BRAID GROUPS OF TYPE ADE, GARSIDE MONOIDS, AND
THE CATEGORIFIED ROOT LATTICE

arXiv:1703.06011v1 [math.QA] 17 Mar 2017

ANTHONY M. LICATA AND HOEL QUEFFELEC

Abstract. We study Artin-Tits braid groups BW of type ADE via the action
of BW on the homotopy category K of graded projective zigzag modules (which
categorifies the action of the Weyl group W on the root lattice). Following
Brav-Thomas [BT11], we define a metric on BW induced by the canonical
t-structure on K, and prove that this metric on BW agrees with the wordlength metric in the canonical generators of the standard positive monoid B+
W
of the braid group. We also define, for each choice of a Coxeter element c in
W , a baric structure on K. We use these baric structures to define metrics on
the braid group, and we identify these metrics with the word-length metrics
in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive
monoid B∨
W.c . As consequences, we give new proofs that the standard and dual
positive monoids inject into the group, give linear-algebraic solutions to the
membership problem in the standard and dual positive monoids, and provide
new proofs of the faithfulness of the action of BW on K. Finally, we use the
compatibility of the baric and t-structures on K to prove a conjecture of Digne
and Gobet regarding the canonical word-length of the dual simple generators
of ADE braid groups.

Contents
Introduction
1. Artin-Tits braid groups in type ADE
2. Zigzag algebras and categorical braid group actions
3. Categorical ping pong
4. Homological interpretations of word-length metrics
5. Standard and dual word-length metrics and a conjecture of Digne-Gobet
References

1
4
10
17
28
37
39

Introduction
A basic tool in the combinatorial study of a Coxeter group W is the linear action
of W on the root lattice. The first important point about the action of W on the
root lattice is that it is faithful. As a result, one can address basic combinatorial
and group-theoretic questions about Coxeter groups, such as the word problem, by
using tools of finite-dimensional linear algebra.
The Artin-Tits braid groups BW associated to W are much less well-understood
than the Coxeter groups themselves; for example, in the generality of arbitrary
Artin-Tits groups, the word problem is still open. Perhaps one feature of the difficulty in studying BW is the lack of a clear substitute for the root lattice: the action
of W on the root lattice can be q-deformed to the Burau representation of BW , but
the Burau representation fails to be faithful outside of small rank.
1

2

ANTHONY M. LICATA AND HOEL QUEFFELEC

In fact, there is a good candidate K for a “root lattice" for BW , though it is not a
lattice but rather a triangulated category. This triangulated category, which we refer
to as the categorified root lattice, has a number of explicit realizations; for example,
when W is simply-laced, one can take for K the derived category of modules over the
associated preprojective algebra. (For more general Coxeter groups, the categorified
root lattice may be constructed as a quotient of the homotopy category of Soergel
bimodules.) At present, the question of whether or not the braid group BW acts
faithfully on the categorified root lattice is still open. However, when the Coxeter
group W is finite, faithfulness is known: this was proven in type A2 by RouquierZimmermann [RZ03], in type An by Khovanov-Seidel [KS02], in type ADE by BravThomas [BT11], and for arbitrary finite W by Jensen [Jen15]. Faithfulness is also
known in affine type A by work of Riche [Ric08], Ishii-Ueda-Uehara [IUU10] and
Gadbled-Thiel-Wagner [GTW15], as well as in the case of the free group by work
of the first author [Lic16]. Thus, at least in these cases, one can attempt to study
BW via its action on the triangulated category K somewhat analogously to the way
one studies W via its action on the root lattice Λ.
The aim of the present paper is to take up such a study when W is a finite Weyl
group of type ADE. We take as our model for K the homotopy category of graded
modules over the zigzag algebra of the Dynkin diagram Γ. Our main goal is to explain how the homological algebra of K can be used to define metrics on BW , and
to then combinatorially describe these metrics. We consider two kinds of homological decompositions of K into positive and negative pieces: t-structures and baric
structures. These decompositions turn out to be closely related to the standard and
dual Garside monoids inside BW .
If a group G acts on a triangulated category T by triangulated auto-equivalences,
then one way to produce a pseudo-length function on G is as follows: first, fix a
t-structure (T ≥0 , T ≤0 ) with heart T 0 ; given g ∈ G, there is a smallest closed interval
[a, b], a, b ∈ R∪{±∞} such that g(T 0 ) ⊂ T [a,b] . We then define the length l(g) of g to
be the length of the interval [a, b]. In good situations, the function d(g, h) = l(h−1 g)
will be a metric on g, though if the action of G on T is not faithful, it will at
best be a pseudo-metric. Analogous definitions give rise to metrics on G using
other decompositions – such as baric structures – on T, rather than t-structures.
For groups with interesting 2-representation theory (such as Artin-Tits groups and
mapping class groups of surfaces), such metrics should carry interesting geometric
information about the group.
In particular, in Theorem 4.11 we identify the metric on BW induced from the
canonical t-structure on K with the canonical word-length metric on BW coming
from the positive lifts from the Weyl group (which are the canonical generators
of the standard Garside monoid). Though we give complete independent proofs of
all the statements required for the proof of Theorem 4.11, this theorem is inspired
by – and is in some sense largely a rederivation of – the foundational work of
Brav-Thomas on braid group actions on the derived category of a resolved Kleinian
singularity [BT11].
Our Theorem 4.1, on the other hand, begins by choosing a Coxeter element c ∈ W ,
and using this choice to define a baric structure (rather than a t-structure) on K.
We use this baric structure to define a metric on BW ; in Theorem 4.1 we identify
this metric with the word-length metric on BW coming from the generators of the
dual Garside monoid on BW . Taken together, Theorems 4.1 and 4.11 explain how
both Garside structures on BW can be studied in a parallel fashion via the action
of BW on the categorified root lattice.

BRAID GROUPS OF TYPE ADE

3

Theorems 4.1 and 4.11 and the constructions that precede them have several important consequences. For one, we obtain new proofs of the faithfulness of the action
of BW on K (Corollary 3.5). As another consequence, we obtain new proofs of the
injectivity of the canonical map from both the standard and dual Garside monoids
∨+
B+
into the group BW (Corollaries 3.3 and 4.7). For the standard posiW , resp. Bc
tive monoid, this injectivity was first established by Deligne [Del72] and BrieskornSaito [BS72], while for the dual positive monoid it is a theorem of Birman-Ko-Lee
[BKL98] in type A and by Bessis [Bes03] and Brady-Watt [BW08] more generally.
We also give a solution to the membership problem in these monoids, by showing
that
• the canonical t-structure (K≥0 , K≤0 ) on K has the property that
β(K≥0 ) ⊂ K≥0 ⇐⇒ β ∈ B+
W , and
• the baric structure (K≥0 , K≤0 ) on K associated to a Coxeter element c ∈ W
has the property that
+

β(K≥0 ) ⊂ K≥0 ⇐⇒ β ∈ B∨
c .
To check each of the conditions β(K≥0 ) ⊂ K≥0 and β(K≥0 ) ⊂ K≥0 requires computing the action of β on only finitely many objects of K; in turn, computing the
action of β on these finitely-many objects is equivalent to performing Gaussian
elimination on a finite integer matrix (whose size depends on β). Thus the above
criteria provides a finite algorithm for determining whether or not a fixed word in
the Artin braid generators represents a positive or dual-positive braid. (Of course,
computing the Garside or dual Garside normal form of a braid would also solve the
above membership problem, so the point to emphasize here is the linear-algebraic
nature of our solution.)
An important motivation for our work is to study both Garside structures on BW
using the same categorical action, in an effort to shed some light on the relationship between these two structures. In [DG15], Digne-Gobet conjecture that any
dual simple generator β of the dual positive monoid can be written β = xy −1 ,
where x and y are both simple generators of the standard positive monoid. This
conjecture provides a basic relationship between the natural metrics on BW coming
from the standard and dual Garside monoids. Digne-Gobet prove their conjecture
in all irreducible type except for type Dn , with some of the proofs relying on a
case-by-case computer analysis. In theorem 5.1, we use the compatibility of t- and
baric structures on K to give a uniform proof of their conjecture for all ADE braid
groups, including the open type D. A different proof for type D will also appear in
a paper by Baumeister and Gobet [BG].
The basic insight that triangulated categories with braid group actions should have
t-structures compatible with the action of a distinguished positive monoid is originally due to Bezrukavnikov [Bez06], who exploits this compatibility to prove a
number of deep results in modular representation theory. The baric structures of
the current paper give an example of another kind of homological decomposition of
a triangulated category compatible with the action of a positive monoid.
A more detailed outline of the contents of the paper is as follows:
• In Section 1, we collect some preliminary information about ADE Weyl
groups and their braid groups, including the definitions of the standard
and dual positive monoids and the associated word-length metrics. We also
recall ping-pong and dual ping-pong lemmas for braid groups (Lemmas

4

ANTHONY M. LICATA AND HOEL QUEFFELEC

1.8 and 1.10); these lemmas are used later in Section 3 to establish the
faithfulness of the action of BW on the categorified root lattice K.
• In Section 2 we recall the definition of the zigzag algebra AΓ , its homotopy
category K, and the action of the braid group on K. We also define the tand baric structures on K, whose compatibility with the braid group action
is the central theme in the remaining sections. We also define root complexes
in K, which are important indecomposable complexes lying in the hearts
of both the baric and t-structures. The main new result in this section is
Theorem 2.7, which establishes a bijection between root complexes and positive roots in the associated root system. We also explain the relationship
between this bijection and Gabriel’s theorem, which concerns the representation theory of the undoubled quiver obtained by orienting the Dynkin
diagram Γ (see Proposition 2.6).
• Section 3 is the technical heart of the paper. We define explicit subsets
of objects in K and use these subsets to establish the assumptions of the
ping-pong and dual ping-pong lemmas, giving new proofs of the faithfulness
of the action of BW on K. The proofs of the injectivity of the maps from
the standard and dual monoids into the group are also contained in this
section.
• The faithfulness proofs imply that the length functions defined by the t and
baric structures on K induce metrics on BW . In Section 4, we identify these
metrics with word length metrics in standard and in dual generators (see
Theorems 4.1 and 4.11).
• In Section 5, we use the compatibility of t and baric structures on K to
prove the conjecture of Digne-Gobet [DG15, Conjecture 8.7] for ADE braid
group (see Theorem 5.1).
Acknowledgements: We would like to thank Christian Blanchet, Thomas Gobet,
Luis Paris, Emmanuel Wagner and Anne-Laure Thiel for interesting discussions
and comments. This work was partially supported by the ARC DP 140103821 and
the ANR Quantact.
1. Artin-Tits braid groups in type ADE
1.1. Weyl groups and braid groups. In this section, we collect the definitions
and classical facts about Weyl and braid groups, that we will freely use throughout
the paper. For more details, we direct the reader to [BB05], [DDG+ 15], and [DP99].
Let Γ be a Dynkin diagram of type ADE, with vertex set I of n elements. The Weyl
group W associated to Γ has a Coxeter presentation:
W = hsi , i ∈ I | si sj si = sj si sj , i adjacent to j in Γ, si sj = sj si , i j, s2i = 1i
Denoting S = {si , i ∈ I} the set of generators, (W, S) is a Coxeter system. Recall
that one can associate to it a root system as follows. Let V ∗ be a real vector space
of dimension n, with basis Π = {αi }i∈I . The basis vectors αi are referred to as the
simple roots. We define a pairing < −, − > : V ∗ × V ∗ 7→ R by


0 if i j in Γ;
< αi , αj >= −1 if i adjacent to j in Γ;


2 if i = j.

BRAID GROUPS OF TYPE ADE

5

An action of W on V ∗ is defined by si (x) = x− < αi , x > αi , so that:


−αi if i = j;
si (αj ) = αj + αi if i adjacent to j in Γ;


αj otherwise.

The root system Φ = {w(αi )}w∈W,i∈I decomposes as Φ = Φ+ ⊔ Φ− , with Φ+ = NΠ
the positive roots and Φ− = Φ − Φ+ = −Φ+ the negative roots. The root lattice
is the free Z-module generated by the αi . The above action on V ∗ restricts to an
action on the root lattice.
Let l denote the word-length on W in the Coxeter generators si . The left and right
descents sets of w ∈ W , denoted DR (w) and DL (w), are defined as follows
DR (w) = {si : l(wsi ) < l(w)}, DL (w) = {si : l(si w) < l(w)}.
A reflection in W is any conjugate of a Coxeter generator si . We denote by T the
set of reflections. There is a one-to-one correspondence between Φ+ and T given
by:
w(αi ) ∈ Φ+ ↔ wsi w−1 ∈ T
The braid group B = BW associated to Γ can be defined via the presentation
B = hσi , i ∈ I | σi σj σi = σj σi σj , i adjacent to j in Γ, σi σj = σj σi , i ji
There is a surjective group homomorphism π : B 7→ W given by σi 7→ si .
1.2. Monoids and Garside structures in type ADE. We recall the definition
of a Garside structure here, and refer to [DDG+ 15, I, 2.1] for further details.
Definition 1.1. A Garside monoid is a pair (M, ∆), with M a monoid, such that
• M is left- and right-cancellative, that is, ∀a, b, c ∈ M , ab = ac ⇒ b = c and
ba = bc ⇒ a = c;
• there exists l : M 7→ N satisfying l(f g) ≥ l(f ) + l(g) and g 6= 1 ⇒ l(g) 6= 0;
• any two elements of M have a left and right lcm and a left and right gcd;
• ∆ is a Garside element of M , meaning that the left and right divisors of ∆
coincide and generate M ;
• the family of divisors of ∆ in M is finite.
A braid group of type ADE has two interesting Garside structures on it. The underlying monoid M of the first Garside structure is the classical braid monoid B+
W:
B+
W = hσi , i ∈ I | σi σj σi = σj σi σj , i adjacent to j in Γ, σi σj = σj σi , i jimonoid
It is a theorem of Deligne [Del72] and Brieskorn-Saito [BS72] that the canonical
map of monoids B+
W −→ BW is injective (and the categorical considerations later
in the paper will give an alternative proof of that fact, see Corollary 3.3).
The Garside element ∆ of the first Garside structure is the positive lift from the
longest element w0 in the Weyl group, while the length function l is the word length
in the Artin generators {σi }i∈I . An important fact about the positive monoid B+
W,
which we will make use of in later sections, is that B+
is
a
lattice
for
left
divisibility,
W
so that any non-empty finite subset of B+
W has both a greatest common divisor (gcd)

6

ANTHONY M. LICATA AND HOEL QUEFFELEC

and a least common multiple (lcm), both of which are themselves in the positive
monoid B+
W.
In writing the Coxeter presentation of the Weyl group W and its braid group BW ,
we have chosen a set of simple roots in order to give generators in our presentation.
More natural, from some points of view, is to choose for generators of W the
entire conjugacy class of reflections, and to present BW by suitably lifting these
reflections to the braid group. This is the starting point of a very rich “dual” point
of view on the braid group, originating in the work of Birman-Ko-Lee [BKL98] and
Bessis [Bes03] and further developed in work of Bessis-Digne-Michel [BDM02] and
Brady and Watt [BW08]. Part of this dual point of view produces a second Garside
structure on BW , whose definition we now recall. (See also [DG15, Section 3].)
To begin, we choose an orientation Γ~o of Γ. The orientation Γ~o specifies a Coxeter
element c ∈ W , defined by taking the product of all generators si in such a way
that if there is an oriented edge from i to j in Γ~o , then sj is on the left of si in the
expression of c (see Example 1.7).
Once we have fixed a Coxeter element c = si1 · · · sin , we let γ = σi1 · · · σin be the
positive lift of c to the braid group. We then lift the reflections T ⊂ W to their
“dual positive lifts" T ⊂ BW , given by (see [DG15, Proposition 3.13]):
· · · σi−1
T := {γ k σi1 · · · σij σij+1 σi−1
γ −k },
j
1

(1.1)

with j ∈ {1, 2, . . . , n}.
+
The positive monoid B∨
associated to the dual Garside structure on BW is the
c
monoid generated by the dual positive reflections T. In order to give a presentation
of this monoid, and to discuss the length function associated to the dual Garside
structure, we let[1, c]T ⊂ W denote the interval between 1 and c in the absolute
partial order on W :

[1, c]T = {u ∈ W | c = uv, lref l (u) + lref l (v) = lref l (c) = n.}
Here lref l : W → N is the word length function in the generating set T of all
reflections.
This generating set gives rise to an alternative presentation of the braid group: we
have a generator τt for each reflection t ∈ T ⊂ W , and these generators are subject
to those relations between distinct reflections in W which are visible in the interval
[1, c]T of the Weyl group. More precisely, if we define
+

B∨
= hτt | τt1 τt2 = τt2 τt3 if t1 t2 = t2 t3 ∈ [1, c]T imonoid
c
and
B∨
c = hτt | τt1 τt2 = τt2 τt3 if t1 t2 = t2 t3 ∈ [1, c]T igroup
then the map which takes τt to the dual positive lift (1.1) of the reflection t in the
braid group BW defines a group isomorphism
B∨ ∼
= BW .
c

Therefore, this second presentation gives rise to a morphism (of monoids)
B∨
c

+

−→ BW

It is a theorem of Birman-Ko-Lee in type A [BKL98] and Bessis and Brady-Watt
[Bes03, BW08] in type ADE that this morphism is injective. The categorical constructions of this paper will be another proof of this fact (see Corollary 4.7).
+

The second Garside structure on BW then consists of the monoid B∨
c , the length
function given by the word-length lref l in the reflections {τt }, and the Garside

BRAID GROUPS OF TYPE ADE

7

element given by γ. The definition of the dual Garside structure depends on the
choice of Coxeter element (or, equivalently, on the choice of orientation of Γ); however, since all Coxeter elements are conjugate, the resulting dual Garside structures
are also all equivalent to each other.
A key subset of the dual positive monoid, which plays a large role in the rest of the
+
consisting of all dual positive divisors of
paper, are the Bessis braids Be+ ⊂ B∨
c
γ:
(1.2)

Be+ = {β : γ = αβ, with α, β ∈ B∨
c

+

and lref l (α) + lref l (β) = lref l (γ)}.

In fact the apparent left-right asymmetry in the above definition is illusory: for β ∈
Be+ , there exists α ∈ Be+ such that γ = αβ and lref l (α)+ lref l (β) = lref l (γ) if and
only if there exists α′ ∈ Be+ such that γ = βα′ with lref l (β) + lref l (α′ ) = lref l (γ).
Note that the braid reflections T, which generate the dual positive monoid, are
themselves Bessis braids. We denote by Be− = (Be+ )−1 the inverses of the Bessis
braids.
Bessis proves the following elementary property [Bes03, Fact 2.2.4].
τt1 τt2 ∈ T.
Lemma 1.2. Let τt1 and τt2 be in T. Then τt1 τt2 ∈ Be+ =⇒ τt−1
2
Moreover, this property extends to Bessis braids.
β, β ′ , ββ ′ ∈ Be+ =⇒ β ′−1 ββ ′ ∈ Be+ .
Definition 1.3. Given β ∈ Be+ a Bessis braid, we define the descent set of β,
denoted Dc∨ (β), to consist of those reflections τt such that β = ατt , with α ∈ Be+ .
Remark 1.4. The definition above is of r ight descents, but, unlike the notion of
descents associated with the standard positive monoid, Bessis braids have complete
left-right symmetry in their descent sets. That is, Dc∨ (β) may be equivalently defined
as those reflections τt such that β = τt α′ with α′ ∈ Be+ .
For β ∈ Be+ , a reflection τt is a descent of β if and only if τt appears in some
minimal lref l -length expression for β in the generating set T; moreover, in any
minimal length expression for β, a given reflection τt will appear at most once.
+

∨
As with the original positive monoid B+
is a comW , the dual positive monoid Bc
+
binatorial lattice. The following lemmas relate the lattice structure on B∨
to the
c
descent sets of a Bessis braid.

Lemma 1.5. Let β ∈ Be+ . Then lcm(Dc∨ (β)) = β.
Proof. Since β is a multiple of all its descents, it follows that lcm(Dc∨ (β)) divides β,
and we may write β = β ′ lcm(Dc∨ (β)), with β ′ ∈ Be+ . Now, any right descent τ of β ′
is also a left descent of β ′ , and hence τ is also a left descent of β. But if τ is a descent
of β ′ , then τ is also a descent of β, and hence a descent of lcm(Dc∨ (β)); but this
implies that there must be a minimal length expression for β with two appearances
of τ - one in a minimal length expression for β ′ and one in the expression for
lcm(Dc∨ (β)). Since this is not possible, we conclude that β ′ has no descents, and
thus that β ′ = 1.

Lemma 1.6. Let γ = β ′ β with β, β ′ ∈ Be+ and lref l (γ) = lref l (β) + lref l (β ′ ), and
suppose that τt ∈ Be+ divides neither β nor β ′ . Then there exists τt′ ∈ T dividing
/ Be+ .
β ′ such that τt′ τt ∈

8

ANTHONY M. LICATA AND HOEL QUEFFELEC

Proof. Let τt be a reflection which divides neither β nor β ′ . For another reflection
τt′ ∈ T, if τt′ τt divides γ for all τt′ dividing β ′ , then every such τt′ divides (γτt−1 ).
From this it follows that β ′ = lcm({τt′ dividing β ′ }) divides (γτt−1 ), and we can
write (γτt−1 ) = β ′ β ′′ with lref l (γ) − 1 = lref l (β ′ ) + lref l (β ′′ ). But this implies that
β = β ′′ τt , with β ′′ ∈ Be+ , which contradicts the assumption that τt does not divide
β.

Example 1.7. We give an example of the set Be+ in type A3 , using the following
orientation
1

2

•

•

3

•

The associated Coxeter element is c = s1 s3 s2 = s3 s1 s2 , so that the dual Garside
element γ = σ1 σ3 σ2 . The set T is then:
T = {σ1 , σ2 , σ3 , σ3 σ2 σ3−1 , σ1 σ2 σ1−1 , σ1 σ3 σ2 σ3−1 σ1−1 },
while the lattice Be+ is given as follows (the edges below denote the partial order
on Be+ ):
c = σ1 σ3 σ2

σ1 σ3

σ1 σ2

σ3 σ2

σ1 σ3 σ2 σ3−1

σ3 σ2−1 σ1 σ2

σ2−1 σ1 σ3 σ2

σ1

σ2

σ3

σ1 σ2 σ1−1

σ3 σ2 σ3−1

σ1 σ3 σ2 σ3−1 σ1−1

1
1.3. Standard and dual ping pong in type ADE. Ping pong lemmas are
ubiquitous in geometric group theory as tools to establish the faithfulness of (notnecessarily linear) actions of certain groups. We will use two such lemmas – one
for each kind of Garside structure – in establishing the faithfulness of the action of
BW on K; we record these lemmas here. A ping pong proof of the faithfulness of a
categorical action of the free group is appears in work of the first author, [Lic16].
Lemma 1.8. Let Y be a set and suppose Ψ : BW 7→ {bijections bij : Y 7→ Y } is a
group homomorphism. Denote by Ψβ the image of an element β ∈ BW under this
homomorphism. Suppose that there exist disjoint non-empty subsets {Xw }w∈Be+
of Y such that for all simple Bessis elements u and w, Ψu (Xw ) ⊂ Xlf (uw) , where
lf (uw) denotes the greatest left factor lying in Be+ . Then the group homomorphism
Ψ is injective.
Proof. See, for example, [Lic16, Section 3.2].



Remark 1.9. Though we have not included the proof here, one can show that
actually enough to check the inclusion condition for u a reflection.

BRAID GROUPS OF TYPE ADE

9

The positive monoid for the standard Garside structure also has a ping pong lemma
associated to it. The proof is almost identical to that of Lemma 1.8.
Lemma 1.10. Let Y be a set and suppose Ψ : BW 7→ {bijections bij : Y 7→ Y } is a
group homomorphism. Denote by Ψβ the image of an element β ∈ BW under this
homomorphism. Suppose that there exist disjoint non-empty subsets {Xw }w∈W+ of
Y such that for all elements u, w ∈ W+ , Ψu (Xw ) ⊂ Xlf (uw) , where lf (uw) denotes
the greatest left factor lying in W+ . Then the group homomorphism Ψ is injective.
1.4. Standard and dual word-length metrics in type ADE. In this section we
define the relevant word length metrics on BW . Two of the theorems in later sections
will show how these metrics can be recovered from homological considerations: see
Theorems 4.1 and 4.11.
The two Garside structures considered in Section 1.2 give rise to two word-length
metrics on the braid group: the first is the word-length in the generating set consisting of positive Weyl braids W+ and their inverses, and the second is the word-length
in the generating set of positive Bessis braids Be+ and their inverses. We briefly
recall here some facts about geodesics in the braid group with respect to these two
metrics.
Given a braid β, we denote by LW (β) the minimal number of elements from W+ ∪
(W+ )−1 required to express β; LW (β) is called the canonical length, or sometimes
the Charney length of the braid β.
An important special fact about the braid groups associated to finite Coxeter groups
(e.g. the braid groups of type ADE) is that any braid β can be written as a product
of a positive and a negative braid:
β = β + β − = β ′− β ′+ , with β + , β ′+ , β −

−1

, β ′−

−1

∈ B+
W.

Moreover, the expressions for β above can be chosen with
LW (β) = LW (β + ) + LW (β − ) = LW (β ′+ ) + LW (β ′− ).
Such expressions can be further refined into normal forms for β by writing β + and
β − in greedy form: we say that an expression β + = β1 · · · βk , with βi ∈ W+ , is
left-greedy if every left descent of βi is a right descent of βi−1 :
DL (π(βi )) ⊂ DR (π(βi−1 )) for all i = 1, . . . , k − 1.
Similarly, a right-greedy expression for β will be:
β = β1 · · · βk , with DR (π(βi )) ⊂ DL (π(βi+1 )) for all i = 1, . . . , k − 1.
For a general braid β ∈ BW , we define a reduced minimal expression for β to be an
expression of the following kind:
+
+
β = β − β + , β − ∈ B−
W , β ∈ BW

with
β − = βl− · · · β1− right-greedy, β + = β1+ · · · βk+ left-greedy,
LW (β) = k + l, and
DR (π(β1− ) ∩ DL (π(β1+ )) = ∅.
Every braid β has a unique reduced minimal expression in the above sense.
There is an entirely analogous story for geodesics in the word-length with respect
to the Bessis braids Be+ and their inverses in Be− = (Be+ )−1 . We define LBe (β)
to be the minimal number of elements from Be+ ∪ Be− needed to write β. LBe is
the word-length metric in the generators from Be+ ∪ Be− .

10

ANTHONY M. LICATA AND HOEL QUEFFELEC

Any braid β can be written as a product of a dual positive and a dual negative
braid:
−1
−1
+
β = β + β − = β ′− β ′+ , with β + , β ′+ , β − , β ′− ∈ B∨
c .
Moreover, the expressions for β above can be chosen with
LBe (β) = LBe (β + ) + LBe (β − ) = LBe (β ′+ ) + LBe (β ′− ).
Such expressions can be further refined into normal forms for β by writing β + and
β − in greedy form: we say that an expression β + = β1 · · · βk , with βi ∈ Be+ , is
left-greedy if every descent of βi is a descent of βi−1 :
Dc∨ (βi ) ⊂ Dc∨ (βi−1 ) for all i = 1, . . . , k − 1.
Similarly, a right-greedy expression for β will be:
β = β1 · · · βk , with Dc∨ (βi ) ⊂ Dc∨ (βi+1 ) for all i = 1, . . . , k − 1.
For a general braid β ∈ BW , we define a dual reduced minimal expression for β an
expression of the following kind:
−

+
∨
β = β − β + , β − ∈ B∨
c , β ∈ Bc

+

with
β − = βl− · · · β1− right-greedy, β + = β1+ · · · βk+ left-greedy,
LBe (β) = k + l, and
Dc∨ (β1− ) ∩ Dc∨ (β1+ ) = ∅.
Every braid β has a unique reduced minimal expression in the above sense.
1.5. The Digne-Gobet conjecture. In relating the standard and dual Garside
structures, a basic question is how to express simple dual elements, that is, elements from Be+ , in terms of generators of the classical monoid. Digne and Gobet
investigated this question in [DG15], and proposed the following conjecture.
Conjecture 1.11 ([DG15]). Let β ∈ Be+ , then β ∈ [∆−1 , ∆].
In other words, dual simple braids can be expressed as the product of a simple
braid and the inverse of a simple braid. In Section 5, we use the action of B on the
categorified root lattice to give a uniform proof of this conjecture in type ADE. We
also note here that the conjecture was already established via case-by-case analysis
in types A and E (as well as in a number of other non-simply-laced types) in [DG15].
2. Zigzag algebras and categorical braid group actions
2.1. Zigzag algebras. Let Γ̄ denote the oriented double of Γ, obtained by doubling
all edges and orienting them in the two opposite directions. Let P ath(Γ̄) denote
the path algebra of Γ̄, which we regard as a C-algebra. We draw basis elements of
◦
P ath(Γ̄) as paths, so that, for example, a b c denotes a length two path from
◦
◦
vertex a to vertex c through vertex b.
Then the zigzag algebra AΓ is the following quotient of P ath(Γ̄):


◦
◦b
◦c
(2.1) AΓ := P ath(Γ̄)/ a b c = 0 if a 6= c, a
= a
∀a, b, c ∈ I
◦
◦
◦
◦
So length two paths which start and end at distinct vertices are zero in AΓ , and
all length 2 paths which start and end at the same vertex i are equal in AΓ . The

BRAID GROUPS OF TYPE ADE

11

above definition does not technically cover the type A1 and A2 cases: by convention,
we take AΓ = C[x]/x2 in type A1 ; in type A2 we define AΓ to be the quotient of
P ath(Γ̄) consisting of all length three paths.

2.2. Gradings on AΓ and adjunctions. There are a number of important nonnegative gradings on the zigzag algebra. One non-negative grading on AΓ is given
by path length: all edges are assigned degree 1; loops at a vertex are then of degree
two. Shifts for this path-length grading will be denoted with angle brackets h−i in
what follows.
Another non-negative grading on AΓ arises from the choice of an orientation ~o
of Γ, resulting in the quiver denoted Γ~o . In this grading, we declare edges in Γ̄
agreeing with the orientation Γ~o to be of degree 0, and edges of Γ̄ not agreeing
with the orientation Γ~o to have degree 1. In order to avoid confusion with the
path-length grading on projective modules, we use the notation {−} to denote a
shift in orientation degree. Though we will not always include it in the notation,
it is important to remember that the orientation grading depends on a choice of
orientation ~o of the Dynkin diagram, or, equivalently, on a choice of Coxeter element
c in the Weyl group W . By convention in type A1 where AΓ = C[x]/(x2 ), x has
path length degree 2 and orientation degree 1.
After fixing an orientation ~o, we endow AΓ with a bigrading by considering both the
path-length and orientation gradings. We denote by ei the length 0 path at vertex
i, which is an idempotent in AΓ . The left AΓ -module Pi = AΓ ei is indecomposable
and projective, as is the right AΓ -module ei AΓ . For bigraded AΓ modules M, N ,
we denote by HomAΓ (M, N ) the bidegree 0 module homomorphisms from M to N ,
and we define
M
HOMAΓ (M, N ) =
HomAΓ (M, N hki{l})
k,l

to be the space of all bigraded, but not necessarily degree 0, module homomorphisms. For the indecomposable projective left modules {Pi }, we have:

C ⊕ C{1}h2i if i = j;



Ch1i if i 7→ j in Γ ;
~
o
∼
(2.2)
HOMAΓ −mod (Pi , Pj ) =

C{1}h1i
if
j
→
7
i
in
Γ~o ;



0 if i j.
We set Qi = ei AΓ h−1i, which is an indecomposable projective right AΓ -module.
We have

Ch−1i ⊕ C{1}h1i if i = j;




∼ C if i 7→ j in Γ~o ;
(2.3)
Qi Pj := Qi ⊗AΓ Pj =

C{1} if j 7→ i in Γ~o ;


0 if i j.
From these computations, the following lemma follows immediately.

Lemma 2.1. The functor given by tensoring on the left with Qi h1i is right adjoint
to tensoring with Pi , and tensoring with Qi {−1}h−1i is left adjoint to it. In other
words, for any left C-module X and any left AΓ -module Y , there are isomorphisms

12

ANTHONY M. LICATA AND HOEL QUEFFELEC

of bigraded vector spaces
HOM(Pi X, Y ) ≃ HOM(X, Qi Y h1i),

and

HOM(Y, Pi X) ≃ HOM(Qi Y {−1}h−1i, X)

Similarly, we have:
Lemma 2.2. The functor given by tensoring on the right with Qi {−1}h−1i is
right adjoint to tensoring with Pi , and tensoring with Qi h1i is left adjoint to it.
In other words, for any right C-module X and any right AΓ -modules Y , there are
isomorphisms of bigraded vector spaces
HOM(XQi h1i, Y ) ≃ HOM(X, Y Pi )

and

HOM(Y, XQi {−1}h−1i) ≃ HOM(Y Pi , X).
2.3. Categories of complexes. We denote by K the homotopy category of bounded
complexes of finitely-generated bigraded projective AΓ -modules, where the bigrading on AΓ is given by the path-length grading and the orientation grading for a fixed
orientation ~o of Γ. Thus, an object of K is a bounded complex of finitely generated
bigraded projective modules
Y = (Y m , ∂ m ), ∂ m : Y m → Y m+1 , ∂ m+1 ◦ ∂ m = 0.
A morphism f from X to Y is a collection of AΓ module maps f i : X i → Y i
intertwining the differentials. Two maps f, g : X → Y are equal in K if f − g is
nullhomotopic. We let [k] denote the auto-equivalence which shifts a complex k
degrees to the left:
Y [k]m = Y k+m , ∂Y [k] = (−1)k ∂Y .
The pair (K, [1]) is a finitely-generated, linear triangulated category.
Given a map of complexes f : X → Y , the cone of f is the complex X[1] ⊕ Y with
the differential
∂(x, y) = (−∂X (x), f (x) + ∂Y (y)).
2.4. Minimal complexes. Let Y be a complex of graded projective AΓ -modules.
We say that Y is a minimal complex if Y is indecomposable in the additive category
Com(AΓ − mod) of bounded complexes of graded projective AΓ -modules, that is:
Y ∼
= 0.
= 0 or Y2 ∼
= Y1 ⊕ Y2 ∈ Com(AΓ − M od) =⇒ Y1 ∼
The important point about minimal complexes is that if Z ∈ K is indecomposable
in the homotopy category K, then Z has a representative Y in the additive category Com(AΓ − mod) which is minimal. In particular, when regarded as a complex
of AΓ -modules, Y has no contractible summands. Moreover, any two minimal representatives Y1 , Y2 of such a Z ∈ K are isomorphic in Com(AΓ ) − mod, so that
the chain groups of a minimal representative are determined up to isomorphism as
graded projective AΓ modules. We thus refer to Y as the minimal complex associated to its homotopy class, with the understanding that the chain groups of Y
are only determined up to non-canonical isomorphism. As every indecomposable
Z ∈ K is homotopy equivalent to a minimal complex, in some arguments it will be
convenient to study Z by considering its associated minimal complex and studying
the chain groups of that complex.

BRAID GROUPS OF TYPE ADE

13

2.5. The canonical t-structure on K. The goal of this section is to briefly describe two ways to use a grading on AΓ to “slice" objects of the homotopy category of
projective modules into homogeneous pieces. The first way, which uses well-known
homological machinery, is via the canonical t-structure on K. We assume some familiarity with the notion of a t-structure here, and refer the reader to [GM13]. We
also refer the reader to [MOS09] for a detailed discussion of hearts of t-structures on
the homotopy category of complexes of projective modules over finite dimensional
non-negatively-graded algebras.
Let Y ∈ K, and assume that Y is minimal. The chain groups of Y are direct
sums of shifts of graded projective modules Pj {l}hki[m] for various j ∈ I and
l, k, m ∈ Z. A chain summand Pj {l}hki[m] appearing in Y is said to have level
m − k. (Note that the orientation grading shift {l} does not enter in the definition
of the level.) Given N ∈ Z, we may define KN (Y ) ∈ K to be the complex obtained
by considering only those terms of the minimal complex of level N ; the differential
in the complex KN (Y ) is the restriction of the differential of the minimal complex of
Y to those terms which have level N . We denote the full subcategory of K consisting
of complexes of the form KN (Y ) by KN ⊂ K.
For N ∈ Z, the composition τ N = KN [−N ] : K → K is the N th truncation
functor associated to the canonical t-structure on K. The heart K0 of this t-structure
consists of linear complexes, that is, complexes Y ∈ K such that Y ∼
= τ 0 (Y ). Note
N
that for any complex Y ∈ K and any N ∈ Z, the complex τ (Y ) is linear. Given
Y ∈ K, we will refer to the complexes τ N (Y ) as the t-slices of Y .
At one point in the sequel we will also make use of the canonical t-structure on
a homotopy category of (AΓ , AΓ )-bimodules. The definition of this t-structure is
almost identical to that given above for AΓ modules. In particular, the complexes
of bimodules we will consider have chain groups isomorphic to AΓ and bimodules
of the form Pi ⊗C Qi {l}hki[m]; the level of such a summand is defined to be m − k,
just as for modules.
2.6. The ~o-baric structure on K. In addition to t-structures, there is another
way to slice a complex Y ∈ K into homogeneous pieces, which is to ignore the
homological degree completely and slice the minimal complex of Y using only the
internal orientation grading on AΓ -modules. The resulting decomposition of Y arises
not from a t-structure, but rather from a baric structure, a notion introduced by
Achar and Treumann in [AT10] for general triangulated categories.
In our specific situation, the relevant baric structure is straightforward to describe:
for k ∈ Z, let Kk denote the full subcategory of K consisting of complexes Y =
(Y m , ∂ m ) such that, in the minimal complex of Y , in all homological degrees all
of the indecomposable projective summands in that homological degree have their
orientation grading shifted exactly by −k. In other words, in the minimal complex
of Y ∈ Kk , the chain groups of the minimal complex are direct sums of projective
modules of the form Pi hni{k}[m] for various n, m ∈ Z. (There is no condition
on the internal path-length grading shifts hni or the homological shifts [m] in the
definition of the categories Kk .) The subcategories {Kk } define a baric structure
in the sense of [AT10]. When we wish to emphasize the choice of the orientation ~o
in the definition of the ~o-baric structure, we will add it to the baric notation, and
write, for example, K~oN (Y ).
Given N ∈ Z and Y ∈ K, we define KN (Y ) to be the complex – well-defined up to
homotopy – obtained by considering only those terms in the minimal complex of Y
whose orientation degree shift is {−N }, and restricting the boundary map to such

14

ANTHONY M. LICATA AND HOEL QUEFFELEC

terms. We refer to the complexes τN (Y ) = KN (Y ){N } ∈ K0 as the o-baric slices
of the object Y . To avoid confusion between t-slices and baric slices, we will denote
the N th baric slices of Y with N as a subscript and the N th t-slice of Y with N
as a superscript.
An important difference between t− and baric- structures is in the relationship
these structures have to the homological shift functor [1]; for example, the heart of
a t-structure K0 satisfies K0 [1] = K1 , while the baric heart K0 satisfies K0 [1] = K0 .
Given an interval [k, l] ⊂ Z, we set K[k,l] to be the full subcategory of K consisting
of those complexes Y whose non-zero baric slices live in Km for k ≤ m ≤ l. Similarly,
K[k,l] is the full subcategory of K whose non-zero t-slices live in Km for k ≤ m ≤ l.
The subcategories K≤k , K≥k , K≤k , K≥k are defined similarly.
2.7. The braid group action on K. Following Khovanov-Seidel [KS02], the braid
group BW of Γ acts by triangulated autoequivalences of K. The braid generators
σi and σi−1 act by tensoring with complexes of bimodules:
(2.4)

σi 7→ (AΓ → Pi ⊗ Qi {−1}h−1i) , σi−1 7→ (Pi ⊗ Qi h1i → AΓ ) ,

where the bimodule AΓ is in homological degree 0 in both complexes.
The bimodule maps which define the differentials above are as follows:

1 7→
i′

X

i′

•

i
i′
• ⊗ •

i
• +

i
i
• ⊗ •

adjacent to i

+ •i ⊗ •i
and
i
i
i
• ⊗ • 7→ •

Proposition 2.3. The assignment (2.4) defines a homomorphism from the braid
group BW to the group of (homotopy classes of ) complexes of bigraded (AΓ , AΓ )
bimodules.
The proof that these complexes of bimodules satisfy the braid relations is given in
[KS02, Proposition 2.4, Theorem 2.5] (see also [HK01]). For future use, we note
some particular computations of braids acting on the modules Pi here.

Pi [−1]{−1}h−2i if i = j;



P → P {−1}h−1i with P in homological degree zero, if i → j;
j
i
j
(2.5) σi Pj =

P
→
P
h−1i
with
P
in
homological
degree zero, if j → i;
j
i
j



Pj if i j.

Pi [1]{1}h2i if i = j;



P h1i → P with P in homological degree zero, if i → j;
i
j
j
(2.6) σi−1 Pj =

Pi {1}h1i → Pj with Pj in homological degree zero, if j → i;



Pj if i j.
Above we kept track of all gradings independently, and only indicated the shifts
when they are non-zero. Note that σi (Pj ) ∈ K0 unless j = i, in which case σi (Pi ) ∈
K1 .

BRAID GROUPS OF TYPE ADE

15

This immediately implies the following lemma, that we will freely use throughout
the paper.
≥0
Lemma 2.4. If β ∈ B+
) ⊂ K≥0 , and β −1 9K≤0 ) ⊂ K≤0 .
W , then β(K

As mentioned in the introduction, one of the results of this paper is that the converse
also holds: this follows, for example, from Proposition 4.18.
Remark 2.5. As explained in [KS02], the action of BW on the Grothendieck group
is equivalent to the Burau representation of the braid group. Since our action is on
a category of bigraded chain complexes, passing to the Grothendieck group gives a
2-variable version of this Burau representation.
2.8. Root complexes in type ADE. The goal of this section is to describe a
relationship between the isomorphism classes of indecomposable linear complexes
and the corresponding ADE root system. To that end, fix an orientation ~o of the
ADE Dynkin diagram Γ. Recall that the choice of orientation gives rise to both a
non-negative grading on the zigzag algebra AΓ , and also to a baric structure on
K, with baric heart denoted K0 (~o). (For notional ease, we will sometimes drop the
orientation ~o from the notation for the baric structure, and write K0 instead of
K0 (~o).) Let K00 = K0 ∩ K0 denote the intersection of the baric heart K0 and the
heart of the canonical t-structure K0 . Objects of K00 are complexes homotopic to
a complex of indecomposable projectives whose underlying chain group is a direct
sum of projective modules of the form Pi {0}h−ki lying in cohomological degree k.
Note that the shift h1i[1] preserves K00 .
In order to connect the representation theory of AΓ to the combinatorics of the
associated root system, we will explain a relationship between the representation
theory of the zigzag algebra AΓ (in particular, the category K00 ) and the representations of the oriented quiver Γ~o . To do this, we regard the path algebra of Γ~o as a
Z-graded algebra, where the grading is by path-length. A representation (Vi , fij ) of
Γ~o is said to be graded if the corresponding representation of the path algebra of Γ~o
is graded. Thus, in a graded representation of Γ~o , the vector spaces Vi at each vertex
i are themselves Z-graded, and the linear maps assigned to edges of the quiver are
of degree 1. Let Rep(Γ~o ) denote the abelian category of graded representations of
Γ~o .
We will define a functor
Let (Vi , fij )i,j

F~o : Rep(Γ~o ) −→ K00 .
be a graded representation of Γ~o . If
Vi = ⊕m Vi (m)

where the Vi (m) are the graded pieces, then the underlying projective module of
the complex F~o ((Vi , fij )i,j ) is defined to be
M
⊕m Pi h−mi[−m] ⊗C Vi (m),
i

so that the underlying graded vector spaces Vi of the representation become the
multiplicity spaces of the projective Pi . The differential d on F~o ((Vi , fij )i,j )) is
defined by setting its i, j component to be fij xij (where xij is the edge from i to j
in the zigzag algebra):
d = ⊕i,j fij xij .

It is clear from the definition that the resulting complex F~o ((Vi , fij )i,j ), d is an
object of K00 . Moreover, we have the following.

16

ANTHONY M. LICATA AND HOEL QUEFFELEC

Proposition 2.6. The functor F~o : Rep(Γ~o ) −→ K00 is an equivalence of abelian
categories.
Proof. We construct the inverse G~o of F~o explicitly as follows. Let C ∈ K00 be a
minimal complex. The underlying chain group of C is isomorphic to
⊕i ⊕k Pi h−ki[−k] ⊗ Ni (k),
where the multiplicity space Ni (k) is a C-vector space. We define a graded representation G~o (C) by putting the graded vector space ⊕k Ni (k) at vertex i. (Note
that, since homotopic minimal complexes have isomorphic underlying projective
modules, the isomorphism class of the above vector space is independent of the
choice of minimal complex C.) For neighboring vertices i, j of Γ~o , let dij denote the
component of differential d of C which maps from the Pi -isotypic component of C
to the Pj -isotypic component. Since C ∈ K0 , it follows that dij is of the form
X
dij =
fi,j (k)xij ,
k

where
fij (k) : Ni (k) −→ Nj (k + 1).
P
We complete the definition of G~o by putting the linear map fij = k fij (k) on the
edge from i to j in Γ~o . It is then clear from the construction that G~o and F~o are
inverse functors.

Let L̃ denote the set of isomorphism classes of indecomposable objects of K00 , and
let L = L̃/h1i[1] denote the set of equivalence classes of objects in L̃ up to the shift
h1i[1]. The assignment χ(Pi ) = αi extends to a well-defined map
dim : L −→ spanN {αi }
from L to the non-negative cone in the root lattice of the corresponding ADE root
system. (Here αi are the simple roots.) We denote by
Φ+ ⊂ spanN {αi }
the set of positive roots. As a corollary of Proposition 2.6, together with Gabriel’s
theorem [Gab72], we obtain the following.
Theorem 2.7. For Γ a Dynkin diagram of type ADE, the map χ defines a bijection
between L and Φ+ .
Remark 2.8. Though we will not use them in the sequel, we mention here how the
Bernstein-Gelfand-Ponomarev reflection functors [BGP73] between the categories
Rep(Γ~o ) for the various orientations ~o appear in relation to the braid group action
on K. For an orientation ~o, let
τ~o : K −→ K00 (~o)
denote the truncation (“0th cohomology") functor for the t-structure whose heart
is K00 (~o).
For i a sink of the orientation ~o, we define
Si+ = τsi ~o σi : K00 (~o) −→ K00 (si ~o).
Note that the target of Si is the intersection of the t-heart K0 and the baric heart
K0 (si ~o) of the quiver with orientation si~o. Similarly, for i a source of ~o, we define
Si− = τsi ~o σi−1 : K00 (~o) −→ K00 (si~o).

BRAID GROUPS OF TYPE ADE

17

Si± are additive functors, but they are not equivalences. Under the equivalences F~o
between the hearts K00 (c) and the categories Rep(Γ~o ) defined above, the functors
Si± are sent to the BGP reflection functors.
For future use, we also record here the basic compatibility between the braid Coxeter element γ defined by an orientation ~o of the Dynkin diagram and the ~o-baric
structure determined by the same orientation: the action of γ on K moves complexes
up in ~o-baric level up by one.
Lemma 2.9. Let C ∈ K0 . Then γ · C ∈ K1 .
Proof. It suffices to show that γ · Pi ∈ K1 for all i. So consider γ · Pi . We have
γ = σ1 . . . σi−1 σi σi+1 . . . σn . Since for j > i the only morphisms from Pi to Pj hki{l}
have l = 1, it follows that σi+1 . . . σn Pi has a minimal complex of the form Pi → X,
where X ∈ K1 is an object whose minimal complex has as underlying chain group a
direct sum of modules of the form Pj {−1}, with j > i. Since σi (Pi ) ∼
= Pi {−1} ∈ K1 ,
applying σi to σi+1 . . . σn Pi results in a complex which lives in K1 , and whose
underlying chain groups are direct sums of Pj {−1} with j ≥ i. But now when we
apply σ1 . . . σi−1 to this, the result will remain in K1 , since all homs from Pk to Pj
for k < j are of orientation degree 0. Thus γ · Pi ∈ K1 , as desired.

2.9. The reflection complexes Ct and Ct . Let ϕ be a positive root, and t ∈ WΓ
the associated reflection in the Weyl group. As explained in Section 1.2, the choice
of orientation ~o (equivalently, the choice of Coxeter element in WΓ ) defines a lift of
t to the braid group, via
. . . σi−1
γ −k .
t 7→ τ = γ k σi1 . . . σij σij+1 σi−1
j
1
The basic compatibility between this lift and Theorem 2.7 is that the indecomposable complex associated to ϕ in Theorem 2.7 is
Ct = γ k σi1 . . . σij Pij+1 {−k}.
Note, however, that under Theorem 2.7, the complex assigned to a positive root is
only well-defined up to a shift. Thus it is perhaps more natural to associate to the
positive root ϕ the direct sum Ct of all shifts of the complex Ct above:
Ct := ⊕l∈Z Ct [l]hli.
The complex Ct is a direct sum of bounded complexes, but is itself not bounded.
Thus technically Ct is not an object of K. However, note that for fixed Y ∈ K,
the morphism spaces between Y ∈ K and Ct is a well-defined, finite-dimensional
bigraded vector space.
In the definition of Ct we have included shifts hli in the path length grading so that
the summands lie in the heart intersection K00 = K0 ∩ K0 . In latter parts of the
paper which are not concerned with the canonical t-structure – for example in the
proof of Proposition 3.9, we will omit these path-length grading shifts.
3. Categorical ping pong
3.1. Standard ping pong and the canonical t-structure. The orientation
shifts {k} have no bearing on the statements in this subsection, so we choose to
omit them here. For w ∈ W , let w+ denote the positive lift to the braid group. For
the purposes of this section, we define a negative braid complex to be a complex of
the form βPi for some negative braid β ∈ B−
W . Negative braid complexes live in the
non-negative part K≤0 of the canonical t-structure on K.

18

ANTHONY M. LICATA AND HOEL QUEFFELEC

We denote by rf (β) and lf (β) the right, respectively, left factor in the greedy normal
form of a braid. If β is a positive braid, the factor lf (β) is the longest positive lift
from W which divides β on the left in the positive braid monoid; moreover, lf (β)
is the least common multiple of all positive lifts from W which divide β on the
left in the positive monoid. Similarly, if β is a negative braid, the factor rf (β) is
the longest negative lift from W which divides β on the right; in particular, for β
positive, lf (β) = rf (β −1 )−1 .
For w ∈ W , we define sets X w as follows: a complex C ∈ K≥0 is in X w if for all
negative braid complexes Y ,
Hom(C, Y ) = 0 ⇐⇒ w+ −1 (Y ) ∈ K<0 .
So, for example, the complex C = ⊕i∈I Pi is in the set X id , since
Hom(⊕i∈I Pi , Y ) = 0 ⇐⇒ Y ∈ K<0 ⇐⇒ id(Y ) ∈ K<0 .
We have the following.
Proposition 3.1. Let C ∈ X u . Then, for w ∈ W , w+ (C) ∈ X lf (w+ u+ ) .
Note that Proposition 3.1 has as a consequence the fact that the sets X w are nonempty, as the set X id is itself non-empty. In order to prove Proposition 3.1, we need
a couple of preliminary results.
Lemma 3.2. Let β ∈ B−
W be an element of the classical negative braid monoid.
Then
βPi ∈ K<0 ⇔ β = ασi−1 ,
where α ∈ B−
W and l(α) + 1 = l(β).
(In the statement of the above lemma, for γ ∈ BW , l(γ) denotes the word-length of
γ in the Artin generators σi±1 .)
Proof. The proof is by induction on l(β). Suppose that βPi ∈ K<0 . Let β = β ′ σj−1 ,
with l(β ′ ) + 1 = l(β). If i = j, we are done, so suppose that i 6= j.
If σi σj = σj σi , then σj−1 Pi ∼
= Pi , whence β ′ Pi ∈ K<0 . Thus, by, induction, β ′ =
β ′′ σi−1 , with l(β ′′ ) + 1 = l(β ′ ); setting α = β ′′ σj−1 , it follows that β = ασi−1 , with
l(α) + 1 = l(β), as desired.
So now suppose that σi σj σi = σj σi σj and that we have
β ′ (Pj h1i → Pi ) ∈ K<0 .
Since β ′ is a negative braid, β ′ (Pj ) ∈ K≤0 . From this it follows that β ′ Pi ∈ K<0 ; for
if β ′ Pi has a non-zero t-slice in K≥0 , then by assumption any terms in K0 (β ′ Pi ) must
cancel under Gaussian elimination with terms appearing in K≥1 (β ′ Pj ). But since β ′
is a negative braid, there are no terms appearing in K≥1 (β ′ Pj ). Thus β ′ Pi ∈ K<0 ,
and therefore by induction hypothesis, β ′ = β ′′ σi−1 with l(β ′ ) = l(β ′′ ) + 1. Now
β ′′ σi−1 σj−1 Pi ∼
= β ′′ Pj [1]h1i ∈ K<0 . Since the shift [1]h1i preserves K<0 , it follows
that β ′′ Pj ∈ K<0 , whence β ′′ = β ′′′ σj−1 with l(β ′′ ) = l(β ′′′ ) + 1 by the induction
hypothesis.
Now β = β ′′′ σj−1 σi−1 σj−1 = (β ′′′ σi−1 σj−1 )σi−1 , as desired.

BRAID GROUPS OF TYPE ADE

19

The converse claim is clear, since if β = β ′ σi−1 , with β ′ a negative braid, then
βPi ∼
= β ′ Pi [1]h2i; as Pi [1]h2i ∈ K<0 , and β ′ is a negative braid, it follows that
′
β Pi [1]h2i ∈ K<0 .

An immediate corollary of the above Lemma that we leave as an exercise to the
interested reader is that the sets X w are pairwise disjoint.
The above lemma also has the following important consequence:
−
Corollary 3.3. The canonical morphisms of monoids B+
W −→ BW and BW −→
BW are injective.
−1
Proof. By Lemma 3.2, an expression for β ∈ B−
W in the monoid generators {σi }
can be read off inductively from the action of β on the indecomposable projective
modules {Pi }. Thus the morphism of monoids

B−
W −→ [Aut(K)]
is injective, where here [Aut(K)] is the group of isomorphism classes of autoequivalences of K. Since the above monoid morphism factors through BW via the canonical
map B−
W −→ BW , this canonical map must therefore also be injective. The proof
of injectivity for the map from positive monoid B+
W to BW follows by a similar
argument.

Lemma 3.4. Let Y be a negative braid complex and let β be a negative braid. Then
βY ∈ K<0 ⇐⇒ rf (β)(Y ) ∈ K<0 .
Proof. Write Y = αPi with α a negative braid. Then
βY ∼
= βαPi ,
so, by Lemma 3.2, βα has σi−1 as a right descent. Thus rf (βα) has σi−1 as a right
−1
<0
descent, so one can write rf (βα) = ησi−1 with η ∈ B−
and
W . Since σi Pi ∈ K
<0
<0
ηK ⊂ K , then
rf (βα)Pi ∈ K<0 .
Since
rf (βα) = rf (rf (β) · α),
it follows that rf (rf (β) · α)Pi ∈ K<0 , and hence that rf (β) · αPi ∈ K<0 .
Thus rf (β)Y ∈ K<0 . The converse is clear.

Now we prove Proposition 3.1.
Proof. Suppose C ∈ X w . We show that σi C ∈ X lf (σi w) . If Y is a negative braid
complex,
Hom(σi C, Y ) ∼
= Hom(C, σi−1 Y ).
Since C ∈ X w , and σi−1 Y is a negative braid complex, we have
−1 −1
C ∈ X w ⇐⇒ w+
σi Y ∈ K<0 .

But now, by the previous Lemma 3.4,
−1 −1
−1 −1
w+
σi Y ∈ K<0 ⇐⇒ rf (w+
σi )Y ∈ K<0 .
−1 −1
Since rf (w+
σi ) = lf (σi w+ )−1 , this completes the proof.



20

ANTHONY M. LICATA AND HOEL QUEFFELEC

The sets X w satisfy the requirements to apply Lemma 1.10, and thus one obtains
a new proof of the main theorem of Brav-Thomas in [BT11]:
Corollary 3.5. For W a Weyl group of type ADE, the action of the braid group
BW on K is faithful.
3.2. Dual ping pong and the ~o-baric structure. In the previous subsection,
we used the compatibility between the classical positive and negative monoids and
the canonical t-structure on K to define complexes for standard ping pong. In this
subsection, we use the compatibility of the dual positive and negative monoids with
the ~o-baric structure on K to give analogous dual ping pong constructions.
Recall that K0 denotes the heart of the ~o-baric structure on K. For X ∈ K0 , we
define a positive Bessis braid ν + (X) ∈ Be+ and a negative Bessis braid ν − (X) ∈
Be− by
(3.1)

ν+ (X) = min{β ∈ Be+ |β · X ∈ K[1,∞) },

and
(3.2)

ν− (X) = min{β ∈ Be− |β · X ∈ K(−∞,−1] }.

Thus ν + (X) is the greatest common divisor of all the positive Bessis braids which
lift X into the strictly positive part of the ~o-baric structure, while ν − (X) is the
greatest common divisor of all the negative Bessis braids which lower X into the
strictly negative part of the ~o-baric structure.
Remark 3.6. Let X ∈ K0 and β ∈ Be+ . By the minimality of ν+ (X), if β · X ∈
K[1,∞) then ν+ (X) divides β in the dual positive monoid. In fact, as we will show
later, ν+ (X)(X) ∈ K[1,∞) (see Corollary 3.15). Thus the set of Bessis braids which
lift a fixed complex from the baric heart K0 into the positive part K≥1 is closed
under greatest common divisors.
The following propositions are the main results needed in the construction of dual
ping pong.
Proposition 3.7. Let τu , τt ∈ T ⊂ Be+ be associated to reflections u, t ∈ T with
u 6= t. Then:
(1) τt τu ∈ Be+ ⇔ τt τu τt−1 ∈ T ⇔ τt Cu ∼
= Ctut−1 ;
(2) τt τu ∈
/ Be+ ⇔ τt τu τt−1 ∈
/ T ⇔ τt · Cu ∈ K[0,1] with the following conditions
on the baric slices
• the top baric slice K1 (τt · Cu ) is isomorphic to a direct sum of shifts of
Ct ;
• the bottom baric slice K0 (τt · Cu ) satisfies
ν+ (K0 (τt · Cu ))τt = lcm(τt , τu ) ∈ Be+ .
In the above, the statement tCu ∼
= Ctut−1 is equivalent to the statement that τt Cu
is isomorphic to Ctut−1 hki[−k] for some k ∈ Z. Informally, the second item in the

BRAID GROUPS OF TYPE ADE

21

above proposition says that τt · Cu looks like
L
Ct {−1}
τt · Cu =

X

where X = K0 (τt · Cu ).
Proposition 3.7 has a direct translation for the reflections in Be− .
Proposition 3.8. Let τu−1 , τt−1 ∈ T−1 with u 6= t ∈ T . Then:
(1) τt−1 τu−1 ∈ Be− ⇔ τt−1 τu−1 τt ∈ T−1 ⇔ τt−1 Cu = Ct−1 ut ;
(2) τt−1 τu−1 ∈
/ Be− ⇔ τt−1 τu−1 τt ∈
/ T−1 ⇔ τt−1 · Cu ∈ K[−1,0] with the following
conditions on the baric slices
• the bottom baric slice K−1 (τt−1 · Cu ) is isomorphic to a direct sum of
shifts of Ct ;
• the top baric slice K0 (τt−1 · Cu ) satisfies
ν− (K0 (τt−1 · Cu ))τt−1 = lcm(τt−1 , τu−1 ) ∈ Be− .
Proposition 3.9. Let τu , τv be positive reflections in Be+ . Then:
τu τv ∈ Be+ ⇔ Hom(Cu , Cv ) = 0.
The proof of Propositions 3.7 and 3.8 will be postponed until Section 3.4, while
Proposition 3.9 will be established in Section 3.5.
Example 3.10. We give an example to illustrate the content of Proposition 3.9.
Consider the following orientation of the A3 Dynkin diagram, whose corresponding
Coxeter element is c = s1 s3 s2 .
1

•

2

•

3

•

Now,
Hom(P1 , P2 {k}) 6= 0 =⇒ k = −1, and
Hom(P2 , P1 {k}) 6= 0 =⇒ k = 0.
From this we see, via Proposition 3.9, that σ1 σ2 ∈ Be+ , but σ2 σ1 ∈
/ Be+ . Similarly,
since
Hom(P3 , P1 {k}) = Hom(P3 , P1 {k}) = 0 for all k,
we have that σ1 σ3 = σ3 σ1 ∈ Be+ .
On the other hand, consider τu = σ2 and τv = σ1 σ3 σ2 σ3−1 σ1−1 ; the associated
reflection complexes are:
P2 and P2

P1
⊕
P3

22

ANTHONY M. LICATA AND HOEL QUEFFELEC

Then there are non-trivial orientation-degree 0 maps f and g in both directions
between these two complexes:
P2

P2

P1
⊕
P3

g

and

f

P1
⊕
P3

P2

P2

and indeed, neither τu τv nor τv τu is in Be+ .
3.3. Dual ping pong. For w ∈ Be+ , we define Xw ⊂ K≥0 as follows:
(3.3)

Xw := {C ∈ K≥0 | Hom(C, Ct ) = 0 ⇔ t|w}

We claim that the sets Xw satisfy the conditions of the dual ping pong lemma 1.8.
It is clear from the definition that the sets {Xw }w∈Be+ are disjoint, as w ∈ Be+
is determined by its reflection factors. Now, using Remark 1.9, it suffices to prove
that for τu a Bessis reflection and w ∈ Be+ , we have τu Xw ⊂ Xlf (τu w) .
To see this, write τu w = lf (τu w)w′ , and let t be a reflection.
• If τu−1 τt τu ∈ Be+ , then using Proposition 3.8, item 1, we have that
∼ Cu−1 tu .
τ −1 Ct =
u

Then, for C ∈ Xw ,
Hom(τu C, Ct ) ∼
=0
= 0 ⇐⇒ Hom(C, Cu−1 tu ) ∼
= 0 ⇐⇒ Hom(C, τ −1 Ct ) ∼
⇐⇒

u
−1
τu τt τu |w ⇐⇒

τt |τu w ⇐⇒ τt |lf (τu w).

• If τu−1 τt τu ∈
/ Be+ , then by Proposition 3.8, we have that τu−1 Ct ∈ K[−1,0]
with bottom baric slice in K−1 (τu−1 Ct ) isomorphic to a direct sum of shifts
of Cu . Let us denote the top baric slice K0 (τu−1 Ct ) by X0 ; the complex X0
satisfies ν+ (X0 )−1 τu−1 = lcm(τt−1 , τu−1 ). In a picture, we have:

τu−1 · Ct =

X0
⊕Cu {1}

Since C ∈ K≥0 , Cu {1} ∈ K<0 , and there are no AΓ -module maps from
minimal complexes in K≥0 to minimal complexes in K<0 , we see that
Hom(C, τu−1 Ct ) ∼
= Hom(C, X0 ).
Now, by Theorem 2.7, X0 is isomorphic to a direct sum of reflection complexes,
X0 = ⊕i∈J Cti hki i[ki ],
so that
Hom(C, τu−1 Ct ) = 0 ⇐⇒ Hom(C, Cti ) = 0 ∀i ∈ J ⇐⇒ τti |w ∀i ∈ J.
Note that the last item is equivalent to lcmi∈J {τti }|w. Now, by Corollary
3.15 (which will be proven along with Proposition 3.7 in the next subsection)
ν+ (X0 ) = lcm(τti ),

BRAID GROUPS OF TYPE ADE

23

and we arrive at
Hom(C, τu−1 Ct ) = 0 ⇐⇒ ν+ (X0 )|w.
Finally, we note that lcm(τt , τu ) = τu ν+ (X0 ), so that
ν+ (X0 )|w ⇐⇒ lcm(τt , τu )|τu w ⇐⇒ τt |τu w ⇐⇒ τt |lf (τu w).
The last condition we need to check is that the sets Xw are non-empty, and we do
this by constructing an explicit complex in each Xw .
L
To do this, write γ = w′ w, with lref l (w)+lref l (w′ ) = lref l (γ), and let C = τt |w′ Ct .
We claim that C ∈ Xw . Indeed, if τt0 |w, then ∀τt |w′ , we have that τt τt0 ∈ Be+ and
by Lemma 3.9, it follows that Hom(C, Ct0 ) = 0. On the other hand, if τt0 |w′ , then
Ct0 is one of the summands C and thus Hom(C, Ct0 ) 6= 0. Finally, if τt0 divides
neither w nor w′ , then Lemma 1.6 implies that there exists a reflection t1 with
/ Be+ . By Lemma 3.9, we have Hom(Ct1 , Ct0 ) 6= 0. Thus we
τt1 |w′ and τt1 τt0 ∈
conclude that
Hom(C, Ct ) = 0 ⇐⇒ τt |w,
so that C ∈ Xw . This completes the proof that the sets {Xw }w∈Be+ satisfy the
requirements of Lemma 1.8 for dual ping pong. As a consequence, we obtain another
proof of Corollary 3.5, that is, of the main theorem of Brav-Thomas in [BT11].
3.4. Proof of Proposition 3.7. We begin with the following lemma.
Lemma 3.11. Let β ∈ Be+ and let Cu ∈ K0 be a reflection complex. We have
β · Cu ∈ K[0,1] .
· · · σi−1
Proof. Recall that any τt ∈ T is of the form τt = γ k σi1 · · · σij σij+1 σi−1
γ −k ,
j
1
−k
with γ = σi1 · · · σin . Using Lemma 2.9, we have that γ Cu ∈ K−k . Now, we successively apply letters σi−1 in the order they appear in the reduced expression for
γ. When σi−1 appears to the left of σl−1 in the expression for γ, the definition of
the orientation grading on the zigzag algebra AΓ implies that Hom(Pi , Pl {m}) = 0
for m 6= 0. From this it follows that σi−1
· · · σi−1
γ −k Cu ∈ K[−k−1,−k] . Now. applyj
1
· · · σi−1
ing σij+1 will yield a complex in K[−k−1,−k+1] , with Kk+1 (σij+1 σi−1
γ −k Cu )
j
1
isomorphic to a direct sum of shifts of Pij+1 (or equal to zero). Moreover, the fact
that
Hom(Pim , Pij+1 {r}) 6= 0 =⇒ r = 1 for m ≤ j
implies that the bottom baric slice does not change when we apply σij+1 :
Kk−1 (σij+1 σi−1
· · · σi−1
γ −k Cu ) ∼
· · · σi−1
γ −k Cu ).
= Kk−1 (σi−1
j
1
j
1
Now, note that if Y is any minimal complex in K−k−1 whose underlying chain
group only contains projective modules of the form Pi1 {−k − 1}, . . . , Pij {−k − 1},
· · · σi−1
then σi1 · · · σij (Y ) ∈ K−k . Since the bottom slice of σij+1 σi−1
γ −k Cu is of
j
1
this form, it follows that
· · · σi−1
σi1 · · · σij σij+1 σi−1
γ −k Cu ∈ K[−k,−k+1] .
j
1
· · · σi−1
Thus γ k σi1 · · · σij σij+1 σi−1
γ −k Cu ∈ K[0,1] , and we have that βCu ∈ K≥0 .
j
1
Thus we see that for any reflection t, and any C ∈ K0 , τt (C) ⊂ K≥0 . From this
it follows that for any β ∈ Be+ and any C ∈ K0 , βC ∈ K≥0 . Moreover, for any

24

ANTHONY M. LICATA AND HOEL QUEFFELEC

β ∈ Be+ , there exists β ′ ∈ Be+ with γ = β ′ β. Since γ(C) ∈ K1 ⊂ K≤1 for all
C ∈ K0 , we must therefore have βC ∈ K≤1 . This shows that βC ∈ K[0,1] .

Remark 3.12. It also follows from the proof of Lemma 3.11 that if β, β ′ , β ′ β ∈ Be+ ,
and X ∈ K0 , but K1 (βX) 6= 0, then K1 (β ′ βX) 6= 0.
The characterization from Lemma 3.11 actually gives a criterion for determining
whether or not a product of reflections is a Bessis braid:
Lemma 3.13. Let τt1 , . . . , τtk ∈ T. Then
τt1 . . . τtk ∈ Be+ ⇐⇒ (τt1 · · · τtk )Y ∈ K[0,1] for all Y ∈ K0 .
Proof. It follows immediately from Lemma 3.11 that
τt1 . . . τtk |γ =⇒ (τt1 · · · τtk )K0 ⊂ K[0,1] .
For the converse, we induct on k. If k = 1, the claim is clear. For k > 1, by induction
hypothesis and Remark 3.12, both τt2 · · · τtk and (τt−1
τt1 τt2 ) · · · τtk are divisors of
2
−1
γ, so α = lcm(τt2 · · · τtk , (τt2 τt1 τt2 ) · · · τtk ) divides γ as well. The braid τt1 τt2 · · · τtk
has length k in the reflection generators, and is a multiple of both the length k − 1
τt1 τt2 ) · · · τtk . Since t1 6= t2 , it follows
braid τt2 · · · τtk and the length k −1 braid (τt−1
2
that these two length k −1 braids are distinct, and thus that α = τt1 τt2 · · · τtk . Since
α divides γ, this proves the claim.

Lemma 3.14. Let β ∈ Be+ be a Bessis braid and let Cu be a reflection complex.
We have the following:
(1) β · Cu ∈ K0 ⇐⇒ βτu β −1 ∈ T and τu does not divide β;
(2) β · Cu ∈ K1 ⇐⇒ τu |β.
Proof. For (1), let τu = µσj µ−1 as in 1.1, so that Cu ∼
= µPj . Suppose that βCu =
βµPj ∈ K0 . Since βCu ∈ K0 , it follows that τu does not divide β, for if β = ατu
with α ∈ Be+ , then
βCu ∼
= αCu {−1} ∈ K≥1 .
We will use Lemma 3.13 to show that βτu β −1 is in T. To see this, let Y ∈ K0 , so
that τu Y ∈ K[0,1] . The top slice K1 (τu Y ) is either 0 or is isomorphic to a direct sum
of shifts of Cu . Now, since βCu ∈ K0 and β ∈ Be+ it follows that βτu · Y ⊂ K[0,1] ,
too, and by Lemma 3.13 it follows that βτu ∈ Be+ . This implies that βτu β −1 ∈ T.
For the converse, suppose βτu β −1 ∈ T and τu does not divide β. Then βτu ∈ Be+ ,
whence βτu Cu ∈ K[ 0, 1]. Since τu Cu ∼
= Cu {−1} ∈ K1 , this shows that βCu must be
in K0 .
We now prove (2). Suppose first that βCu ∈ K1 , so that γ −1 βCu ∈ K0 . Since
γ −1 β ∈ Be− , we can use the analog of item 1 for dual negative braids to conclude
that γ −1 βτu−1 ∈ Be− . Thus βτu−1 is in Be+ , and τu is a factor of β. For the converse,
write β = β ′ τu with β ′ ∈ Be+ and τu not dividing β ′ . Then, by item (1), β ′ Cu ∈ K0 .
Thus
βCu = β ′ τu Cu = β ′ Cu {−1} ∈ K1 ,
as desired.



BRAID GROUPS OF TYPE ADE

25

As a consequence, we get the following corollary, which establishes the link between
+
the lattice structure on the interval [1, γ] = Be+ in the dual positive monoid B∨
c
and the action of BW on K, equipped with its ~o-baric structure.
Corollary 3.15. For X ∈ K0 , recall that ν+ (X) = gcd{β ∈ Be+ : βX ∈ K>0 }.
We have
• ν+ (X)X ∈ K1 , and
• ν+ (⊕t∈J Ct ) = lcm{τt }t∈J .
Proof. Let X ∈ K0 . After possibly applying homological shifts [k] to the indecomposable summands of X, we may assume that X ∈ K00 (c). By Theorem 2.7, we may
then write X = ⊕j∈J Ctj . Now, consider the set {β ∈ Be+ |β · X ∈ K1 }. By Lemma
3.14 (2) above, if β ∈ {β ∈ Be+ |β · X ∈ K1 } and j ∈ J, then τtj divides β. Thus
lcm{τtj }j∈J divides β, and therefore lcm{τtj }j∈J divides ν+ (X). This shows that
ν+ (X)X ∈ K1 . Since lcm{τtj }j∈J ∈ {β ∈ Be+ |β · X ∈ K1 }, it now follows that

ν+ (X) = lcm{τtj }j∈J .
As a consequence of Corollary 3.15, together with the relationship between gcds
and lcms of subsets of Be+ , we obtain the following relationship between ν+ (X)
and ν− (X) for complexes X ∈ K0 .
Lemma 3.16. ν− (X) = ν+ (X)−1
The following Lemma will now complete the proof of Proposition 3.7.

Lemma 3.17. Let τt , τu ∈ T. Let X = K0 (tCu ). Then ν+ (X)τt = lcm(τt , τu ) ∈
Be+ .
Proof. Let us first prove that ν+ (X)τt ∈ Be+ . As in the proof of Corollary 3.15, we
may assume that X = ⊕s∈J Cs . From τt−1 (τt Cu ) = Cu , we deduce that τt−1 · X ∈ K0 .
Now, by Lemma 3.14, this implies that τt−1 τs−1 ∈ Be− for all s ∈ J. Using Lemma
3.14 again, we see that τs Ct ∈ K0 , and thus γ −1 τs Ct ∈ K−1 . By Corollary 3.15),
gcd{γ −1 τs }s∈J Ct ∼
= γ −1 lcm{τs }s∈J Ct ∈ K<0 ,
and thus lcm({τs }J ) · Ct ∈ K0 . This implies that lcm({τs }s∈J )t · K0 ⊂ K[0,1] , and
by Lemma 3.13, we obtain ν+ (X)τt ∈ Be+ .
Clearly ν+ (X)τt Cu ∈ K>0 , so that τu |ν+ (X)t. Since τt divides ν+ (X)t as well, we
have that lcm(τt , τu ) divides ν+ (X)τt . Now, writing lcm(τt , τu ) = ζτu , with ζ ∈ Be+ ,
we have that ζX ∈ K>0 , so that ν+ (X) divides ζ. Thus ν+ (X)t divides lcm(τt , τu ).
This shows that ν+ (X)t = lcm(τt , τu ).

This completes the proof of Proposition 3.7.

26

ANTHONY M. LICATA AND HOEL QUEFFELEC

3.5. Proof of Proposition 3.9. Up to this point we have described some information about a braid β that can be obtained by considering the action of β
on complexes of AΓ -modules. In this section we collect some information directly
about the complex of bimodules associated β. This will be of use in the proof of
Proposition 3.9 and later again in Section 5.
Outside of homological degree 0, the chain groups of a minimal complex associated
to a braid β are direct sums of bigraded (AΓ , AΓ ) bimodules of the form Pi ⊗
Qj {k}hli. In homological degree 0, the minimal complex is a direct sum of such
bimodules, along with a single copy of the bimodule AΓ {0}h0i itself, which by
the conventions of this paper sits in homological degree 0. The following lemma
concerns the baric slices of such a complex of bimodules. In particular, note that
the existence of the term AΓ {0}h0i in homological degree 0 of the minimal complex
of B implies that K0 (B) 6= 0.
Lemma 3.18. Let B be a complex of (AΓ , AΓ ) bimodules associated to a braid.
Suppose that for some k ≥ 0, K−k (B) 6= 0, but K(−∞,−k−1] (B) = 0. Moreover, in
the case k = 0, so that K<0 (B) = 0, assume that B is not isomorphic to AΓ . Then
∃i ∈ I such that K−k−1 (B · Pi ) 6= 0.
Note that in the statement of the above Lemma, K−k (B) is a complex of (AΓ , AΓ )
bimodules, while K−k−1 (B · Pi ) is a complex of left AΓ modules.
Proof. Consider B bot := K−k (B) the degree −k slice of B. We claim that the
minimal complex of B bot has a bimodule Pj ⊗ Qi {k} as a subcomplex. This is clear
if k > 0. If k = 0, then by assumption the chain groups of the minimal complex of
K0 (B) have at least one summand other than the summand AΓ . Moreover, since
all nonzero maps from AΓ to Pj ⊗ Qi are of strictly positive degree, it follows that
no other summand of the chain group of K0 (B) is mapped into from AΓ via the
differential. It then follows that the minimal complex of K0 (B) has a shift of Pj ⊗Qi
as a subcomplex.
The fact that Pj ⊗ Qi is a subcomplex implies that for some s ∈ Z,
Hom(B bot , Pj ⊗ Qi [s]{k}) ∼
= C 6= 0,
as the above hom space contains the identity map out of the Pj ⊗ Qi [s] summand
of B, and when Pj ⊗ Qi [s] is a summand of a minimal complex, this map cannot
be homotopic to zero.
Now by Lemma 2.2, we have:
Hom(B bot ⊗AΓ Pi , Pj [s]{k + 1}) ≃ Hom(B bot ⊗AΓ Pi , Pj [s]{k}){1}
≃ Hom(B bot , Pj ⊗C Qi [s]{k}) 6= 0.
The non-vanishing of Hom(B bot ⊗AΓ Pi , Pj [s]{k + 1}) implies that K−k−1 (B ⊗ Pi ) 6=
0, which concludes the proof.

An analogous argument proves a similar statement for top baric slices:
Lemma 3.19. Let B be a minimal complex of bimodules associated to a braid.
Assume that for some k > 0, Kk (B) 6= 0 but for all l > k, Kl (B) = 0. Then ∃i ∈ I
such that Kk (B ⊗ Pi ) 6= 0.

BRAID GROUPS OF TYPE ADE

27

As a consequence, we have the following relationship between bounds on the baric
slices of the complex of bimodules B associated to a braid β and the baric slices of
complex of left modules B ⊗AΓ (⊕i Pi ).
Corollary 3.20. Let B be a complex of bimodules associated to a braid β. Then
the complex of left modules B ⊗AΓ Pi ∈ K[0,1] for all i if and only if the complex of
bimodules B has only non-zero baric slices in degree 0 and 1, with
K0 (B) ∼
= AΓ
Since we have already shown that a braid β ∈ Be+ if and only if β(Pi ) ∈ K[0,1] for
all i, it follows that β ∈ Be+ if and only if the complex of bimodules B associated
to β has only non-zero baric slices in degree 0 and 1, with
K0 (B) ∼
= AΓ
We may now complete the proof of Proposition 3.9.
Proof of Proposition 3.9. Recall the statement of Proposition 3.9: if u, v ∈ T , then
τu τv ∈ Be+

⇔

Hom(Cu , Cv ) = 0.

By Corollary 3.20, in order to show that τu τv ∈ Be+ , we must show that the
complex of bimodules assigned to τu τv has non-trivial baric slices only in degrees 0
and 1, with the degree 0 baric slice K0 (τu τv ) ∼
= AΓ .
Let us consider τu = µσi µ−1 , so that Cu = µPi ∈ K0 , and τv = δσj δ −1 with
Cv = δPj ∈ K0 . The complex B of bimodules associated to τu τv can then be
described as follows:
µPi ⊗ Qi µ−1 {−1}
B∼
= AΓ

µPi ⊗ Qi µ−1 δPj ⊗ Qj δ −1 {−2}
δPj ⊗ Qj δ −1 {−1}

From the fact that µPi and δPj are in K0 (and similarly the fact that Qi µ−1 and
Qj δ −1 are in the baric heart for right AΓ modules), we see that:
• Kn (B) = 0 for all n ∈
/ {0, 1, 2};
• K0 (B) ∼
= AΓ ;


• K2 (B) ∼
= K2 µPi ⊗ Qi µ−1 δPj ⊗ Qj δ −1 {−2} .

Furthermore, since µPi and Qj δ −1 are complexes of left (resp. right) AΓ modules in the baric heart, it follows that K2 (B) = 0 if and only if the degree 0
component of the graded vector space Qi µ−1 δPj vanishes; that is, if and only if
HomgrV ect (C, Qi µ−1 δPj [k]) = {0} for all k ∈ Z.
Then, by adjunction, we see that:
HomgrV ect (C, Qi µ−1 δPj [k]) ≃ HomAΓ (Pi , µ−1 δPj [k]) ≃ HomAΓ (µPi , δPj [k]).
for all k ∈ Z. Thus, we conclude that uv ∈ Be+ if and only if
Hom(Cu , Cv ) = {0},
as desired.



28

ANTHONY M. LICATA AND HOEL QUEFFELEC

4. Homological interpretations of word-length metrics
4.1. The dual metric and the o-baric structure.
Word-length. We now would like to give a better description of how the orientation grading we have considered on the category of AΓ -modules interacts with the
structure of the braid group. The final result, Theorem 4.1, will allow one to read
off the word-length of a braid in terms of generators from Bessis’ braids.
For β a braid in the braid group BW , define:
(4.1)

Lγ+ (β) := max{k ∈ Z : ∀X ∈ K0 , Kk (βX) 6= 0}

(4.2)

Lγ− (β) := max{k ∈ Z : ∀X ∈ K0 , K−k (βX) 6= 0}

(4.3)

Lγ (β) := max(Lγ+ (β), 0) + max(Lγ− (β), 0)

Theorem 4.1. Let β ∈ BW . Then we have:
(1) d(β1 , β2 ) = l(β2−1 β1 ) is a metric on BW ;
(2) Lγ+ (β) equals the number of letters from Be+ used in any minimal length
expression for β in the alphabet Be+ ∪ Be− ;
(3) Lγ− (β) equals the number of letters from Be− used in any minimal length
expression for β in the alphabet Be+ ∪ Be− ;
(4) −Lγ+ (β) equals the number of times γ −1 appears in any minimal length
expression for β in the alphabet Be+ ∪ Be− ;
(5) −Lγ− (β) equals the number of times γ appears in any minimal length expression for β in the alphabet Be+ ∪ Be− .
In any of the above statement, if the considered number is negative, it stands for
zero letters appearing in the expression. Combined together, these statements tell
that Lγ is the word-length metric Lc in the generators Be+ ∪ Be− .
The proof will be based on a succession of technical results that will give a step-bystep description of the image of linear complexes under the action of Bessis braids,
and braids that one can form from them. Starting from an expression of a braid β
as β = β − β + , with β − ∈ hBe− i and β + ∈ Be+ , written as β + = β1+ · · · βk+ leftgreedy (that is, β1+ is the lowest common multiple of the descents of β, etc.) and
β − = βl− · · · β1− right-greedy, the goal will be to control well enough the different
layers of the image of a complex under the action of the first terms of β so that we
know what result applying one more term will yield. In that context, Lemma 3.17
already gives us some control, that we will want to extend first to any Bessis braid,
then to any braid lying in Bessis’ positive monoid, and finally to any braid.
Bessis’ braids. We first introduce a useful piece of notation.
Definition 4.2. For β ∈ Be+ , define:
∨

β = β −1 γ

,

β ∨ = γβ −1

The following are basic facts about the lattice Be+ that we will use freely in what
follows.
•

∨

(β ∨ ) = β = (∨ β)∨ ;

BRAID GROUPS OF TYPE ADE

•

∨

•

(∨ β)(∨ α) = ∨ (αγ −1 β);

29

(αβ) = (∨ β)γ(∨ α);

• gcd(β, β ∨ ) = 1;
• lcm(β, β ∨ ) = γ.
We can then state the main statement describing the action of a braid β ∈ Be+ on
a complex in the baric heart K0 .
Lemma 4.3. For β ∈ Be+ , β(⊕t∈J Ct ) ∈ K[0,1] with:
• the bottom baric slice X0 = K0 (β ⊕t∈J Ct ) ∈ K0 satisfies
ν+ (X0 )β = lcm(ν+ (⊕t∈J Ct ), β);
• the top baric slice X1 {−1} = K1 (β(⊕t∈J Ct ){−1} ∈ K1 satisfies
β = ν+ (X1 ) gcd((ν+ (⊕t∈J Ct ))∨ , β).
Proof. Note that ν+ (X0 )β(⊕t∈J Ct ) ∈ K[1,∞) , so we have
ν+ (⊕t∈J Ct )|ν+ (X0 )β.
The right hand side is obviously also divisible by β, so we get:
lcm(ν+ (⊕t∈J Ct ), β)|ν+ (X0 )β.
Now, lcm(ν+ (⊕t∈J Ct ), β)(⊕t∈J Ct ) ∈ K[1,∞) . We may write
lcm(ν+ (⊕t∈J Ct ), β) = δβ,
with δ ∈ Be+ and
lref l (lcm(ν+ (⊕t∈J Ct ), β) = lref l (δ) + lref l (β).
From this we see that δ(X0 ) ∈ K1 , so ν+ (X0 )|δ. Thus we see that
ν+ (X0 )β| lcm(ν+ (⊕t∈J Ct ), β)
and also that
lcm(ν+ (⊕t∈J Ct ), β)|ν+ (X0 )β.
Thus the two braids are equal, proving the first statement.
The second statement follows from the first (or, rather, its analog for braids in Be− )
once we replace β by γ −1 β ∈ Be− .

We state some special cases of the above as corollaries for later use.
Corollary 4.4. Let β ∈ Be+ , and let J be such that lcm{τt , t ∈ J} = γ. Then
ν + (K0 (β(⊕t∈J Ct ))) = β ∨ .
Corollary 4.5. Let β ∈ Be+ , and suppose that gcd(ν+ (⊕t∈J Ct )∨ , β) = 1. Then
β = ν+ (K1 (β(⊕t∈J Ct )).

30

ANTHONY M. LICATA AND HOEL QUEFFELEC

The dual positive monoid. The following proposition is the main result describing
the top baric slice of a general dual positive braid acting on the baric heart K0 .
Proposition 4.6. Let β = βk · · · β1 with βi ∈ Be+ be right-greedy, and let
Xk = Kk (β ⊕t∈T Ct )
Then
ν+ (Xk ) = βk .
Proof. When k = 1, the claim follows from Lemma 4.3.
Let k > 1. By induction, the claim is true for βk−1 · · · β1 , hence Xk−1 , the degree
k − 1 baric slice βk−1 · · · β1 (⊕t∈T Ct ), satisfies
ν+ (Xk−1 ) = βk−1 .
Since the decomposition is greedy, βk βk−1 ∈
/ Be+ . We want to claim that this
implies that βk Xk−1 ∈
/ K0 .
If βk Xk−1 ∈ K0 , then Lemma 4.3 implies that
−1
βk = gcd(γβk−1
, βk ).
−1
In particular, this gives that that βk |γβk−1
and thus that βk βk−1 |γ, which contradicts the greediness of the decomposition βk βk−1 . . . β1 .

Since βk Xk−1 ∈
/ K0 , we have that
Xk ∼
= K1 (βk Xk−1 ) 6= 0.
Now, clearly, βk−1 (Xk ) ∈ K−1 , whence βk is divisible by α = ν+ (Xk ), and we may
write βk = αδ, with δ ∈ Be+ . We want to show that δ = 1, as that will imply that
βk = ν+ (Xk ).
To see that δ = 1, we observe that
δβk−1 · · · β1 · K0 ⊂ K[0,k−1] .
But, from what we have shown above, this implies that δβk−1 · · · β1 has a greedy
decomposition with k − 1 Bessis braids. Thus this greedy decomposition must be
βk−1 · · · β1 , which shows that δ = 1.

As an immediate corollary, we obtain a theorem of Birman-Ko-Lee [BKL98], Bessis
[Bes03] and Brady-Watt [BW08].
Corollary 4.7. The map of monoids
+

B∨
c W −→ BW
is injective.
+
Proof. By Proposition 4.6, the map from B∨
c W to the group of isomorphism classes
of autoequivalences of K is injective. Since this map factors through BW , the result
follows.


BRAID GROUPS OF TYPE ADE

31

Proposition 4.6 described the relationship between the top baric slice and the rightgreedy normal form of a dual positive braid. It turns out that the bottom baric
slice is analogously related to the left-greedy normal form. We will deduce this from
Proposition 4.6 together with Lemma 4.8 below, which explains how to pass between
left-greedy decompositions of dual positive braids and right-greedy decompositions
of dual negative braids. We leave the proof of Lemma 4.8 as an exercise for the
reader.
Lemma 4.8. If β = β1 · · · βk with βi ∈ Be+ is left-greedy, then
γ −k β = (γ −k β1 γ k−1 )(γ −k+1 β2 γ k−2 ) · · · (γ −1 βk )
is a right-greedy expression of γ −k β in the generators Be− .
Corollary 4.9. If β = β1 · · · βk with βi ∈ Be+ is left-greedy, and let
X0 = K0 (β(⊕t∈T Ct )).
Then ν+ (X0 ) = β1∨ .
Proof. Consider the right-greedy expression for γ −k β coming from Lemma 4.8. Let
X−k = K−k (γ −k β(⊕t∈T Ct )). Then by Proposition 4.6 and Lemma 3.16, we have
that
ν+ (X−k ) = γ −k+1 β1−1 γ k .
Furthermore, if X0 = K0 (β(⊕t∈T Ct )), then clearly
∼ γ −k X0 .
X−k =
Thus,
ν+ (X0 ) = γ k ν+ (X−k )γ −k = γβ1−1 = β1∨ .

General braids. We now generalize the results from the previous section from dual
positive braids to arbitrary braids. Recall from Section 1.4 that every braid β can
be written as
β = β− · β+,
+
+
+
−
∨−
+
with β + ∈ B∨
c , β ∈ Bc . Furthermore, we may write β = β1 · · · βk left-greedy,
−
−
+
−
−
and β = βl · · · β1 right-greedy, and gcd(β1 , β1 ) = 1.
Proposition 4.6, together with Corollary 4.5, now imply the following result, which
gives a linear-algebraic algorithm for computing the greedy normal form of any
braid.
Proposition 4.10. Let β = β − · β + be a reduced minimal decomposition, with
β − = βl− · · · β1− right-greedy. Let X−l = K−l (β ⊕t∈T Ct ). Then we have ν− (X−l ) =
βl− .
We can now prove Theorem 4.1.
Proof of Theorem 4.1. Let us first prove that d is a metric. The only non-trivial
thing to check is that d(β1 , β2 ) = 0 implies β1 = β2 , or in other words that
β2−1 β1 · K0 ⊂ K0 =⇒ β2−1 β1 = 1.
Writing the braid β2−1 β1 in a reduced minimal decomposition, Proposition 4.10
shows that l, the length of the dual negative braid in such a decomposition, is equal

32

ANTHONY M. LICATA AND HOEL QUEFFELEC

to 0. Thus β2−1 β1 is a dual positive braid, and Proposition 4.6 then implies that the
braid must be trivial.
The remaining items follow from Proposition 4.10. For example, to prove 3, write
β = β − β + in a reduced minimal decomposition. Then, by Proposition 4.10, Lγ− (β)
equals the minimal number of generators from Be− requested to write β − .
Item 2 follows similarly. For item 5, suppose that β(⊕t∈T Ct ) ∈ K[a,∞) with a =
−Lγ− (β) > 0. Then both γ −a β is a dual positive braid. Moreover, since
K0 (γ −a β ⊕t∈T Ct ) 6= 0,
it follows that γ −a β does not have γ as one of its letters in a minimal length decomposition. Thus β has exactly a letters equal to γ in a minimal length decomposition.

Item 4 is proven similarly.
4.2. The standard metric and the canonical t-structure. As in the oriented
graded case, for β a braid in the braid group BW , define:
(4.4)

L′+ (β) := max{k ∈ Z : ∀X ∈ K0 , Kk (βX) 6= 0}

(4.5)

L′− (β) := max{k ∈ Z : ∀X ∈ K0 , K−k (βX) 6= 0}

(4.6)

L′ (β) := max(L′+ (β), 0) + max(L′− (β), 0)

We now prove an analog of Theorem 4.1 for the word-length metric in the Weyl
generators L and the canonical t-structure. The content of the theorem below is
very similar to the main result of Brav-Thomas in [BT11].
Theorem 4.11. Let β ∈ BW . Then we have:
(1) d′ (β1 , β2 ) = L′ (β2−1 β1 ) is a metric on BW ;
(2) L′+ (β) equals the number of letters from W+ used in any minimal length
expression for β in the alphabet W+ ∪ W− ;
(3) L′− (β) equals the number of letters from W− used in any minimal length
expression for β in the alphabet W+ ∪ W− ;
(4) −L′+ (β) equals the number of times ∆−1 appears in any minimal length
expression for β in the alphabet W+ ∪ W− ;
(5) −L′− (β) equals the number of times ∆ appears in any minimal length expression for β in the alphabet W+ ∪ W− .
In any of the above statement, if the considered number is negative, it stands for
zero letters appearing in the expression. Combined together, these statements tell
that L′ is the word-length metric in the Weyl generators W+ ∪ W− .
As before, this theorem will be a consequence of some technical results.
Understanding the action. For i ∈ I, set Pi = ⊕k∈Z Pi [k]hki. For X ∈ K and r ∈ Z,
we define the following subsets of the vertex set I:
(4.7)

Er+ (X) = {i : Hom(Pi h−ri, X) 6= 0},

(4.8)

Er− (X) = {i : Hom(X, Pi h−ri) 6= 0}.

BRAID GROUPS OF TYPE ADE

33

For example, let Γ be of type A2 , so that I = {1, 2}. Let X be the complex:
P2 h−3i
σ22 σ1 P2 =

P1

P2 h−1i

Then we have E0− = {1}, E0+ = ∅ and E1− = ∅, E1+ = {2}.
We first prove some facts about the Er± sets for positive Weyl braids β ∈ W+ . This
essentially recovers part of [BT11, Proposition 3.1].
Proposition 4.12. Let β ∈ W+ , with π(β) = w ∈ W . Then
E1+ (β ⊕r∈I Pr ) = DL (w).
Proof. The proof goes by induction on the length l(w). When l(w) = 1, the result
is immediate, as E1+ (σi ⊕r∈I Pr ) = {i} = DL (si ).
Suppose now that the claim is proven for elements in W of length k, and let β ∈ W+
with π(β) = w and l(w) = k + 1.
We first prove that E1+ (β(⊕r∈I Pr )) ⊂ DL (w). Suppose that
Hom(Pi h−1i, β(⊕Pr )) 6= {0}.
That si ∈ DL (w) follows easily by induction hypothesis if β = σj µ with σi σj = σj σi .
For in that case:
Hom(Pi h−1i, σj µ(⊕Pr )) = Hom(Pi h−1i, µ(⊕Pr )) 6= {0},
+
and by induction hypothesis, we have that si ∈ DL
(π(µ)); since si sj = sj si , we
+
conclude that si ∈ DL
(w).

So suppose that β = σj µ with l(π(β)) = l(π(µ)) + 1, and σi σj σi = σj σi σj . In that
case, we have:
(4.9)

0 6= Hom(Pi h−1i, σj µ(⊕Pr )) = Hom((Pj → Pi h−1i), µ(⊕Pr )).

Denote µ(⊕Pr ) = ⊕Xr the decomposition of the complex µ(⊕Pr ) in terms of the homological grading, and consider a non-zero representative (f, g) of Hom((Pj h−ki →
Pi h−k − 1i)[−k], µ(⊕Pr )) as illustrated below:
Pj h−ki

p

g

f
···

Xk−2

∂

Xk−1

Pi h−k − 1i

∂

Xk

∂

Xk+1

···

Then, g can be restricted to a chain map in Hom(Pi h−1i, µ(⊕Pr )). If it is not homotopic to zero, then this proves that Hom(Pi h−1i, µ(⊕Pr )) 6= {0} and by induction
that si is a descent of π(µ). Thus β = σj σi β ′ with β ′ ∈ W+ and l(w) = l(π(β ′ ) + 2
and
Hom(Pi h−1i, σj σi β ′ (⊕Pr )) = Hom(Pj h−1i, β ′ (⊕Pr )) 6= 0.
This shows that sj is a left descent of π(β ′ ) and, using the braid relation, that si
indeed is a left descent of π(β).

34

ANTHONY M. LICATA AND HOEL QUEFFELEC

Now, if g were homotopic to zero with homotopy map h : Pi [−k]h−k − 1i →
Xk−1 [−(k − 1)], then f + hp ∈ Hom(Pi h−1i, µ(⊕Pr ) would be a chain map nonhomotopic to zero, for if it were, (f, g) would also have been homotopic to zero. But
since sj is not a descent of π(µ), by induction hypothesis we know that
Hom(Pj h−1i, µ(⊕Pr )) = 0.
We now prove the inclusion DL (w) ⊂ E1+ (β ⊕r∈I Pr ). Suppose β = σi β ′ , π(β) = w,
π(β ′ ) = w′ , and so that w = si w′ , k + 1 = l(w) > l(w′ ). We want to show that
i ∈ E1+ (β). Suppose that β ′ = σj µ, with l(w′ ) > l(π(µ). If σi and σj commute,
the claim follows by induction hypothesis, as then β = σi σj µ = σj σi µ, with i ∈
DL (si π(µ)) = E1+ (σi µ). Now, since σj Pi ∼
= Pi , it follows that i ∈ E1+ (β).
Now suppose β ′ = σj µ with si sj si = sj si sj in W . Then, since sj is a descent of
π(β ′ ), we have by induction hypothesis that:
Hom(Pj h−1i, β ′ (⊕Pr )) 6= {0}.
This implies that
Hom((Pj h−1i → Pi h−2i), σi β ′ (⊕Pr )) 6= {0}.
Let (f, g) be chain map between minimal complexes representing a non-zero element
of Hom((Pj h−1i → Pi h−2i), σi β ′ (⊕Pr )), as illustrated below.
Pj h−ki

p

Pi h−k − 1i
g

f
σi β ′ (⊕Pr ) :

Xk−2

∂

Xk−1

∂

Xk

∂

Xk+1

Then g is a well-defined chain map. If g is not homotopic to 0, then we are done,
as then Hom(Pi h−1i, β(⊕Pr )) 6= 0. On the other hand, if g is homotopic to 0, then
following the same argument as in the first half of the above proof, it follows that
Hom(Pj h−1i, β(⊕Pr )) 6= 0, in which case sj is a descent of w. In that case, since
si and sj are both descents of w, it follows that
β = σi σj σi α = σj σi σj α
+

for α ∈ W with l(π(α)) + 3 = l(w).
By induction hypothesis, we have:
Hom(Pj h−1i, σj α(⊕Pr )) 6= {0},
from which we deduce:
Hom(σj σi Pj h−1i, β(⊕Pr )) = Hom(Pi h−1i, β(⊕Pr )) 6= {0}
which concludes the proof.

The following lemma is the analog for the classical Garside monoid of the dual
Garside monoid’s Lemma 3.11.
Lemma 4.13. Let β ∈ W+ . Then β · (⊕Pr ) ∈ K[0,1] .

BRAID GROUPS OF TYPE ADE

35

Proof. The proof goes by induction. When l(π(β)) = 1, the claim follows by explicit
computation.
Let us now take β = σk β ′ , with l(π(β)) = l = l(π(β ′ )) + 1. Note that this implies
that sk ∈
/ DL (π(β ′ )). By induction hypothesis, β ′ (⊕Pr ) ∈ K[0,1] . Now, if
K2 (σk β ′ (⊕Pr )) 6= 0,
then we must have
Hom(Pk h−2i, β ⊕ Pr ) ∼
= Hom(Pk h−1i, β ′ ⊕ Pr ) 6= 0,
whence we would have sk ∈ DL (π(β ′ )). From this it follows that β(⊕Pr ) ∈ K[0,1] ,
as desired.

The above, together with (the positive analog of) Lemma 3.2 now implies the following.
Lemma 4.14. Let β ∈ W+ . Then βPi ∈ K1 ⇐⇒ si ∈ DR (π(β)).
Lemma 4.15. ∆Pi ∼
= Pj h−1i ∈ K1 , with ∆σi ∆−1 = σj .
Proof. Lemma 4.13 ensures that ∆Pi ∈ K[0,1] , and Lemma 4.14 shows that since
σi is a descent of ∆, we have ∆Pi ∈ K1 .
Now, after passing to the Grothendieck group and setting q = −1, ∆ acts as the
long element of W . In particular, the action of ∆ on the q = −1 Grothendieck group
permutes the simple roots according to an automorphism of the Dynkin diagram.
Since the complex ∆Pi ∈ K1 , the minimal complex is completely determined by
its image in the Grothendieck group. Since the image of ∆Pi in the Grothendieck
group is a simple root αj , it follows that ∆Pi is isomorphic to (a shift of) Pj . 
The following Lemma also essentially appears in [BT11].
Lemma 4.16. Let β = βk · · · β1 ∈ B+
W be right-greedy. Then
Ek+ (β · (⊕σi ∈D+ (β1 ) Pi )) = DL (π(βk )).
R

Moreover, the conclusion remains the same if we replace be the same if we replace
⊕i∈D+ (π(β1 )) Pi by ⊕r∈I Pr .
R

Proof. The equality follows by induction once the base case k = 1 case is proven,
and that case is a direct consequence of Lemma 4.14.

As a corollary, we obtain an analogous statement relating the greedy decomposition
of β ∈ B+
W to the action of β on (⊕i Pi ).
Corollary 4.17. Let β = β1 · · · βk ∈ B+
W be left-greedy. Then
E0− (β · ⊕Pi ) = I − DL (π(β1 )).

36

ANTHONY M. LICATA AND HOEL QUEFFELEC

Proof. First, notice that ∆−k β ∈ B−
W . Set
αi = ∆−k+i (∆−1 βi )∆k−i ,
and note that
∆−k β = α1 α2 . . . αk
is right-greedy. From the negative braid analog of Lemma 4.16, we see that
−
E−k
(∆−k β(⊕Pi )) = DL (α1 ).

Now, it is also straightforward to see that
−
E−k
(∆−k β(⊕Pi )) = w0−k (E0− (β ⊕ Pi ))w0k ,

where w0 is the long element of W . Thus
E0− (β ⊕ Pi ) = w0k DL (α1 )w0−k .
Since
DL (π(α1 )) = w0k−1 DL (w0−1 π(β1 ))w0−k+1 ,
it follows that
E0− (β ⊕ Pi ) = w0 DL (π(β1 ))w0−1 = I − DL (π(β1 )),
as desired.



General braids. We have a situation very similar to Section 4.1. Indeed, recall from
Section 1.4 that every braid β can be written as
β = β− · β+,
−
+
+
−
+
with β + ∈ B+
W , β ∈ BW . Furthermore, we may write β = β1 · · · βk left-greedy,
−
−
+
−
−
and β = βl · · · β1 right-greedy, and gcd(β1 , β1 ) = 1.

We can now deduce the following statement for arbitrary β ∈ BW . We also reemphasize the point that in order to read the descents of a braid β, it suffices to
compute the minimal complexes of β on a finite number of objects (namely, the
indecomposable projective modules Pr ; the computation of minimal complexes, in
turn, requires a finite number of linear algebraic computations.
Proposition 4.18. Let β = β − β + be a reduced minimal (−, +) decomposition,
with
β − = βl− . . . β1−
right-greedy. Then El− (β ⊕ Pr ) = DL (π(βl− )).
Proof. Corollary 4.17 shows that DR (β1− ) ⊂ E0− (β + ⊕ Pr ). Then the conclusion
follows from Lemma 4.16.

This completes the proof of Theorem 4.11.

BRAID GROUPS OF TYPE ADE

37

5. Standard and dual word-length metrics and a conjecture of
Digne-Gobet
It appears as a very natural question to ask what kind of mutual control both
gradings can have, giving, for example, more information about the length in the
usual or the dual generators. We give here a specific result that solves a conjecture
of Digne and Gobet [DG15, Conjecture 8.7], that can be restated as follows.
Theorem 5.1. Let β ∈ Be+ , then β ∈ [∆−1 , ∆] (that is, β = β + β − with β + ,
(β − )−1 ∈ W+ ).
Proof. To prove the above theorem, rather than studying the action of β on linear complexes of AΓ -modules, we will focus on the gradings on the category of
complexes of bimodules. We wish to show that the complex of bimodules associated to β is in the heart of the canonical t-structure on the homotopy category of
(AΓ , AΓ ) bimodules; that is, that in the minimal complex of β, all terms of the form
Pi Qj {m}hni lie in homological degree −n.
The proof has two main steps. The first point is to note that the assumption β ∈ Be+
implies that in the minimal complex of bimodules, all terms of the form Pi Qj {m}hni
have m = −1. (This is a direct consequence of Lemmas 3.18 and 3.19.)
Having established this, it now follows that the only maps which contribute to
the boundary in the minimal complex of β are either maps into or out of the
identity bimodule AΓ , or are bimodule maps of orientation degree {0}. The possible
orientation degree {0} bimodule maps are then the following:
• Pi Qj {−1}hni → Pi Qj {−1}hni;
• Pi Qj {−1}hni → Pi Qj ′ {−1}hn−1i, with the edge between j and j ′ oriented
from j ′ to j;
• Pi Qj {−1}hni → Pi′ Qj {−1}hn − 1i, with the edge between i and i′ oriented
from i to i′ ;
• Pi Qj {−1}hni → Pi′ Qj ′ {−1}hn − 2i, with oriented edges i to i′ and j ′ to j.
The complex of bimodules for β is in the heart of the canonical t-structure if and
only if the first and last kinds of maps do not occur in the minimal complex. Since
the space of orientation-degree {0} maps is such that
dim Hom(Pi Qj , Pi Qj ) = 1,
all maps of the first kind are multiplies of the identity map 1 : Pi Qj {−1}hni →
Pi Qj {−1}hni, and hence do not occur in the minimal complex. Thus we must show
that maps of the last kind
Pi Qj {−1}hni → Pi′ Q′j {−1}hn − 2i
do not appear in the boundary map of the minimal complex of β.
In order to show that these maps do not appear in the minimal complex, we need a
little bit of setup. Starting from the Dynkin diagram Γ, we construct a larger graph
b by adding a single new vertex x, and connecting this new vertex to every other
Γ
vertex of Γ. The indecomposable projective (right, respectively, left) AΓb modules
are Qi , respectively Pi for i ∈ Γ and Qx , respectively Px . Note that, by construction,
Qi Px ∼
= C for all i ∈ Γ.

38

ANTHONY M. LICATA AND HOEL QUEFFELEC

b induces an inclusion of zigzag algebras
The inclusion of graphs Γ ֒→ Γ
AΓ ֒→ AΓb .

The induction functor
Ind : AΓ − mod −→ AΓb − mod,
sends the indecomposable projective AΓ module Pi = AΓ ei to the indecomposable
projective AΓb module AΓb ei = Pi . (Note that we have abused notation slightly here
by using the same notation for modules over different algebras.) Under the induction functor, the homotopy category of graded projective AΓ -modules embeds as
a full triangulated subcategory of the homotopy category of graded projective AΓb
modules; similarly, the homotopy category of graded projective (AΓ , AΓ )-bimodules
embeds as a full triangulated subcategory of the homotopy category of graded projective (AΓb , AΓb )-bimodules. Moreover, a complex of graded, projective AΓ modules
(or bimodules) is a minimal complex if and only if it is minimal when induced and
regarded as a complex of graded, projective AΓb modules.
Now, in order to show that maps of (AΓ , AΓ ) bimodules of the form
Pi Qj {−1}hni → Pi′ Q′j {−1}hn − 2i
do not appear in the minimal complex of β, we will consider the complex for β
bΓ , A
bΓ ) bimodules, and show that there are no such
in the homotopy category of (A
maps there. The reason for this consideration is that we can study the complex of
bΓ , A
bΓ ) by letting it act on the new indecomposable project module
bimodules (A
Px .
Now we consider the complex of projective AΓb modules βPx . The minimal complex of βPx has a unique summand of the chain group isomorphic to Px {0}h0i,
coming from the unique summand AΓb in the minimal complex of bimodules for β
tensored with Px . We consider what happens to each of the kinds of bimodule maps
considered above when we tensor with the left AΓb -module Px .
• Consider first bimodule maps of the form Pi Qj {−1}hni → Pi Qj ′ {−1}hn −
1i, with the edge between j and j ′ oriented from j ′ to j. When we act on
Px , we obtain the 0 map
0

Pi {−1}hni −
→ Pi {−1}hn − 1i.
• Consider bimodule maps of the form Pi Qj {−1}hni → Pi′ Qj {−1}hn − 1i,
with the edge between i and i′ oriented from i to i′ . When we act on Px ,
we obtain
f

Pi {−1}hni −
→ Pi′ {−1}hn − 1i.
where
f ∈ Hom(Pi , Pi′ h−1i) ∼
= C.
• Finally, consider maps of the form Pi Qj {−1}hni → Pi′ Q′j {−1}hn − 2i, with
oriented edges i to i′ and j ′ to j. When we act on Px , we obtain
0

Pi {−1}hni −
→ Pi′ {−1}hn − 2i.
Note that this map must be zero, because
Hom(Pi , Pi′ h−2i) ∼
=0
when i 6= i′ .

BRAID GROUPS OF TYPE ADE

39

There are two important things to note regarding the above maps. The first is
that none of them are isomorphisms, and thus the complex of modules obtained
by taking the minimal complex for β and acting on Px term-by-term is already
minimal. The second thing to note is that, since no non-zero maps of the form
Pi → Pi {−1}h−2i are obtained when we act on Px , the complex βPx is isomorphic
to a direct sum of its t-slices:
βPx ∼
= ⊕l Kl (βPx )
Since Px itself is an indecomposable complex, and β is an equivalence, it follows
that βPx is an indecomposable complex, and so all but one of the summands above
must be 0; thus βPx ∼
= Kl (βPx ) for some l ∈ Z. Since the unique term of the form
Px appears in the minimal complex of βPx , we must have l = 0. Now from this it
follows that no map of the form Pi Qj {−1}hni → Pi′ Q′j {−1}hn − 2i can appear in
the minimal complex for β.
This proves that the minimal complex of bimodules associated to β is in the heart
of the canonical t-structure on the homotopy category of bigraded bimodules. Using
Lemma 4.11, it follows that β = β + β − with β + , (β − )−1 ∈ W+ .

References
[AT10]

P. N. Achar and D. Treumann, Baric structures on triangulated categories and
coherent sheaves, International Mathematics Research Notices (2010), 3688–3743,
arXiv:0808.3209.
[BB05]
A. Björner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005.
[BDM02] D. Bessis, F. Digne, and J. Michel, Springer theory in braid groups and the
Birman–Ko–Lee monoid, Pacific journal of mathematics 205 (2002), no. 2, 287–309,
arXiv:math/0010254.
[Bes03]
D. Bessis, The dual braid monoid, Annales scientifiques de l’Ecole normale supérieure,
vol. 36, 2003, arXiv:math/0101158, pp. 647–683.
[Bez06]
R. Bezrukavnikov, Noncommutative counterparts of the Springer resolution, International Congress of Mathematicians (Madrid), vol. II, Eur. Math. Soc., 2006,
arXiv:math/0604445, pp. 1119–1144.
[BG]
B. Baumeister and T. Gobet, Simple dual braids, noncrossing partitions and Mikado
braids of type Dn , in preparation.
[BGP73] I. N. Bernšteı̆n, I. M. Gel′fand, and V. A. Ponomarev, Coxeter functors, and Gabriel’s
theorem, Uspehi Mat. Nauk 28 (1973), no. 2(170), 19–33.
[BKL98]
J. Birman, K. H. Ko, and S. J. Lee, A new approach to the word and conjugacy
problems in the braid groups, Advances in Mathematics 139 (1998), no. 2, 322–353,
arXiv:math/9712211.
[BS72]
E. Brieskorn and K. Saito, Artin-gruppen und Coxeter-gruppen, Inventiones mathematicae 17 (1972), no. 4, 245–271.
[BT11]
C. Brav and H. Thomas, Braid groups and Kleinian singularities, Mathematische
Annalen 351 (2011), no. 4, 1005–1017, arXiv:0910.2521.
[BW08]
T. Brady and C. Watt, Non-crossing partition lattices in finite real reflection groups,
Transactions of the American Mathematical Society 360 (2008), no. 4, 1983–2005,
arXiv:math/0501502.
[DDG+ 15] P. Dehornoy, F. Digne, E. Godelle, D. Kramer, and J. Michel, Foundations of Garside
theory, Zürich: European Mathematical Society (EMS), 2015 (English).
[Del72]
P. Deligne, Les immeubles des groupes de tresses généralisés, Inventiones mathematicae 17 (1972), no. 4, 273–302.
[DG15]
F. Digne and T. Gobet, Dual braid monoids, Mikado braids and positivity in Hecke
algebras, 2015, arXiv:1508.06817.
[DP99]
P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations
of Artin groups, Proceedings of the London Mathematical Society 79 (1999), no. 3,
569–604.
[Gab72]
P. Gabriel, Unzerlegbare darstellungen i, Manuscripta mathematica 6 (1972), no. 1,
71–103.

40

ANTHONY M. LICATA AND HOEL QUEFFELEC

[GM13]

S. I. Gelfand and Y. I. Manin, Methods of homological algebra, Springer Science &
Business Media, 2013.
[GTW15] A. Gadbled, A.-L. Thiel, and E. Wagner, Categorical action of the extended braid group
of affine type A, Communications in Contemporary Mathematics (2015), 1650024,
arXiv:1504.07596.
[HK01]
R. S. Huerfano and M. Khovanov, A category for the adjoint representation, Journal
of Algebra 246 (2001), no. 2, 514–542.
[IUU10]
A. Ishii, K. Ueda, and H. Uehara, Stability conditions on an-singularities, Journal of
Differential Geometry 84 (2010), no. 1, 87–126, arXiv:math/0609551.
[Jen15]
L. T. Jensen, The 2-braid group and Garside normal form, 2015, arXiv:1505.05353.
[KS02]
M. Khovanov and P. Seidel, Quivers, Floer cohomology, and braid group actions,
J. Am. Math. Soc. 15 (2002), no. 1, 203–271.
[Lic16]
A. M. Licata, On the 2-linearity of the free group, 2016, arXiv:1606.06444.
[MOS09] V. Mazorchuk, S. Ovsienko, and C. Stroppel, Quadratic duals, koszul dual functors,
and applications, Transactions of the American Mathematical Society 361 (2009),
no. 3, 1129–1172, arXiv:math/0603475.
[Ric08]
S. Riche, Geometric braid group action on derived categories of coherent sheaves, Represent. Theory 12 (2008), 131–169, With a joint appendix with Roman Bezrukavnikov.
[RZ03]
R. Rouquier and A. Zimmermann, Picard groups for derived module categories, Proceedings of the London Mathematical Society 87 (2003), no. 01, 197–225.
Mathematical Sciences Institute, Australian National University, John Dedman Building, 27 Union Lane, Canberra ACT 2601
E-mail address: amlicata@gmail.com
CNRS and Institut Montpelliérain Alexander Grothendieck, Université de Montpellier, Place Eugène Bataillon, 34095 Montpellier, France
E-mail address: hoel.queffelec@umontpellier.fr

