inner sep=.1cm}]
\coordinate (p01) at (0,0); \node[anchor=north east] (pp01) at (p01) {};\node[anchor=north west] (pp011) at (p01) {};
\coordinate (p11) at (-1,1);\node[anchor=north east] (pp11) at (p11) {};
\coordinate (p21) at (-1,2); \node[anchor=south east] (pp21) at (p21) {};
\coordinate (p31) at (0,3); \node[anchor=south east] (pp31) at (p31) {};\node[anchor=south west] (pp311) at (p31) {};
\coordinate (p02) at (.5,0.25); \node[anchor=north east] (pp02) at (p02) {};\node[anchor=north west] (pp02bis) at (p02) {};
\coordinate (p32) at (.5,3.25); \node[anchor=south east] (pp32) at (p32) {};\node[anchor=south west] (pp32bis) at (p32) {};
\coordinate (p42) at (2.5,2.25); \node[anchor=south west] (pp42) at (p42) {}; \node[anchor=north west] (pp42bis) at (p42) {};
\coordinate (p03) at (1,0); \node[anchor=south east] (pp03) at (p03) {};\node[anchor=north east] (pp03bis) at (p03) {};
\coordinate (p43) at (3,2); \node[anchor=south east] (pp43) at (p43) {};\node[anchor=south west] (pp43bis) at (p43) {};
\coordinate (p53) at (4,1); \node[anchor=south west] (pp53) at (p53) {};\node[anchor=north west] (pp53bis) at (p53) {};
\coordinate (p63) at (3,0); \node[anchor=north west] (pp63) at (p63) {};
\fill[green!40](p01)--(p11)--(p21)--(p31)--cycle;
\fill[green!40](p02)--(p32)--(p42)--cycle;
\fill[green!40](p03)--(p43)--(p53)--(p63)--cycle;
\draw[green](p01)--(p11)--(p21)--(p31)--cycle;
\draw[green](p02)--(p32)--(p42)--cycle;
\draw[green](p03)--(p43)--(p53)--(p63)--cycle;
\draw[->,thick] (pp01)--(pp11);
\draw[->,thick] (pp11)--(pp21);
\draw[->,thick] (pp21)--(pp31);
\draw[->,thick,red] (pp311)--(pp011);
\draw[->,thick,red] (pp02)--(pp32);
\draw[->,thick] (pp32bis)--(pp42);
\draw[->,thick,red] (pp42bis)--(pp02bis);
\draw[->,thick,red] (pp03)--(pp43);
\draw[->,thick] (pp43bis)--(pp53);
\draw[->,thick] (pp53bis)--(pp63);
\draw[->,thick] (pp63)--(pp03bis);
\end{tikzpicture}
\]

we can restrict to the case where $P$ is the boundary of a single polygon $\sigma$.
Let $e_1,\dots,e_r$ be the consecutive oriented edges forming $P$, and let
$q_0,\dots, q_{r-1},q_r=q_0$ be its vertices, so that $e_i$ goes from $q_{i-1}$ to
$q_i$. Then

\begin{equation}\label{eq_pathpolygon}
\Phi(P)=\Phi(e_r)\dots\Phi(e_1)=(-1)^r \phi_{q_r,e_r}\phi_{q_{r-1},e_r}^{-1}\dots\phi_{q_1,e_1}\phi_{q_0,e_1}^{-1}.
\end{equation}

The square of the differential map vanishes, so, computing the component
$A_{\sigma}\rightarrow A_{q_i}$ of $d^2$, we have

\[
\phi_{q_i,e_{i+1}}\phi_{e_{i+1},\sigma}+\phi_{q_{i},e_i}\phi_{e_i,\sigma}=0.
\]

This implies

\begin{equation}\label{eq_replacement}
\phi_{q_i,e_{i+1}}^{-1}\phi_{q_{i},e_i}=-\phi_{e_{i+1},\sigma}\phi_{e_i,\sigma}^{-1}
\end{equation}

and, applying (<ref>) to (<ref>), we get the identity.

A face of a polytope of $\Pi$ is called free if it is not the face of
any other polytope of $\Pi$. A collapse is the operation of removing
from $\Pi$ a polytope $\sigma$ of maximal dimension together with a free face $\tau$ and all the faces of $\sigma$ containing $\tau$.
A collapse is called elementary if the face $\tau$ has codimension 1 (so we are only removing $\sigma$ and $\tau$). One can show
that any collapse can be obtained by a sequence of elementary collapses.
Then $\Pi$ is called collapsible if there is a sequence of (elementary) collapses that reduces it to a single point.

We can now give the key lemma for large scale Gaussian elimination.

Let $A$ be a complex left-described by a polytopal set $\Pi$.
If $\Pi$ is collapsible to the point $p$, then $A_p[0]$ is a Gaussian summand of $A$.
Furthermore, if one chooses a path $P_q$ from each point $q$ to $p$, then
the projection morphism $\pi$ is given componentwise by

\[
\begin{cases}
\pi_{q}=\Phi(P_q) & \text{if $q\in \Pi^0$,} \\
\pi_{\sigma} = 0 & \text{if $\sigma\notin \Pi^0.$}
\end{cases}
\]

Notice that the elementary collapse of a face $\sigma$ with a free face $\tau$ in $\Pi$
corresponds to the elimination of the isomorphism $\phi_{\tau,\sigma}$ in $A$.
The condition on $\tau$ to be free corresponds to the condition $\phi_{\sigma',\tau}=0$
for any $\sigma'\neq \sigma$.
In other words, the complex is of the form

\[
\begin{tikzcd}[ampersand replacement=\&]
\dots \ar[r] \&
\Big(\bigoplus_{\sigma'\in\Pi^i\setminus\{\sigma\}}A_{\sigma'}\Big)\oplus A_{\sigma}
\arrow{r}{\begin{psmallmatrix}
* & * \\ 0 & \phi_{\tau,\sigma}
\end{psmallmatrix}} \&
\Big(\bigoplus_{\tau'\in\Pi^{i-1}\setminus\{\tau\}}A_{\tau'}\Big)\oplus A_{\tau}
\ar[r] \&\dots
\end{tikzcd}
\]

Hence, by Lemma <ref>, we can eliminate $A_\sigma$ and $A_\tau$ and
the projection, according to (<ref>), behaves as follows:

* it is the identity on all $A_{\sigma'}$ for $\sigma'\neq\sigma,\tau$;

* it is zero on $A_\sigma$;

* it sends $A_{\tau}$ on each $A_{\tau'}$ via

The Gaussian summand is now the complex left-described by $\Pi\setminus \{\sigma,\tau\}$.
Then we can proceed by induction.

If we compose all the projections thus obtained, we see that all $A_\sigma$ with $\sigma\notin \Pi^0$ are
eventually sent to zero, whereas the $A_q$ for $q\in \Pi^0$ are sent to $A_p$ by a composition of
$-\phi_{q',e}\phi_{q,e}^{-1}$ which gives precisely a path from $q$ to $p$.

Any two paths connecting $q$ to $p$ are homotopic by contractibility (collapsible polytopal sets are clearly contractible).
Hence the morphism $\Phi(P_q)$ does not depend on the choice of the path, by Lemma <ref>.

Of course one can reverse all the arrows in the lemma, and consider inclusion morphism instead of
projection, and obtain an analogous statement for
right-described complexes.

§.§ Exact structure for complexes

We want now to give a precise meaning to filtrations in the category $\Cb(\CC)$, following <cit.>.

We say that a sequence $A\rightarrow B \rightarrow C$
is exact in $\Cb(\CC)$ if it is split exact in the additive category $\mathcal{G}(\CC)$.
In this case we say that $A$ is a subcomplex of $B$ and $C$ is a quotient.

This defines an exact structure on $\Cb(\CC)$ in the sense of <cit.>. Notice that when we pass to
the homotopy category, exact sequences become distinguished triangles.

We say that complexes $A_i$ ($i=0,\dots,n$) filter $A$ by subcomplexes when
$A_i$ is a subcomplex of $A_{i+1}$, $A_0=0$ and $A_n=A$. Then the quotients
of $A_{i+1}$ by $A_i$ are called subquotients of the filtration.

If $A$ and $B$ are complexes filtered by $A_i$ and $B_i$ respectively,
and $f:A\rightarrow B$ a morphism compatible with the filtrations. Then $f$ is
the projection to (inclusion of) a Gaussian summand if and only if it is so on all subquotients.

It suffices to prove that, given a commutative diagram

\[
\begin{tikzcd}
X\ar[r]\ar[d]	& Y \ar[r]\ar[d]	& Z \ar[d] \\
X'\ar[r]		& Y' \ar[r]			& Z'
\end{tikzcd}
\]

where the rows are exact, if two of the vertical arrows are projections to a Gaussian summand, then
the third is too.
Complete the diagram to a square

\[
\begin{tikzcd}
X''\ar[r]\ar[d]	& Y''\ar[r]\ar[d]	& Z''\ar[d]\\
X\ar[r]\ar[d]	& Y\ar[r]\ar[d]		& Z \ar[d]\\
X'\ar[r]		& Y'\ar[r]			& Z'
\end{tikzcd}
\]

where all rows and columns are exact (this is possible by the exact category versions of
the $3\times 3$ lemma and the snake lemma). Now the first and last columns are split by hypothesis and
$X''$ and $Z''$ are null-homotopic, because $X'$ and $Z'$ are Gaussian. Then also $Y''$ is null-homotopic
(it suffices to see the top row as a distinguished triangle in the homotopy category).
But then the middle	column has to be split too (contractible complexes are injective/projective
for the Frobenius structure on $\Cb(\CC)$: see for example <cit.>).
The case of inclusions is symmetric.

§.§ Reduced representatives

Let us now describe the reduced version for positive Rouquier complexes more generally.
If $\uw$ is a Coxeter word, we write $\Roudg{\uw}$ for the corresponding positive Rouquier
complex (we should replace $\uw$ with its positive word lift). We now describe
a new complex $F_{\uw}$ that we will obtain after Gaussian elimination.

If $s\in S$ and $\ux$ is the word consisting
of $n$ repetitions of the letter $s$, we set

\[
\worsum{\ux}:=B_s(-n+1).
\]

If $\ux$ is any Coxeter word, first write it as

\[
\ux_1\ux_2\dots \ux_k
\]

where each $\ux_i$ is monotonous but $\ux_i\ux_{i+1}$ is not.

Then set

\[
\worsum{\ux}:=\worsum{\ux_1}\worsum{\ux_2}\cdots \worsum{\ux_k}\underset{\oplus}{\subset} B_{\ux}.
\]

We also set $\worsum{\emptyset}:=\un$.

Let $\ux=sssttusuu$, then

\begin{align*}
&\ux_1=sss,& &\ux_2=tt,& &\ux_3=u,& &\ux_4=s,& &\ux_5=uu.
\end{align*}

Hence $\worsum{\ux}=B_s(-2)\otimes B_t(-1) \otimes B_u \otimes B_s \otimes B_u(-1)=B_{\underline{stusu}}(-4)$.

Let now $\uw$ be a Coxeter word:
the graded object corresponding to $F_{\uw}$ is

\[
\grad{(F_{\uw})}=\bigoplus_{\ux\preceq \uw} \worsum{\ux} \Tate{q_{\uw,\ux}},
\]

where $q_{\uw,\ux}=\ell(\uw)-\ell(\ux)$.

We have to describe the differential map $d$.
We will specify the components

\[
d_{\ux',\ux}:\worsum{\ux}\Tate{q_{\uw,\ux}}\rightarrow \worsum{\ux'}\Tate{q_{\uw',\ux'}},
\]

for $\ux,\ux'\preceq \uw$ and $\ell(\ux)=\ell(\ux')+1$.

This map is nonzero only if $\ux'\preceq \ux$, which means that $\ux'$ is obtained
from $\ux$ by eliminating one letter.
The simplest such situation is when $\ux$ and $\ux'$ are monotonous of color $s$, of lengths $k+1$
and $k$ respectively. Then we set

\[
\begin{cases}
\def\hdot{.32cm}
\,\,
\begin{tikzpicture}
\pic[red] at (0,0) {dot=\hdot};
\end{tikzpicture} & \text{if $k=0$,}\\
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=east] at (0,.85*\hdot) {$\delta_{\re{s}}$};
\end{tikzpicture}
\hphantom{\delta_s}
\hphantom{\delta_s}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=west] at (0,.85*\hdot) {$\delta_{\re{s}}$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=east] at (0,.85*\hdot) {$\alpha_{\re{s}}$};
\end{tikzpicture}
\hphantom{\delta_s}- \hphantom{\delta_s}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,.6*\hdot);\fill[red] (0,0.6*\hdot) circle (1.5pt);
\draw[red] (0,1.7*\hdot)--(0,1.1*\hdot);\fill[red] (0,1.1*\hdot) circle (1.5pt);
\end{tikzpicture} & \text{if $k>0$ is odd,}
\\[1em]
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=east] at (0,.85*\hdot) {$\delta_{\re{s}}$};
\end{tikzpicture}
\hphantom{\delta_s}-\hphantom{\delta_s}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=west] at (0,.85*\hdot) {${\re{s}}(\delta_{\re{s}})$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,.6*\hdot);\fill[red] (0,0.6*\hdot) circle (1.5pt);
\draw[red] (0,1.7*\hdot)--(0,1.1*\hdot);\fill[red] (0,1.1*\hdot) circle (1.5pt);
\end{tikzpicture} & \text{if $k>0$ is even.}
\end{cases}
\]

In general, one can write $\ux$ and $\ux'$
in the form

\begin{align*}
&\ux=\uz_1 \underbrace{sss\dots s}_{k+1} \uz_2,&
&\ux'=\uz_1 \underbrace{ss\dots s}_{k} \uz_2,
\end{align*}

where $\uz_1$ and $\uz_2$ are (possibly empty) Coxeter words such that $\uz_1$ does not
end with $s$ and $\uz_2$ does not start with $s$. Hence we set

\[
d_{\ux',\ux}:=(-1)^{\ell(\uz_1)} \id_{\worsum{\uz_1}}\otimes d_{s,k} \otimes \id_{\worsum{\uz_2}}.
\]

If $k=0$ and $\uz_1$ ends with same letter as $\uz_2$ starts, then we also compose on top
with the corresponding trivalent:

\[
\begin{tikzpicture}[y=.3cm,x=.4cm,baseline=.5cm]
\draw[gray] (-1,0)--(7,0);
\draw[blue] (2,0)--(3,2)--(4,0);\draw[blue](3,2)--(3,4);
\draw[violet] (1,0)--(1,4);\draw[violet] (5,0)--(5,4);
\pic[red] at (3,0) {dot={1cm}};
\node at (0,2) {\dots};\node at (6,2) {\dots};
\end{tikzpicture}
\]

Let $\uw=\bl{t}\re{ss}\bl{t}$, then $F_{\uw}$ is

\[
\begin{tikzcd}[row sep=tiny]
& 															& \worsum{\underline{tt}}(2)\ar[rdd]			& 						&						\\
& \worsum{\underline{tst}}(1)\ar[ru,"d_2"]\ar[rd]\ar[rddd]	& \oplus										&						&						\\
& \oplus													& \worsum{\underline{ts}}(2)\ar[r]\ar[rdd]		& \worsum{t}(3)\ar[rd]	&						\\
\worsum{\underline{tsst}}\ar[ruu,"d_1"]\ar[r]\ar[rdd]	& \worsum{\underline{tss}}(1)\ar[ru]\ar[rddd]				& \oplus 										& \oplus				& \worsum{\emptyset} 	\\
& \oplus													& \worsum{\underline{st}}(2)\ar[r]\ar[ruu]		& \worsum{s}(3)\ar[ru]	& 						\\
& \worsum{\underline{sst}}(1)\ar[ru]\ar[rd]					& \oplus										& 						&						\\
& 															& \worsum{\underline{ss}}(2)\ar[ruu]			& 						&
\end{tikzcd}
\]

We describe the arrows $d_1$ and $d_2$, as an example, and we leave the others to the reader.
We have $\worsum{\underline{tsst}}=B_tB_sB_t(-1)$ and $\worsum{\underline{tst}}(1)=B_tB_sB_t(1)$. We are canceling the letter $s$
and passing from two occurrences to one. Furthermore the length of the word preceding the
group of $s$'s is one, hence the morphism $d_1$ is:

\[
\begin{tikzpicture}[x=.5cm,baseline=.4cm]
\draw[gray] (0.5,0)--(3.5,0); \draw[gray] (.5,1)--(3.5,1);
\draw[blue] (1,0)--(1,1); \draw[blue] (3,0)--(3,1);
\draw[red] (2,0)--(2,1);
\node at (1.5,0.5) {$\delta_{\re{s}}$};
\end{tikzpicture}
\begin{tikzpicture}[x=.5cm,baseline=.4cm]
\draw[gray] (0.5,0)--(3.5,0); \draw[gray] (.5,1)--(3.5,1);
\draw[blue] (1,0)--(1,1); \draw[blue] (3,0)--(3,1);
\draw[red] (2,0)--(2,1);
\node at (2.5,0.5) {$\delta_{\re{s}}$};
\end{tikzpicture}
\]

Next, we have $\worsum{\underline{tt}}(2)=B_t(1)$ and the morphism $d_2$ is eliminating the only
$s$ from $\worsum{\underline{tst}}$. Notice that the two adjacent letters are both $t$'s, hence $d_2$ is

\[
\begin{tikzpicture}[x=.4cm,y=.5cm,baseline=.3cm]
\draw[gray] (0.5,0)--(3.5,0); \draw[gray] (0.5,2)--(3.5,2);
\draw[blue] (1,0)--(2,1)--(2,2); \draw[blue] (3,0)--(2,1);
\draw[red] (2,0)--(2,0.5);\fill[red] (2,0.5) circle (1.5pt);
\end{tikzpicture}
\]

The differential map $d$ defined above satisfies $d^2=0$.

One can prove this directly but it will follow from Claim <ref> below
(which is proven independently of this statement).

Here is the statement of the main result of this section.

Let $\uw$ be a Coxeter word. Then the complex $\Roudg{\uw}$ admits
$F_{\uw}$ as a Gaussian summand. In particular $\Roudg{\uw} \simeq F_{\uw}$.

If $\uw=\underline{s}^n$, the theorem gives the minimal complex for $F_s^n$. See

The rest of this section is devoted to the proof of this result.
We will explicitely construct a projection from $\Roudg{\uw}$ to $F_{\uw}$ and then show
that it is a homotopy equivalence.

§.§ Multiword decomposition

The first step is a convenient decomposition of the original complex, generalizing that
of Example <ref>.
We will need the following notion.

We call multiword an expression of the form

\[
\mu=s_1^{n_1} \dots s_k^{n_k}
\]

with $s_i\in S$ and $n_i\in \ZZ_{>0}$. Notice that $s_i$ need not be different from $s_{i+1}$.
If all the $s_i$ are equal we still say that $\mu$ is monotonous.
Equivalently, a multiword is an element of the free monoid
generated by $S\times \ZZ_{>0}$. Let $\mword$ denote the set of multiwords.
To each multiword we can associate a Coxeter word in two natural ways: one can either
expand the exponents, or forget them. This gives two maps

\[
\begin{tikzcd}
\mword \ar[r,shift left,"e"]\ar[r,shift right,swap,"f"] & \cword,
\end{tikzcd}
\]

\begin{align*}
& e(\mu)=\underbrace{s_1\dots s_1}_{n_1} \dots \underbrace{s_k\dots s_k}_{n_k}, \\
& f(\mu)=s_1\dots s_k.
\end{align*}

We say that a multiword is simple if all multiplicities are 1's, or equivalently

Recall now the decomposition of Example <ref>.
Iterating, we get

\begin{align*}
& B_s^{\otimes 3}= B_s(-2)\oplus B_s^{\oplus 2} \oplus B_s(2), \\
& B_s^{\otimes 4}= B_s(-3)\oplus B_s(-1)^{\oplus 3}\oplus B_s(1)^{\oplus 3} \oplus B_s(3), \\
& \dots,
\end{align*}

and in general

\[
B_s^{\otimes (n+1)}=\bigoplus_{k=0}^{n} B_s(n-2k)^{\oplus {\binom{n}{k}}}.
\]

The inclusion morphisms of the $\binom{n}{k}$ summands of the form $B_s(n-2k)$
are all possible combinations of inclusions $\iota_1$ and $\iota_2$ from
Example <ref>. Hence they are of the form

\[
\begin{tikzpicture}[x=.3cm,y=.7cm]
\draw[red] (0,0)--(0,1);
\draw[red] (-15,2) ..controls (-15,1).. (0,1) ..controls (15,1).. (15,2);
\draw[decorate,decoration={brace}] (-15,2.2) to node[above] {$n_1$} (-9,2.2);
\draw[decorate,decoration={brace}] (-7,2.2) to node[above] {$n_2$} (-1,2.2);
\node at (4,2.3) {\dots};
\draw[decorate,decoration={brace}] (9,2.2) to node[above] {$n_k$} (15,2.2);
\clip (-15,2) ..controls (-15,1).. (0,1) ..controls (15,1).. (15,2);
\foreach \i in {-13,-9,-7,-5,-1,1,3,7,9,11}{
\draw[red] (\i,0)--(\i,2);
\foreach \i in {-8,0,8}{
\node[anchor=south] at (\i,1) {$\delta_{\re{s}}$};
\foreach \i in {-11,-3,5,13}{
\node[anchor=south,red] at (\i,1.5) {\dots};
\end{tikzpicture}
\]

where $n_1+n_2+\dots +n_k=n+1$, and there is no decoration in between each group of $n_i$ strands.
We label these summands by monotonous multiwords: the summand corresponding to the inclusion above
will be denoted by $C_{\mu}$ with $\mu=s^{n_1}s^{n_2}\dots s^{n_k}$.
The corresponding projection is obtained by appropriately combining projections
$\pi_1$ and $\pi_2$ from Example <ref>: one reflects the diagram vertically,
and replaces empty decorations with $-s(\delta_s)$ and $\delta_s$ with empty decorations.

There are $\binom{3}{2}=3$ copies of
$B_s(-1)$ inside $B_s^{\otimes 4}$. Their inclusion	mophisms are:

\begin{align*}
\draw[red] (0,0)--(0,1)--(-3,4);
\foreach \i in {0,1,2}{%
\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
\foreach \i in {1,2}{%
\node[anchor=south, inner sep=.2cm] at ({-\i},{1+\i}) {$\delta_{\re{s}}$};
\end{tikzpicture}&
\draw[red] (0,0)--(0,1)--(-3,4);
\foreach \i in {0,1,2}{%
\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
\foreach \i in {0,2}{%
\node[anchor=south, inner sep=.2cm] at ({-\i},{1+\i}) {$\delta_{\re{s}}$};
\end{tikzpicture}&
\draw[red] (0,0)--(0,1)--(-3,4);
\foreach \i in {0,1,2}{%
\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
\foreach \i in {0,1}{%
\node[anchor=south, inner sep=.2cm] at ({-\i},{1+\i}) {$\delta_{\re{s}}$};
\end{tikzpicture}
\end{align*}

and the corresponding projection morphisms are respectively:

\begin{align*}
\draw[red] (0,0)--(0,1)--(-3,4);
\foreach \i in {0,1,2}{%
\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
\foreach \i in {0}{%
\node[anchor=north west, inner sep=.1cm,font=\small,rotate=-45] at ({-\i},{1+\i}) {$-\re{s}(\delta_{\re{s}})$};
\end{tikzpicture}&
\draw[red] (0,0)--(0,1)--(-3,4);
\foreach \i in {0,1,2}{%
\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
\foreach \i in {1}{%
\node[anchor=north west, inner sep=.1cm,font=\small,rotate=-45] at ({-\i},{1+\i}) {$-\re{s}(\delta_{\re{s}})$};
\end{tikzpicture}&
\draw[red] (0,0)--(0,1)--(-3,4);
\foreach \i in {0,1,2}{%
\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
\foreach \i in {2}{%
\node[anchor=north west, inner sep=.1cm,font=\small,rotate=-45] at ({-\i},{1+\i}) {$-\re{s}(\delta_{\re{s}})$};
\end{tikzpicture}
\end{align*}

With the multiword notation the three summands are, respectively,

\begin{align*}
&C_{sss^2}& &C_{ss^2s}& &C_{s^2ss}.
\end{align*}

Let us now pass to the general case. For an arbitrary multiword
$\mu$, first we decompose it into monotonous multiwords $\mu_1\dots\mu_k$
with $\mu_i$ and $\mu_{i+1}$ of different colors, then we set

\[
C_{\mu}:=C_{\mu_1}\dots C_{\mu_k}.
\]

Notice that when the multiword $\mu$ is simple and $e(\mu)=f(\mu)=\ux$ then
$C_{\mu}=C_{\ux}$, so our notation is consistent.

Let $\ux$ be an arbitrary Coxeter word.	We can apply the one color case to
each sequence of repeated letters and decompose the Bott-Samelson object $B_{\ux}$
accordingly. Hence we get

Let $\ux$ be a Coxeter word, then:

\[
B_{\ux}=\bigoplus_{e(\mu)=\ux} C_{\mu}.
\]

Let us apply this to the graded object $\grad{(\Roudg{\uw})}$.
For a subexpression $\boi$ of $\uw$, let $\uw_{\boi}$ be the subword of $\uw$ corresponding
to it. For a given subword $\ux$, let $M_{\ux}$ denote the following set:

\[
M_{\ux}:=\{ (\boi,\mu) \in  \{0,1\}^{\ell(\uw)}\times\mword \mid f(\mu)=\ux, e(\mu)=\uw_{\boi}\}.
\]

Then, using Proposition <ref>, we have:

\[
\grad{(\Roudg{\uw})}=\bigoplus_{\boi\in \{0,1\}^{\ell(\uw)}} B_{\boi}\Tate{q_{\boi}}=
\bigoplus_{\boi}\,\bigoplus_{e(\mu)=\uw_{\boi}} \mulsum{\mu}\Tate{q_{\boi}}=
\bigoplus_{\ux\preceq \uw}\,\,\underbrace{%
\bigoplus_{(\boi,\mu)\in M_{\ux}} \mulsum{\mu}\Tate{q_{\boi}} }_{:=G_{\ux}}.
\]

We will use the notation $C_{\boi,\mu}:=C_{\mu}\Tate{q_{\boi}}$.

In Example <ref>, the subwords of $\uw=sss$ are

\[
\underline{sss},\underline{ss}, \underline{s}, \emptyset.
\]

So the corresponding sets are the following:

\begin{align*}
& M_{\underline{sss}} = \{ (111,sss) \}, \\
& M_{\underline{ss}} = \{ (110,ss),(101,ss),(011,ss),(111,s^2s),(111,ss^2)\}, \\
& M_{\underline{s}} = \{ (100,s),(010,s),(001,s),(110,s^2),(101,s^2),(011,s^2),(111,s^3)\}, \\
& M_{\emptyset} = \{ (000,\emptyset) \}.
\end{align*}

Then $G_{\underline{sss}},G_{\underline{ss}},G_{\underline{s}}$ and $G_{\emptyset}$ are
respectively the violet, blue, green and black parts in (<ref>).

These $G_{\ux}$ allow us to define a filtration of the complex as in the example.

Choose an total ordering $\ux_0,\ux_1,\dots, \ux_N$ of the subwords of $\uw$, refining
Then the objects

\[
G_{\le n} := \bigoplus_{i=0}^n F_{\ux_i}
\]

define a filtration of $\Roudg{\uw}$ by subcomplexes.

It is sufficient to prove that the component $G_{\ux}\rightarrow G_{\uz}$ of the
differential map can be nonzero only if $\uz=\ux$ or $\uz \preceq \ux$. Consider
the composition:

\begin{equation}\label{eq_compdiff}
\begin{tikzcd}
\mulsum{\boi,\mu}\ar[r,hook]	& B_{\boi}\ar[r,"d"]	& B_{\boi'}\ar[r,two heads]		& \mulsum{\boi',\mu'}.
\end{tikzcd}
\end{equation}

This is nonzero only if $\boi'$ is obtained from $\boi$ by turning a 1 to a 0, by definition
of the differential. Then $e(\mu')\preceq e(\mu)$ and $\ell(e(\mu))=\ell(e(\mu'))+1$. Then
one can easily see that either $f(\mu')=f(\mu)$ or $f(\mu')\preceq f(\mu)$.

As in Example <ref>, each of the subquotients $G_{\ux}$ will shrink to a single object $C_{\ux}$, via
large scale Gaussian elimination, and we will assemble these reductions together to a
reduction of the entire complex.

§.§ Colored polytopal complexes

To reduce the subquotients we will use large scale Gaussian elimination.
More precisely, the subquotient $G_{\ux}$ is left-described by a collapsible polytopal set
$\Pi(\uw,\ux)$ that we now describe.

Let $\Delta^i$ denote the $i$-simplex. The polytopes of $\Pi(\uw,\ux)$
are labeled by the set $M_{\ux}$.
The polytope labeled $(\boi,\mu)$ is a multisimplex
$\Delta^{n_1-1}\times \dots \times \Delta^{n_k-1}$, where $\mu=s_1^{n_1}\dots s_k^{n_k}$.
The faces of this multisimplex are those corresponding
to pairs $(\boi',\mu')$, where $\boi'\le \boi$ (componentwise) and $\mu'$ is of the form $s_1^{m_1}\dots s_k^{m_k}$
with $1\le m_i\le n_i$.
For instance, the vertices are labeled by all the subexpressions for $\ux$ (and the only
simple multiwords for them).

Here are two examples of these polytopal sets.

(st0000) at (1,0); (s00t00) at (0,-1);
(00st00) at (-1,-2); (s0000t) at (1,-2);
(00s00t) at (0,-3); (0000st) at (1,-4);
[green!50] (st0000)–(00st00)–(0000st)–cycle;
ıin st0000,s00t00,s0000t,00st00,00s00t,0000st
(ı) circle (1.2pt);

[anchor=south east] at (st0000) 110000;
[anchor=south west] at (s00t00) 100100;
[anchor=west] at (00st00) 001100;
[anchor=east] at (s0000t) 100001;
[anchor=north west] at (00s00t) 001001;
[anchor=north east] at (0000st) 000011;

1 .8pt  0.866
(A1) at (0,0,0); (A2) at (0,1,0); (A3) at (,0.5,0);
(B1) at (0,0,1); (B2) at (0,1,1); (B3) at (,0.5,1);
(C1) at (0,0,2); (C2) at (0,1,2); (C3) at (,0.5,2);
(D) at (/2,0.5,3);
[yellow!50] (D)–(C1)–(B1)–(A1)–(A3)–(B3)–(C3)–cycle;
[yellow!70] (D)–(C2)–(B2)–(A2)–(A3)–(B3)–(C3)–cycle;
(A3)–(A1); (A2)–(A3); [gray,dashed] (A1)–(A2);
(B3)–(B1); (B2)–(B3); [gray,dashed] (B1)–(B2);
(C3)–(C1); (C2)–(C3); [gray,dashed] (C1)–(C2);
ıin A1,A2,A3,B1,B2,B3,C1,C2,C3,D
(ı) circle ();

[anchor=south] at (D) 00000111;
[anchor=east] at (C1) 00100011;
[anchor=east] at (B1) 00110001;
[anchor=east] at (A1) 00111000;
[anchor=west] at (C2) 01000011;
[anchor=west] at (B2) 01010001;
[anchor=west] at (A2) 01011000;
[anchor=north] at (C3) 10000011;
[anchor=north] at (B3) 10010001;
[anchor=north] at (A3) 10011000;

The first is $\Pi(\underline{ststst},\underline{st})$,
the second is $\Pi(\underline{ssstustu},\underline{stu})$.
The vertices are labeled by colored subexpressions: we color the symbols of the 01-sequences
according to the corresponding elements in $S$.

The polytopal set $\Pi(\uw,\ux)$ left-describes the subquotient $G_{\ux}$.

Of course, we associate $\mulsum{\boi,\mu}$ to the polytope labeled $(\boi,\mu)$. We need to check that
the nonzero morphisms correspond to the face relationship and that they are isomorphisms.

Recall the proof of Lemma <ref> and the composition (<ref>).
When $f(\mu')=f(\mu)$, the only possibility is that

\begin{align*}
&\mu=\dots s_{i-1}^{n_{i-1}} s_i^{n_i} s_{i+1}^{n_{i+1}}\dots,&
&\mu'=\dots s_{i-1}^{n_{i-1}} s_i^{n_i-1} s_{i+1}^{n_{i+1}}\dots.
\end{align*}

and one can check that the composition is just the identity, up to sign.

The polytopal set $\Pi(\uw,\ux)$ is collapsible.

Consider the lexicographic order on 01-sequences and take the smallest. This corresponds to
a point and it is easy to see that there is only one maximal polytope containing it. Then we can collapse them. Then we can proceed by induction.

To finally prove Theorem <ref>, we have to glue all the reductions of the subquotients together

§.§ Proof of the theorem

We now define the homotopy equivalence from $F_{\uw}^\bullet$ to $F_{\uw}$.

Given an element of $\{0,1\}^n$, we call a flip the replacement of a subsequence
“01” with “10” or viceversa.
For each subexpression $\boi$ let $|\boi|$ be the minimal number of flips that are necessary
to turn $\boi$ into the sequence $11\dots 100\dots 0$.
Consider the maps

\[
\begin{cases}
C_{\boi,\mu} \overset{(-1)^{|\boi|}}{\longrightarrow} C_{\ux} & \text{if $e(\mu)=f(\mu)=\ux$} \\
C_{\boi,\mu}\rightarrow 0 & \text{otherwise}
\end{cases}
\]

One can glue them together to form a morphism $\pi:\grad{(\Roudg{\uw})}\rightarrow \grad{(F_{\uw})}$.

The morphism $\pi$ is compatible with the differential maps.

As $\pi$ is surjective, this also implies Proposition <ref>.

We will check this over each $\mulsum{\boi,\mu}$. Let first $\mu$ be simple and
$e(\mu)=f(\mu)=\ux$. Let $\ux'$ be a word obtained by $\ux$ by removing a letter. Then
the component $\mulsum{\boi,\mu}\rightarrow \worsum{\ux'}$ of the composition $d\pi$ is

\[
(-1)^{|\boi|} d_{\ux',\ux}
\]

as defined in  <ref>. The same component of the composition $\pi d$
is the sum of the compositions

\begin{equation}\label{eq_compositionspid}
\begin{tikzcd}
\mulsum{\boi,\mu}\ar[r,hook]	& B_{\boi}\ar[r,"d"]	& B_{\boi'}\ar[r,two heads]		& \mulsum{\boi',\mu'}\ar[r] & \worsum{\ux'},
\end{tikzcd}
\end{equation}

for all possible $\boi'$ obtained from $\boi$ by turning a 1 to a 0. Let $\ux=\uz_1 sss\dots s\uz_2$
and $\ux'=\uz_1 ss\dots s\uz_2$, then the only nonzero such composition are those where the new
$0$ corresponds to one of the $s$'s, and we get

\[
\sum_{\boi'}(-1)^{|\boi'|}
\begin{tikzpicture}[x=.3cm,y=.4cm,baseline=.9cm]
\draw[gray] (-12.5,0)--(12.5,0);\draw[gray] (-12.5,4)--(12.5,4);
\draw[red] (0,0)--(0,1);
\draw[red] (0,3)--(0,4);
\draw[violet] (-10,0)--(-10,4); \draw[violet] (10,0)--(10,4);
\node at (-11,2) {\dots};\node at (11,2) {\dots};
\draw[red] (-9,2) ..controls (-9,1).. (0,1) ..controls (9,1).. (9,2)..controls (9,3).. (0,3) ..controls (-9,3).. (-9,2);
\clip (-9,2) ..controls (-9,1).. (0,1) ..controls (9,1).. (9,2)..controls (9,3).. (0,3) ..controls (-9,3).. (-9,2);
\foreach \i in {-7,-5,-1,3,7}{
\draw[red] (\i,0)--(\i,4);
\draw[red](1,0)--(1,2); \fill[red] (1,2) circle (1.5pt);
\foreach \i in {-8,-6,0,2,8}{
\node[anchor=south] at (\i,1) {$\delta_{\re{s}}$};
\foreach \i in {-3,5}{
\node[anchor=south,red] at (\i,1.5) {\dots};
\end{tikzpicture}
\]

and one can check that this gives the same.

If $\mu$ is not simple, we have to check that the composition $\pi d$ is zero.
Consider again the component $\mulsum{\boi,\mu}\rightarrow \worsum{\ux'}$, with
$\ux'$ as above.
The only non trivial case for the compositions (<ref>) is

\begin{align*}
&\mu=s_1 \dots s_{i-1} s_i^2 s_{i+1} \dots s_k,&
&\mu=s_1 \dots s_{i-1} s_i s_{i+1} \dots s_k,&
\end{align*}

and there are only two possibilities for $\boi'$. These give

\[
\pm
\begin{tikzpicture}[x=.6cm,y=.5cm,baseline=.4cm]
\draw[gray] (-0.5,0)--(4.5,0);\draw[gray] (-0.5,2)--(4.5,2);
\draw[violet] (1,0)--(1,2); \draw[violet] (3,0)--(3,2);
\draw[red] (2,0)--(2,1); \draw[red] (2,1)--(1.75,1.5); \draw[red] (2,1)--(2.5,2);
\node at (0,1) {\dots};\node at (4,1) {\dots};
\fill[red] (1.75,1.5) circle (1.5pt);
\end{tikzpicture}
\mp
\begin{tikzpicture}[x=.6cm,y=.5cm,baseline=.4cm]
\draw[gray] (-0.5,0)--(4.5,0);\draw[gray] (-0.5,2)--(4.5,2);
\draw[violet] (1,0)--(1,2); \draw[violet] (3,0)--(3,2);
\draw[red] (2,0)--(2,1); \draw[red] (2,1)--(1.5,2); \draw[red] (2,1)--(2.25,1.5);
\node at (0,1) {\dots};\node at (4,1) {\dots};
\fill[red] (2.25,1.5) circle (1.5pt);
\end{tikzpicture}
\]

Now, consider a total ordering $\ux_1,\dots \ux_N$ on the subwords of $\uw$.
This gives a filtration on $\Roudg{\uw}$ as in Lemma <ref> and also naturally on
$F_{\uw}$: the filtered pieces are just the $\worsum{\ux_i}$.

The map $\pi$ is compatible with the filtrations. Furthermore, on subquotients, it coincides,
up to a sign with the projection coming from Proposition <ref>.

The compatibility with the filtrations is obvious by the definition of $\pi$.
Notice that the edges of $\Pi(\uw,\ux)$ connect vertices that can be obtained
from one another by a colored flip: this is a
replacement of a subsequence
“10…00” with “00…01”
where the symbols in the middle are all zeros (that can have any color).
For example the following is a colored flip:

\[
\ssstustu{\underline{0}}{0}{0}{0}{0}{\underline{1}}{1}{1}\rightsquigarrow \ssstustu{\underline{1}}{0}{0}{0}{0}{\underline{0}}{1}{1}.
\]

Then one can check that the isomorphism associated to an edge is $(-1)^n$ where $n$ is
the number of zeros involved in the flip. In the example above $n=5$. Then one sees that
up to a global sign, the isomoprhisms associated with the paths to a given point agree with

This concludes the proof of Theorem <ref>.

§.§ Negative case

Of course everything works symmetrically for the negative case. Let $\uw$ be a Coxeter word
and $\uom$ its negative word lift. Then one can describe a complex $F_{\uw}^-$ homotopy equivalent to
$\Roudg{\uom}$ as follows.

First we define, for $\ux$ a word consisting of $n$ repetition of the letter $s$,

\[
\worsum{\ux}^-:=B_s(n-1).
\]

Then, as before, with $\ux=\ux_1\ux_2\dots \ux_k$ as before, set

\[
\worsum{\ux}:=\worsum{\ux_1}^-\worsum{\ux_2}^-\cdots \worsum{\ux_k}^-\underset{\oplus}{\subset} B_{\ux}.
\]

The graded object corresponding to $F_{\uw}^-$ is

\[
\grad{(F_{\uw}^-)}=\bigoplus_{\ux\preceq \uw} \worsum{\ux}^- \Tate{-q_{\uw,\ux}}.
\]

The differential is given, again, via its components $d^-_{\ux',\ux}$. These
are nonzero only if $\ux'\succeq \ux$ which implies that $\ux'$ is obtained from $\ux$
by inserting a single occurrence of a letter $s$.
Again, we first consider $\ux$ and $\ux'$ consisting of respectively $k$ and $k+1$ occurrences of the same letter $s$.
In this case

\[
\begin{cases}
\def\hdot{.32cm}
\,\,
\begin{tikzpicture}
\pic[red,yscale=-1] at (0,0) {dot=\hdot};
\end{tikzpicture} & \text{if $k=0$}\\
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=east] at (0,.85*\hdot) {$\delta_{\re{s}}$};
\end{tikzpicture}
\hphantom{\delta_s}-\hphantom{\delta_s}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=west] at (0,.85*\hdot) {$\delta_{\re{s}}$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=east] at (0,.85*\hdot) {$\alpha_{\re{s}}$};
\end{tikzpicture}
\hphantom{\delta_s}-\hphantom{\delta_s}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,.6*\hdot);\fill[red] (0,0.6*\hdot) circle (1.5pt);
\draw[red] (0,1.7*\hdot)--(0,1.1*\hdot);\fill[red] (0,1.1*\hdot) circle (1.5pt);
\end{tikzpicture} & \text{if $k>0$ is odd}
\\[1em]
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=east] at (0,.85*\hdot) {$\delta_{\re{s}}$};
\end{tikzpicture}
\hphantom{\delta_s}-\hphantom{\delta_s}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,1.7*\hdot);
\node[anchor=west] at (0,.85*\hdot) {${\re{s}}(\delta_{\re{s}})$};
\end{tikzpicture}
\begin{tikzpicture}[baseline=.6*\hdot]
\draw[red] (0,0)--(0,.6*\hdot);\fill[red] (0,0.6*\hdot) circle (1.5pt);
\draw[red] (0,1.7*\hdot)--(0,1.1*\hdot);\fill[red] (0,1.1*\hdot) circle (1.5pt);
\end{tikzpicture} & \text{if $k>0$ is even}
\end{cases}
\]

In general, one can write $\ux$ and $\ux'$ in the form

\begin{align*}
&\ux=\uz_1 \underbrace{\underline{ss\dots s}}_{k} \uz_2,&
&\ux'=\uz_1 \underbrace{\underline{ss\dots s}}_{k+1} \uz_2,&
\end{align*}

where $\uz_1$ and $\uz_2$ are as in the previous case. Then we set

\[
d_{\ux',\ux}:=(-1)^{|\uz_1|} \id_{\mulsum{\uz_1}^-}\otimes d^-_{s,k} \otimes \id_{\mulsum{\uz_2}}.
\]

Again we need to treat the case when $k=0$ and $\uz_1$ ends with same letter as
$\uz_2$ starts. In this case we compose on bottom with a trivalent:

\[
\begin{tikzpicture}[y=.3cm,x=.4cm,baseline=.5cm]
\draw[gray] (-1,0)--(7,0);
\draw[blue] (2,4)--(3,2)--(4,4);\draw[blue](3,2)--(3,0);
\draw[violet] (1,0)--(1,4);\draw[violet] (5,0)--(5,4);
\pic[red,yscale=-1] at (3,4) {dot={1cm}};
\node at (0,2) {\dots};\node at (6,2) {\dots};
\end{tikzpicture}
\]

This concludes the definition of $F_{\uw}^-$ and we have the analogue of Theorem

Let $\uw$ be a Coxeter word and $\uom$ its negative word lift. Then the complex
$\Roudg{\uom}$ admits $F_{\uw}^-$ as a Gaussian summand. In particular
$\Roudg{\uom} \simeq F_{\uw}^-$.

For a arbitrary braid word $\uom$ we can first write its alternating decomposition

\[
\uom=\uom_1\uom_2\dots \uom_k
\]

and consider the corresponding $F_{\uw_i}$ and $F_{\uw_j}^-$. Then define $F_{\uom}$ to be their
tensor product. Then we have
$\Roudg{\uom}\simeq F_{\uom}$.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

titleOn Soergel bimodules
eory (math.RT),FOS: Mathematics,FOS: Mathematics,20F55

journaltitleJ. Amer. Math. Soc.
titleKoszul duality for Kac-Moody groups and characters of tilting modules

journaltitleDuke Math. J.
titleModular perverse sheaves on flag varieties II: Koszul duality and formality

journaltitleJ. Knot Theory Ramifications
titleFast Khovanov homology computations

Springer, New York

isbn978-3540-442387; 3-540-44238-3
seriesGraduate Texts in Mathematics
titleCombinatorics of Coxeter groups

Hermann, Paris

seriesActualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337
titleÉléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines
pages288 pp. (loose errata)

journaltitleMath. Ann.
titleBraid groups and Kleinian singularities

journaltitleTrans. Amer. Math. Soc.
titleExtensions of Verma modules

journaltitleInvent. Math.
titleAction du groupe des tresses sur une catégorie

titleGaitsgory's central sheaves via the diagrammatic Hecke category

journaltitleCompos. Math.
titleThe two-color Soergel calculus

journaltitleInt. J. Math. Math. Sci.
titleDiagrammatics for Soergel categories
pagesArt. ID 97863558

Springer, Cham

isbn978-3-030-48825-3; 978-3-030-48826-0
seriesRSME Springer Series
titleIntroduction to Soergel bimodules
year[2020] ©2020

journaltitleRepresent. Theory
titleSoergel calculus

journaltitleAnn. of Math. (2)
titleThe Hodge theory of Soergel bimodules

Cambridge University Press, Cambridge

seriesCambridge Studies in Advanced Mathematics
titleReflection groups and Coxeter groups

journaltitleMath. Z.
titleThe 2-braid group and Garside normal form

Amer. Math. Soc., Providence, RI

booktitleCategorification and higher representation theory
seriesContemp. Math.
titleThe $p$-canonical basis for Hecke algebras

journaltitleJ. Amer. Math. Soc.
titleParity sheaves

Cambridge University Press, Cambridge

isbn0-521-37215-1; 0-521-46693-8
titleInfinite-dimensional Lie algebras

journaltitleAnnales scientifiques de l'École Normale Supérieure
titleDeriving DG categories
volumeSer. 4, 27

journaltitleInternat. J. Math.
titleTriply-graded link homology and Hochschild homology of Soergel bimodules

journaltitleFund. Math.
titleMatrix factorizations and link homology

journaltitleJ. Amer. Math. Soc.
titleQuivers, Floer cohomology, and braid group actions

titleHomotopy reductions of dg morphism spaces between Rouquier complexes

Berlin, Heidelberg

Springer Berlin Heidelberg

booktitleHigher K-Theories
titleHigher algebraic K-theory: I

titleTilting modules and the $p$-canonical basis

Kluwer Acad. Publ., Dordrecht

booktitleFinite-dimensional algebras and related topics (Ottawa, ON, 1992)
seriesNATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci.
titleTranslation functors and equivalences of derived categories for blocks of algebraic groups

Amer. Math. Soc., Providence, RI

booktitleTrends in representation theory of algebras and related topics
seriesContemp. Math.
titleCategorification of ${\mathfrak{sl}}_2$ and braid groups

issn1474-7480, 1475-3030
journaltitleJournal of the Institute of Mathematics of Jussieu
titleKazhdan-Lusztig-Polynome und unzerlegbare Bimoduln über Polynomringen

Walter de Gruyter, Berlin/New York Berlin, New York

titleThe combinatorics of Harish-Chandra bimodules.

Birkhäuser Boston, Inc., Boston, MA

seriesModern Birkhäuser Classics
titleLinear algebraic groups

Berlin, Heidelberg

Springer Berlin Heidelberg

booktitleCohomologie Etale
titleCatégories Derivées, Quelques résultats (Etat 0)

New York, NY

Springer New York

titleLectures on Polytopes: Updated Seventh Printing of the First Edition

[heading = bibintoc]