Complex reflection groups are somehow real

Lyashko-Looijenga morphisms
Cyclic labels, and the trivialization theorem
The Garside nerve
Complex reﬂection groups are somehow real
David Bessis
Lausanne, 23/9/2016
David Bessis Complex reﬂection groups are somehow real

The Garside nerve
Definition
Let W ⊆ GL(V ) be a finite complex reflection group.
The discriminant of W is the algebraic hypersurface
H ⊆ W V
defined as the image of r∈R ker(r − 1) under the quotient map
V → W V .

The Garside nerve
Definition
Let W ⊆ GL(V ) be a finite complex reflection group.
The discriminant of W is the algebraic hypersurface
H ⊆ W V
defined as the image of r∈R ker(r − 1) under the quotient map
V → W V .
Goal: understand the geometry and topology of
V reg
:= V −
r∈R
ker(r − 1)
and
W V reg
= W V − H.

The Garside nerve
r∈R ker(r − 1)

// V

H // W V

The Garside nerve
r∈R ker(r − 1)

// V

H // W V
Theorem (Shephard-Todd)
As an algebraic variety, W V is an aﬃne space of dimension n.

The Garside nerve
r∈R ker(r − 1)

// V

H // W V
Theorem (Shephard-Todd)
As an algebraic variety, W V is an aﬃne space of dimension n.
Theorem (Steinberg)
The restriction of V → W V to V reg := V − r∈R ker(r − 1) is
an unramiﬁed covering.
In other words, H is the branch locus of the quotient map
V → W V .

The Garside nerve
Deﬁnition
A system of basic invariants for a complex reﬂection group
W ⊆ GL(V ) is a tuple (f1, . . . , fn) of algebraically independent
generators of C[V ]W such each fi is homogeneous of degree di and
d1 ≤ · · · ≤ dn.

The Garside nerve
Definition
A system of basic invariants for a complex reflection group
W ⊆ GL(V ) is a tuple (f1, . . . , fn) of algebraically independent
generators of C[V ]W such each fi is homogeneous of degree di and
d1 ≤ · · · ≤ dn.
All complex reflection groups admit systems of basic invariants.
Choosing one amounts to choosing an explicit isomorphism
C[V ]W ∼
−→ C[X1, . . . , Xn]
and an explicit isomorphism
W V = Spec C[V ]W ∼
−→ Spec C[X1, . . . , Xn] = Cn
.

The Garside nerve
Example. The symmetric group Sn is a reﬂection group, via its
permutation representation in GLn(C). The quotient space
En := SnCn
is the space of (non-necessarily centered) conﬁgurations of n
(non-necessarily distinct) points in C.

The Garside nerve
Example. The symmetric group Sn is a reflection group, via its
permutation representation in GLn(C). The quotient space
En := SnCn
is the space of (non-necessarily centered) configurations of n
(non-necessarily distinct) points in C.
The elementary symmetric functions σ1, . . . , σn, defined by
n
i=1
(T − xi ) = Tn
− σ1Tn−1
+ · · · + (−1)n
σn,
form a system of basic invariants. By identifying En with the space
of monic polynomials of degree n in C[T], one recovers the
“discriminant” notion from polynomial theory.

The Garside nerve
Theorem
Let W ⊆ GL(V ) be an irreducible complex reﬂection group. The
following assertions are equivalent:
(1) W is well-generated
(2) there exists a system of basic invariants such that the
equation of the discriminant H ⊆ W V is of the form:
Xn
n + α2(X1, . . . , Xn−1)Xn−2
n + · · · + αn(X1, . . . , Xn−1) = 0

The Garside nerve
Example: type A2 reﬂection group (S3). The discriminant
equation can be written:
X2
2 − X3
1 = 0

The Garside nerve

The Garside nerve
Let us choose a system of basic invariants f for an irreducible
well-generated reflection group W , such that
∆f = Xn
n + α2(X1, . . . , Xn−1)Xn−2
n + · · · + αn(X1, . . . , Xn−1)
Definition
The (extended) Lyashko-Looijenga morphisma is the morphism
LL : W V → En
that maps (x1, . . . , xn) ∈ Spec C[X1, . . . , Xn] W V to the
polynomial in C[T] defined by
(T + xn)n
+ α2(x1, . . . , xn−1)(T + xn)n−2
+ · · · + αn(x1, . . . , xn−1)
a
In my K(π, 1) paper, another version, denoted by LL, is also used. This is
plainly dumb as working with LL is much easier for all purposes.

The Garside nerve
r∈R ker(r − 1) //

V
W

HW
// W V
LL

En

The Garside nerve
Proposition
The Lyashko-Looijenga morphism LL is a (non-Galois) algebraic
covering of degree
n!hn
|W |
,
where h is the Coxeter number of W (i.e., the degree dn.)
Its ramification locus is the discriminant of Sn.
Remark: a finite algebraic morphism between two affine spaces
closely resembles a complex reflection group quotient – if it were
to be Galois, it would be one (by Shephard-Todd theorem). I like
to think of such morphisms as virtual reflection groups. By purity
of the branch locus, we know that they ramify purely in
codimension 1.

The Garside nerve
r∈R ker(r − 1) //

V
W

HW
// W V
LL

Koo
i,j Hi,j
//
{{
Cn
Sn

En HSn
oo // En

The Garside nerve
Proposition
For any v ∈ W V , we have v ∈ H ⇔ 0 ∈ LL(v).

The Garside nerve
Proposition
In other words:
the space W V − H is a ramiﬁed cover of En(C×), the space of
conﬁgurations of n (non-necessarily distinct) points in the
punctured plane.

The Garside nerve
Proposition
In other words:
the space W V − H is a ramified cover of En(C×), the space of
configurations of n (non-necessarily distinct) points in the
punctured plane.
How can we describe the ramification of LL?

The Garside nerve
Deﬁnition
The braid group B(W ) of W is the fundamental group of
W V − H.
When W is real, then B(W ) is isomorphic to the associated Artin
group A(W ).
The unramiﬁed covering V reg W V − H yields an exact
sequence:
1 //π1(V reg) //B(W ) = π1(W V − H) //W //1.

The Garside nerve
How do you choose a basepoint?

The Garside nerve
Deﬁnition (fat basepoint trick)
Let X be a topological space. Let U be a contractible subspace of
X. Let π1(X) be the fundamental groupoid of X.
The fundamental group of X with respect to the “fat basepoint” U
is deﬁned as the transitive limit
π1(X, U) := lim−→
u,v∈U
Homπ1(X)(u, v)
for the transitive system of isomorphisms given by homotopy
classes of paths within U.

The Garside nerve
If you don’t like transitive limits, just remember this:
any path starting in u ∈ U and ending in v ∈ U represents an
element of π1(X, U)
if your intuition requires you to really see a loop, draw a path
within U connecting u and v
the product of an element represented by a path with
endpoints u, v ∈ U with an element represented by a path
with endpoints u , v ∈ U is well-deﬁned
if your intuition requires you to see this product as
concatenation, draw a path within U connecting v and u
because U is contractible, all the paths you can draw within U
are homotopic

The Garside nerve
Definition
The standard fat basepoint for an irreducible well-generated
complex reflection group W is the subspace U ⊆ V W − H
defined by:
U := {v ∈ V W | LL(v) ⊆ C −
√
−1R≥0}.
•
•
•
• •
•
•

The Garside nerve
Lemma
The standard fat basepoint U is contractible.

The Garside nerve
Deﬁnition
The braid group of W is
B(W ) := π1(W V − H, U).

The Garside nerve
We endow W V with the quotient of the scalar action of C on V .
We endow En with the quotient of the scalar action of C on Cn.
r∈R ker(r − 1) //

V
W

HW
// W V
LL

Koo
i,j Hi,j
//
{{
Cn
Sn

En HSn
oo // En
Lemma
For any λ ∈ C, we have LL(λv) = λhLL(v).

The Garside nerve
Let v ∈ U such that LL(v) has k distinct points. Let θ ∈ R0 be
minimal such that e
√
−1θv /∈ U.
We deﬁne a sequence s1, . . . , sk ∈ B as follows:
s1, the “head” of v, is the element of B(W ) associated with
the path:
[0, 1] −→ W V − H
t −→ e
√
−1(θ+ε)t
v
s2 is the head of e
√
−1(θ+ε)v
and so on until all distinct points in LL(v) have been
labelled...

The Garside nerve
•
•
•
• •
•
•
s3
s5
s1
s6 s2
s4
s7
Note: easy desingularizations can deal with the case when there
are several points with the same argument, or when v /∈ U. The
sequence (s1, . . . , sk) can be deﬁned for any v ∈ W V − H.

The Garside nerve
Deﬁnition
The cyclic label of v is the sequence
clbl(v) := (c1, . . . , ck),
where ci is the image of si via B(W ) → W .

The Garside nerve
Deﬁnition
The cyclic label of v is the sequence
clbl(v) := (c1, . . . , ck),
where ci is the image of si via B(W ) → W .
Lemma
The product c := c1 . . . ck is independent of the choice of
v ∈ W V − H.
This product is a Coxeter element of W , i.e., a regular
element of order h.
We have
lR(c1) + · · · + lR(ck) = lR(c) = n.

The Garside nerve
Deﬁnition
We denote by D•(c) the set of lR-additive factorizations of c
in W .
A pair (x, (c1, . . . , ck)) ∈ En(C∗) × D•(c) is compatible if the
multiplicities of the distinct points in x, ordered clockwise
from noon, coincides with (lR(c1), . . . , lR(ck)).
We denote by En(C∗) D•(c) the set of compatible pairs in
En × D•(c).

The Garside nerve
Theorem (facet decomposition, aka the trivialization theorem)
The map
LL × clbl : W V − H −→ En(C∗
) D•(c)
is bijective. Moreover, En(C∗) D•(c) can be equipped with a
natural topology that makes LL × clbl an homeomorphism.
Example:
Generically, LL(v) consists of n distinct points. A generic ﬁber of
LL consists of n!hn/|W | distinct points, indexed by the n!hn/|W |
ways to write c as a product of n reﬂections.

The Garside nerve
The set D•(c) comes equipped with:

The Garside nerve
a simplicial set structure, which consists of:
face operators
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si si+1, . . . , sk)
degeneracy operators:
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si , 1, si+1, . . . , sk)

The Garside nerve
face operators
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si si+1, . . . , sk)
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si , 1, si+1, . . . , sk)
a stratiﬁed Hurwitz action: for each k, the braid group Bk acts
on Dk(c) by
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si si+1s−1
i , si , . . . , sk)

The Garside nerve
face operators
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si si+1, . . . , sk)
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si , 1, si+1, . . . , sk)
a stratiﬁed Hurwitz action: for each k, the braid group Bk acts
on Dk(c) by
(s1, . . . , si , si+1, . . . , sk) → (s1, . . . , si si+1s−1
i , si , . . . , sk)
...and an extra “cycling” operator
(s1, s2, . . . , sk) → (s2, . . . , sk, sc
1 )

The Garside nerve
These three structures satisfy many compatibility axioms.

The Garside nerve
The simplicial set structure on D•(c) encodes the
ramiﬁcation theory of the Lyashko-Looijenga morphism.

The Garside nerve
The Hurwitz structure on D•(c) encodes the stratiﬁed
monodromy theory of the Lyashko-Looijenga morphism.

The Garside nerve
The Hurwitz structure on D•(c) encodes the stratiﬁed
monodromy theory of the Lyashko-Looijenga morphism.
Having a simplicial set structure and a compatible Hurwitz
structure is what is needed to fully understand a ramiﬁed
covering. I have not found in the literature adequate axioms and
theory of this generic situation.

The Garside nerve
Together, the simplicial set structure on D•(c) and the
compatible cycling operator form an helicoidal structure.
This notion is a mild generalization of the notion of cyclic set
structure, in the sense of Connes. (An helicoidal structure is cyclic
if and only if the k-th power of the cycling operator acts trivially
on the k-th component.)

The Garside nerve
Together, the simplicial set structure on D•(c) and the
compatible cycling operator form an helicoidal structure.
This notion is a mild generalization of the notion of cyclic set
structure, in the sense of Connes. (An helicoidal structure is cyclic
if and only if the k-th power of the cycling operator acts trivially
on the k-th component.)
Theorem
The geometric realization of a cyclic set comes equipped with a
natural S1-action.
This is the reason why why everything we’re doing is compatible
with taking fixed points under finite subgroups of S1 (“Springer
theory of regular elements”), yielding a “relative” theory applicable
to non-well-generated complex reflection groups such as G31.

The Garside nerve
Theorem
The space W V − H is a K(B(W ), 1) space.
2 main ingredients in the proof:

The Garside nerve
Theorem
1. Use the beautiful combinatorics of D•(c) (the “dual Garside
structure”, especially the lattice property) to construct a
simplicial K(B(W ), 1) (the “dual Garside nerve”).

The Garside nerve
Theorem
2. Use geometric constructions to compare this particular
simplicial K(B(W ), 1) and W V − H, proving that they have
the same homotopy type.

The Garside nerve
Theorem
2. Use geometric constructions to compare this particular
simplicial K(B(W ), 1) and W V − H, proving that they have
the same homotopy type.
While preparing this talk, I came to realize that the long and
twisted approach to Step 2 used in my paper is unecessarily
complicated. A very direct and much simpler argument was
basically hidden in plain sight.

The Garside nerve
The are (at least) two practical ways of proving that a topological
space X is homotopy equivalent to the geometric realization of a
simplicial space (or simplicial complex.)
Find an open covering Ω of X such that non-empty
intersections are contractible. Then X is homotopy equivalent
to the nerve of Ω.
Find a cellular decomposition of X.
My paper uses the ﬁrst approach. Unfortunately, to construct Ω,
one has to work in the universal cover, which makes everything
complicated.

The Garside nerve
Denote by En(S1) the subspace of En consisting of conﬁgurations
of points on the unit circle in C. Via the trivialization theorem
W V − H En(C∗
) D•(c),
one may view En(S1) D•(c) as a subspace of W V − H.
Lemma
The subspace En(S1) D•(c) is a deformation retract of
W V − H.
Clearly, En(S1) has real dimension n. My gut feeling is that it is
actually smooth, but I’m confused about whether this is true/false,
trivial/non-trivial.

The Garside nerve
The dual Garside nerve is
gar•(c)
is a simplicial set such that
gark(c) = {(c1, . . . , ck) ∈ W |(c1, . . . , ck, (c1 . . . ck)−1
c) ∈ Dk+1(c)}.
The tautological “Kreweras map”
(c1, . . . , ck) → (c1, . . . , ck, (c1 . . . ck)−1
c)
is a degree-shifting bijection between between gar•(c) and D•(c).
The degree shift creates subtle nuances:
A k-simplex (c1, . . . , ck) in the dual Garside nerve is
non-degenerate if none of c1, . . . , ck is 1.
Note that a non-degenerate (c1, . . . , ck) may have a
degenerate Kreweras image (if c1 . . . ck = c, which may or
may not hold.)

The Garside nerve
Theorem
The geometric realization of gar•(c) is a K(B(W ), 1).
The geometric realization of simplicial set is built by glueing
geometric simplices, one for each non-degenerate element.

The Garside nerve
Consider the map
φ : En(S1
) D•(c) → gar•(c)
such that, for all (x, (c1, . . . , ck) ∈ En(S1) D•(c):
if
√
−1 /∈ x, then φ((x, (c1, . . . , ck)) = (c1, . . . , ck),
if
√
−1 ∈ x, then φ((x, (c1, . . . , ck)) = (c1, . . . , ck−1).
Clearly, for any Σ ∈ gark(c), the pre-image φ−1(Σ) is indexed by
conﬁgurations of k distinct points in S1 −
√
−1. Thus φ−1(Σ) is
homeomorphic to the interior of a standard k-simplex.
One obtains a cellular decomposition of En(S1) D•(c) indexed
by non-degenerate simplices in gar•(c).

The Garside nerve
THANKS!

Complex reflection groups are somehow real

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Complex reflection groups are somehow real