Discriminant of a reflection group and factorisations of a ...vripoll/sherbrooke.pdf · finite Coxeter groups); Shephard-Todd’s classification (1954): an infinite series with

Discriminant of a reflection group andfactorisations of a Coxeter element

Vivien RIPOLL

Laboratoire de Combinatoire et d’Informatique de Montreal (LaCIM)Universite du Quebec a Montreal (UQAM)

Colloque Surfaces et RepresentationsSherbrooke

9 octobre 2010

Outline

1 Fuss-Catalan numbers of type W

2 Factorisations as fibers of a Lyashko-Looijenga covering

3 Maximal and submaximal factorisations of a Coxeterelement

Outline




Factorisations in a generated group

Group G, generated by A

length function À on G.

Definition (A-factorisations)

(g1, . . . ,gp) is an A-factorisation of g ∈ G if

g1 . . . gp = g;À(g1) + · · ·+ À(gp) = À(g).

Example: (G,A) = (W ,S) finite Coxeter system

Strict maximal factorisations of w0 ←→ galleries connecting achamber to its opposite. [Deligne]

Definition (Divisibility order 4A)g 4A h if and only if g is a (left) “A-factor” of h.

[1,h]4A = {divisors of h for 4A} ' {2-factorisations of h}.


Group G, generated by A length function À on G.


(g1, . . . ,gp) is an A-factorisation of g ∈ G if

g1 . . . gp = g;À(g1) + · · ·+ À(gp) = À(g).








(g1, . . . ,gp) is an A-factorisation of g ∈ G ifg1 . . . gp = g;

À(g1) + · · ·+ À(gp) = À(g).








(g1, . . . ,gp) is an A-factorisation of g ∈ G ifg1 . . . gp = g;À(g1) + · · ·+ À(gp) = À(g).





























Prototype: noncrossing partitions of an n-gon

G := Sn, with generating set T := {all transpositions}

c := n-cycle (1 2 . . . n)

{T -divisors of c} ←→ {noncrossing partitions of an n-gon}

6

7

8

9

1

2

3

4

5 6

7

8

9

1

3

4

2

5σ = (2 4 6)(3 7 8 9) τ = (1 4 5)(2 3)(6 7 9)

croisee


G := Sn, with generating set T := {all transpositions}c := n-cycle (1 2 . . . n)


6

7

8

9

1

2

3

4

5 6

7

8

9

1

3

4

2

5σ = (2 4 6)(3 7 8 9) τ = (1 4 5)(2 3)(6 7 9)

croisee




6

7

8

9

1

2

3

4

5 6

7

8

9

1

3

4

2

5σ = (2 4 6)(3 7 8 9) τ = (1 4 5)(2 3)(6 7 9)

croisee




6

7

8

9

1

2

3

4

5 6

7

8

9

1

3

4

2

5σ = (2 4 6)(3 7 8 9) τ = (1 4 5)(2 3)(6 7 9)

croisee




6

7

8

9

1

2

3

4

5 6

7

8

9

1

3

4

2

5σ = (2 4 6)(3 7 8 9) τ = (1 4 5)(2 3)(6 7 9)

crossing




6

7

8

9

1

2

3

4

5 6

7

8

9

1

3

4

2

5σ = (2 4 6)(3 7 8 9) τ = (1 4 5)(2 3)(6 7 9)

crossing noncrossing

Complex reflection groups

V : complex vector space (finite dimension).

DefinitionA (finite) complex reflection group is a finite subgroup of GL(V )generated by complex reflections.A complex reflection is an element s ∈ GL(V ) of finite order, s.t.Ker(s − IdV ) is a hyperplane:

s ↔B

matrix Diag(ζ,1, . . . ,1) , with ζ root of unity.

includes finite (complexified) real reflection groups (akafinite Coxeter groups);Shephard-Todd’s classification (1954): an infinite serieswith 3 parameters G(de,e, r), and 34 exceptional groups.




s ↔B


includes finite (complexified) real reflection groups (akafinite Coxeter groups);

Shephard-Todd’s classification (1954): an infinite serieswith 3 parameters G(de,e, r), and 34 exceptional groups.




s ↔B


includes finite (complexified) real reflection groups (akafinite Coxeter groups);Shephard-Todd’s classification (1954): an infinite serieswith 3 parameters G(de,e, r), and 34 exceptional groups.

Noncrossing partitions of type W

Now suppose that W ⊆ GL(V ) is a complex reflection group,irreducible and well-generated (i.e. can be generated byn = dim V reflections).

generating set R := {all reflections of W}.( length `R, R-factorisations, order 4R)c : a Coxeter element in W .

Definition (Noncrossing partitions of type W )

NCPW (c) := {w ∈W | w 4R c}

NCPW (c) ' {2-factorisations of c};the structure does not depend on the choice of the Coxeterelement (conjugacy).



generating set R := {all reflections of W}.( length `R, R-factorisations, order 4R)

c : a Coxeter element in W .


NCPW (c) := {w ∈W | w 4R c}






NCPW (c) := {w ∈W | w 4R c}






NCPW (c) := {w ∈W | w 4R c}






NCPW (c) := {w ∈W | w 4R c}

NCPW (c) ' {2-factorisations of c};

the structure does not depend on the choice of the Coxeterelement (conjugacy).





NCPW (c) := {w ∈W | w 4R c}


Fuss-Catalan numbers

Kreweras’s formulaW := Sn;c : an n-cycle.

The number of T -factorisations of c in p + 1 blocks is theFuss-Catalan number

Cat(p)(n) =n∏

i=2

i + pni

.

Proof: [Athanasiadis, Reiner, Bessis...] case-by-case!Remark: Cat(p)(W ) counts also the number of maximal facesin the “p-divisible cluster complex of type W ” (generalization ofthe simplicial associahedron) [Fomin-Reading]. Related tocluster algebras of finite type if W is a Weyl group.

Fuss-Catalan numbers of type W

Chapoton’s formulaW := Sn;c : an n-cycle.


Cat(p)(n) =n∏

i=2

i + pni

.



Chapoton’s formulaW := an irreducible, well-generated c.r.gp., of rank n;c : an n-cycle.


Cat(p)(n) =n∏

i=2

i + pni

.



Chapoton’s formulaW := an irreducible, well-generated c.r.gp., of rank n;c : a Coxeter element.


Cat(p)(n) =n∏

i=2

i + pni

.




The number of R-factorisations of c in p + 1 blocks is theFuss-Catalan number

Cat(p)(n) =n∏

i=2

i + pni

.




The number of R-factorisations of c in p + 1 blocks is theFuss-Catalan number of type W

Cat(p)(n) =n∏

i=2

i + pni

.





Cat(p)(W ) =n∏

i=1

di + phdi

.





Cat(p)(W ) =n∏

i=1

di + phdi

.

Proof: [Athanasiadis, Reiner, Bessis...] case-by-case!

Remark: Cat(p)(W ) counts also the number of maximal facesin the “p-divisible cluster complex of type W ” (generalization ofthe simplicial associahedron) [Fomin-Reading]. Related tocluster algebras of finite type if W is a Weyl group.




Cat(p)(W ) =n∏

i=1

di + phdi

.

Proof: [Athanasiadis, Reiner, Bessis...] case-by-case!Remark: Cat(p)(W ) counts also the number of maximal facesin the “p-divisible cluster complex of type W ” (generalization ofthe simplicial associahedron) [Fomin-Reading].

Related tocluster algebras of finite type if W is a Weyl group.




Cat(p)(W ) =n∏

i=1

di + phdi

.


Outline




The quotient-space V/W

W ⊆ GL(V ) a complex reflection group. W acts on C[V ].

Chevalley-Shephard-Todd’s theorem: there exist invariantpolynomials f1, . . . , fn, homogeneous and algebraicallyindependent, s.t. C[V ]W = C[f1, . . . , fn].

isomorphism : V/W ∼−→ Cn

v 7→ (f1(v), . . . , fn(v)).

DefinitionThe degrees d1 ≤ · · · ≤ dn = h of f1, . . . , fn do not depend onthe choice of f1, . . . , fn. They are called the invariant degrees ofW .





v 7→ (f1(v), . . . , fn(v)).






v 7→ (f1(v), . . . , fn(v)).






v 7→ (f1(v), . . . , fn(v)).


Discriminant of W

A := {Ker(r − 1) | r ∈ R}(arrangement of hyperplanes of W )

discriminant hypersurface (in V/W ' Cn):

H :=

( ⋃H∈A

H

)/ W

discriminant ∆W : equation of the hypersurface H inC[f1, . . . , fn]. (∆W =

∏H∈A ϕ

eHH ∈ C[V ]W )

Discriminant of W

A := {Ker(r − 1) | r ∈ R}(arrangement of hyperplanes of W )discriminant hypersurface (in V/W ' Cn):

H :=

( ⋃H∈A

H

)/ W


∏H∈A ϕ

eHH ∈ C[V ]W )

Discriminant of W

A := {Ker(r − 1) | r ∈ R}(arrangement of hyperplanes of W )discriminant hypersurface (in V/W ' Cn):

H :=

( ⋃H∈A

H

)/ W


∏H∈A ϕ

eHH ∈ C[V ]W )

Example W = A3: discriminant (“swallowtail”)⋃

H∈A

H ⊆ V


H∈A

H ⊆ V

/W

Example W = A3: discriminant (“swallowtail”)

/W

⋃H∈A

H ⊆ V

hypersurface H (discriminant) ⊆W\V ' C3


H∈A

H ⊆ V

/W

H = {∆W = 0} ⊆W\V ' C3

∆W (f1, f2, f3) = Disc(T 4 + f1T 2 − f2T + f3 ; T )

Lyashko-Looijenga map and geometric factorisations

H ⊆W\V ' C3


H ⊆W\V ' C3

Y

fn

ϕ projection


H ⊆W\V ' C3

ϕ

Y

fn

y


H ⊆W\V ' C3

ϕ

Y

fn

y


H ⊆W\V ' C3

ϕ

Y

fn

y


H ⊆W\V ' C3

ϕ

Y

fn

y


H ⊆W\V ' C3

ϕ

Y

fn

y


{x1, . . . , xn} ∈ En

LLϕ

Y

fn

y y ∈ Y


{x1, . . . , xn} ∈ En

LLϕ

Y

fn

y y ∈ Y

ϕ−1(y) ' C


{x1, . . . , xn} ∈ En

LLϕ

Y

fn

y y ∈ Y

ϕ−1(y) ' C


LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

y y ∈ Y

ϕ−1(y) ' C


LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

y y ∈ Y


LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

y y ∈ Yy ′


LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

y y ∈ Yy ′

y ′′


LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

y y ∈ Yy ′

y ′′

0

Lyashko-Looijenga map of type W

V/W = Y × C.

LL : Y → En := {multisets of n points in C}y 7→ {roots, with multiplicities, of ∆W (y , fn) in fn}

∆W = f nn + a2f n−2

n + a3f n−3n + · · ·+ an−1fn + an.

Definition (LL as an algebraic (homogeneous) morphism)

LL : Cn−1 → Cn−1

(f1, . . . , fn−1) 7→ (a2, . . . ,an)

facto : Y → FACT(c) := {strict R-factorisations of c}Geometrical compatibilities:

length of the factors (↔ multiplicities in the multiset LL(y));conjugacy classes of the factors (↔ parabolic strata in H).

Details


V/W = Y × C.LL : Y → En := {multisets of n points in C}

y 7→ {roots, with multiplicities, of ∆W (y , fn) in fn}

∆W = f nn + a2f n−2

n + a3f n−3n + · · ·+ an−1fn + an.


LL : Cn−1 → Cn−1

(f1, . . . , fn−1) 7→ (a2, . . . ,an)



Details




∆W = f nn + a2f n−2

n + a3f n−3n + · · ·+ an−1fn + an.


LL : Cn−1 → Cn−1

(f1, . . . , fn−1) 7→ (a2, . . . ,an)



Details




∆W = f nn + a2f n−2

n + a3f n−3n + · · ·+ an−1fn + an.


LL : Cn−1 → Cn−1

(f1, . . . , fn−1) 7→ (a2, . . . ,an)

facto : Y → FACT(c) := {strict R-factorisations of c}

Geometrical compatibilities:length of the factors (↔ multiplicities in the multiset LL(y));conjugacy classes of the factors (↔ parabolic strata in H).

Details




∆W = f nn + a2f n−2

n + a3f n−3n + · · ·+ an−1fn + an.


LL : Cn−1 → Cn−1

(f1, . . . , fn−1) 7→ (a2, . . . ,an)



Details

Fibers of LL and strict factorisations of c

Let ω be a multiset in En.

Compatibility ⇒ ∀y ∈ LL−1(ω), the distribution of lengths offactors of facto(y) is the same (composition of n).

Theorem (Bessis’07)

The map facto induces a bijection between the fiber LL−1(ω)and the set of strict factorisations of same “composition” as ω.

Equivalently, the product map:

YLL× facto−−−−−−→ En × FACT(c)

is injective, and its image is the set of “compatible” pairs.

















Outline




Bifurcation locus (K) of LL

LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

y y ∈ Yy ′

y ′′

0

Bifurcation locus (K) of LL

LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

K

y ∈ Y

An unramified covering

Bifurcation locus:K := LL−1(En − E reg

n )= {y ∈ Y | ∆W (y , fn) has multiple roots w.r.t. fn}= {y ∈ Y | DLL(y) = 0}

whereDLL := Disc(∆W (y , fn) ; fn).

Proposition (Bessis)

LL : Y −K� E regn is a topological covering, of degree

n! hn/ |W |;| FACTn(c)| = n! hn/ |W |.

Can we compute | FACTn−1(c)| ?







n! hn/ |W |;| FACTn(c)| = n! hn/ |W |.








n! hn/ |W |;

| FACTn(c)| = n! hn/ |W |.








n! hn/ |W |;| FACTn(c)| = n! hn/ |W |.








n! hn/ |W |;| FACTn(c)| = n! hn/ |W |.


Irreducible components of K

Want to study the restriction of LL : K → En − E regn .

PropositionThere are (canonical) bijections between:

the set of irreducible components of K (or irreduciblefactors of DLL);the set of conjugacy classes of elements of NCPW of length2;the set of conjugacy classes of parabolic subgroups of Wof rank 2.

Explanations

Denote “this” set by L2. Thus: DLL =∏

Λ∈L2

DrΛΛ

(irreducible factors in C[f1, . . . , fn−1]).




the set of irreducible components of K (or irreduciblefactors of DLL);

the set of conjugacy classes of elements of NCPW of length2;the set of conjugacy classes of parabolic subgroups of Wof rank 2.

Explanations


Λ∈L2

DrΛΛ





the set of irreducible components of K (or irreduciblefactors of DLL);the set of conjugacy classes of elements of NCPW of length2;

the set of conjugacy classes of parabolic subgroups of Wof rank 2.

Explanations


Λ∈L2

DrΛΛ






Explanations


Λ∈L2

DrΛΛ






Explanations

Denote “this” set by L2.

Thus: DLL =∏

Λ∈L2

DrΛΛ






Explanations


Λ∈L2

DrΛΛ



LL

facto

{x1, . . . , xn} ∈ En

(w1, . . . ,wp) ∈ FACT(c)ϕ

Y

fn

K

y ∈ Y


H ⊆W\V ' C3

ϕ

Y

fn

K

Λ1

ϕ(Λ1)


H ⊆W\V ' C3

ϕ

Y

fn

K

Λ1

Λ2

ϕ(Λ2)

ϕ(Λ1)

Submaximal factorisations of type Λ

FACTΛn−1(c) := set of factorisations of c in n − 1 factors, with:n − 2 reflections; and1 element of length 2 and conjugacy class Λ.

Remark: FACTΛn−1(c) = facto({“generic” points in {DΛ = 0}}).

The restriction LLΛ : KΛ → En − E regn corresponds to the

extension C[a2, . . . ,an]/(D) ⊆ C[f1, . . . , fn−1]/(DΛ) .

Theorem (R.)

For any Λ in L2,

LLΛ is a finite morphism of degree (n−2)! hn−1

|W | deg DΛ;

the number of factorisations of c of type Λ is

| FACTΛn−1(c)| =

(n − 1)! hn−1

|W |deg DΛ .






Theorem (R.)

For any Λ in L2,


|W | deg DΛ;


| FACTΛn−1(c)| =

(n − 1)! hn−1

|W |deg DΛ .




The restriction LLΛ : KΛ → En − E regn

corresponds to theextension C[a2, . . . ,an]/(D) ⊆ C[f1, . . . , fn−1]/(DΛ) .

Theorem (R.)

For any Λ in L2,


|W | deg DΛ;


| FACTΛn−1(c)| =

(n − 1)! hn−1

|W |deg DΛ .






Theorem (R.)

For any Λ in L2,


|W | deg DΛ;


| FACTΛn−1(c)| =

(n − 1)! hn−1

|W |deg DΛ .






Theorem (R.)

For any Λ in L2,


|W | deg DΛ;


| FACTΛn−1(c)| =

(n − 1)! hn−1

|W |deg DΛ .






Theorem (R.)

For any Λ in L2,


|W | deg DΛ;


| FACTΛn−1(c)| =

(n − 1)! hn−1

|W |deg DΛ .

Submaximal factorisations

Problem: find a general computation of∑

Λ∈L2deg DΛ.

Recall that DLL =∏

Λ∈L2DrΛ

Λ .

Proposition (Saito; R.)

Set JLL := Jac((a2, . . . ,an)/(f1, . . . , fn−1)). Then:

JLL.

=∏

Λ∈L2

DrΛ−1Λ

Virtual reflection groups?

So,∑

deg DΛ = deg DLL − deg JLL = . . .



Λ∈L2deg DΛ.


Λ∈L2DrΛ

Λ .



JLL.

=∏

Λ∈L2

DrΛ−1Λ


So,∑




Λ∈L2deg DΛ.


Λ∈L2DrΛ

Λ .



JLL.

=∏

Λ∈L2

DrΛ−1Λ


So,∑




Λ∈L2deg DΛ.


Λ∈L2DrΛ

Λ .



JLL.

=∏

Λ∈L2

DrΛ−1Λ


So,∑



CorollaryLet W be an irreducible, well-generated complex reflectiongroup, of rank n. The number of strict factorisations of aCoxeter element c in n − 1 factors is:

| FACTn−1(c)| =(n − 1)! hn−1

|W |

((n − 1)(n − 2)

2h +

n−1∑i=1

di

).

We recover what is predicted by Chapoton’s formula;but the proof is more satisfactory and enlightening: wetravelled from the numerology of FACTn(c) (non-ramifiedpart of LL) to that of FACTn−1(c), without adding anycase-by-case analysis.



| FACTn−1(c)| =(n − 1)! hn−1

|W |

((n − 1)(n − 2)

2h +

n−1∑i=1

di

).




| FACTn−1(c)| =(n − 1)! hn−1

|W |

((n − 1)(n − 2)

2h +

n−1∑i=1

di

).

We recover what is predicted by Chapoton’s formula;

but the proof is more satisfactory and enlightening: wetravelled from the numerology of FACTn(c) (non-ramifiedpart of LL) to that of FACTn−1(c), without adding anycase-by-case analysis.



| FACTn−1(c)| =(n − 1)! hn−1

|W |

((n − 1)(n − 2)

2h +

n−1∑i=1

di

).


Conclusion, questions

We recover geometrically some combinatorial resultsknown in the real case [Krattenthaler].

Can we go further (compute the | FACTk (c)|) ? Can weinterpret Chapoton’s formula as a ramification formula forLL ?

Merci !


We recover geometrically some combinatorial resultsknown in the real case [Krattenthaler].Can we go further (compute the | FACTk (c)|) ? Can weinterpret Chapoton’s formula as a ramification formula forLL ?

Merci !



Merci !



Merci !



Merci !



Merci !



Merci !



Merci !



Merci !



Merci !

Outline

4 AppendixStratificationsComparison reflection groups / LL extensions

Stratifications

Stratification of V with the “flats” (intersection lattice):L :=

{⋂H∈AH | B ⊆ A

}.

Bijections [Steinberg]:stratification L

= L/W

↔ {parabolic subgroups of W}

↔ NCPW /conj.codim(Λ) = rank(WΛ) = `R(wΛ)

Remark: H is the union of strata of L of codim. 1.

Conjugacy classes of factors of facto(y)↔ strata containing theintersection points.

Stratifications


{⋂H∈AH | B ⊆ A

}.

Bijections [Steinberg]:stratification L

= L/W

↔ {parabolic subgroups of W}




Stratifications


{⋂H∈AH | B ⊆ A

}.

Bijections [Steinberg]:stratification L = L/W ↔ p.sg.(W )/conj.




Stratifications


{⋂H∈AH | B ⊆ A

}.

Bijections [Steinberg]:stratification L = L/W ↔ p.sg.(W )/conj. ↔ NCPW /conj.

codim(Λ) = rank(WΛ) = `R(wΛ)



Stratifications


{⋂H∈AH | B ⊆ A

}.





Stratifications


{⋂H∈AH | B ⊆ A

}.





Stratifications


{⋂H∈AH | B ⊆ A

}.





Example of W = A3: stratification of the discriminantReturn to facto⋃

H∈A

H ⊆ V

/W

H = {∆W = 0} ⊆W\V ' C3



H∈A

H ⊆ V

/W

H = {∆W = 0} ⊆W\V ' C3


A3

t us


H∈A

H ⊆ V

/W

H = {∆W = 0} ⊆W\V ' C3


A3

A1 (s)

t us


H∈A

H ⊆ V

/W

H = {∆W = 0} ⊆W\V ' C3


A3

A1 × A1 (su)

A1 (s)

t us

A2 (st)


H∈A

H ⊆ V

/W

H = {∆W = 0} ⊆W\V ' C3


A3

A1 × A1 (su)

A1 (s)

t us

A2 (st)

A3 (stu)

Irreducible components of K, details

L2 := {strata of L of codim. 2}.

Steinberg’s theorem⇒ L2 is in bijection with:{conjugacy classes of parabolic subgroups of W of rank 2}{conjugacy classes of elements of NCPW of length 2}

Proposition

The ϕ(Λ), for Λ ∈ L2, are the irreducible components of K(where ϕ is the projection V/W � Y).

Return to Irreducible components ofK




Proposition






Proposition



Reflection group vs. Lyashko-Looijenga extension

Reflection group W Extension LLV → V/W Y → Cn−1

C[f1, . . . , fn] = C[V ]W ⊆ C[V ] C[a2, . . . ,an] ⊆ C[f1, . . . , fn−1]degree |W | degree n! hn/ |W |

V reg � V reg/W Y −K� E regn

Generic fiber 'W ' RedR(c)

ramified on⋃

H∈AH � H K =⋃

Λ∈L2ϕ(Λ)� En − E reg

n

∆W =∏

H∈A αeHH DLL =

∏Λ∈L2

DrΛΛ

JW =∏αeH−1

H JLL =∏

DrΛ−1Λ

eH = |WH | rΛ = pseudo-order ofelements of NCPW of type Λ

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Discriminant of a reflection group and factorisations of a ...vripoll/sherbrooke.pdf · finite Coxeter groups); Shephard-Todd’s classification (1954): an infinite series with