On ideal subarrangements of Weyl arrangements @let@tokenterao/20140728SaoCarlosBrazil.ver.5.p… ·...

......

On ideal subarrangementsof

Weyl arrangements

Hiroaki Terao

(Hokkaido University, Sapporo, Japan)

The 13th International Workshop in Real and Complex Singularities

Sao Carlos, Brazil

2014.07.28

H. Terao (Hokkaido University) 2014.07.28 1/ 29

Credit

Takuro Abe (Kyoto University)

Mohamed Barakat (Katholische UniversitatEichstatt-Ingolstadt)

Michael Cuntz (Leibniz Universitat Hannover )

Torsten Hoge (Leibniz Universitat Hannover)

arXiv:1304.8033

(to appear in J. Euro. Math. Soc.)

Credit

arXiv:1304.8033

Credit

arXiv:1304.8033

Credit

arXiv:1304.8033

(to appear in J. Euro. Math. Soc.)H. Terao (Hokkaido University) 2014.07.28 2/ 29

Weyl arrangements

A reflection arrangement A is called a Weylarrangement if there exists a set Φ of roots such that

2(α, β)/(β, β) ∈ Z

for any α, β ∈ Φ.Then the lattice ZΦ is stable under the Weyl groupW(A).

Φ+ : the set of positive roots

∆ = {α1, α2, . . . , αℓ} : the simple system (=the set ofsimple roots)

Weyl arrangements

2(α, β)/(β, β) ∈ Z

for any α, β ∈ Φ.

Then the lattice ZΦ is stable under the Weyl groupW(A).

Weyl arrangements

2(α, β)/(β, β) ∈ Z

Weyl arrangements

2(α, β)/(β, β) ∈ Z

Weyl arrangements

2(α, β)/(β, β) ∈ Z

the root poset and ideals

.Definition..

......

Introduce a partial order ≥ into the set Φ+ of positive roots by

β1 ≥ β2⇐⇒ β1 − β2 ∈ℓ∑

Z≥0αi .

The poset is called the (positive) root poset.A subset I of Φ+ is called an ideal if, for {β1, β2} ⊂ Φ+,

β1 ≥ β2, β1 ∈ I ⇒ β2 ∈ I .

.Definition..

......

When I is an ideal of Φ+ the arrangement A(I ) := {kerα | α ∈ I } iscalled an ideal subarrangement of A.

.Definition..

......

β1 ≥ β2⇐⇒ β1 − β2 ∈ℓ∑

Z≥0αi .

β1 ≥ β2, β1 ∈ I ⇒ β2 ∈ I .

.Definition..

......

.Definition..

......

β1 ≥ β2⇐⇒ β1 − β2 ∈ℓ∑

Z≥0αi .

The poset is called the (positive) root poset.

A subset I of Φ+ is called an ideal if, for {β1, β2} ⊂ Φ+,

β1 ≥ β2, β1 ∈ I ⇒ β2 ∈ I .

.Definition..

......

.Definition..

......

β1 ≥ β2⇐⇒ β1 − β2 ∈ℓ∑

Z≥0αi .

β1 ≥ β2, β1 ∈ I ⇒ β2 ∈ I .

.Definition..

......

.Definition..

......

β1 ≥ β2⇐⇒ β1 − β2 ∈ℓ∑

Z≥0αi .

β1 ≥ β2, β1 ∈ I ⇒ β2 ∈ I .

.Definition..

......

Examples of ideals/non-ideals of the root poset ofA3

A3: •α1

•α2

•α3

Φ+ = {α1, α2, α3, α1 + α2, α2 + α3, α1 + α2 + α3}

α1 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α2 + α3 ≤ α1 + α2 + α3,

α3 ≤ α2 + α3 ≤ α1 + α2 + α3

Thus {α1, α2, α3, α1 + α2, α2 + α3} is an ideal, while{α1, α2, α3, α1 + α2 + α3} is not.

Note that the entire set Φ+ is always an ideal.

A3: •α1

•α2

•α3

Φ+ = {α1, α2, α3, α1 + α2, α2 + α3, α1 + α2 + α3}

α1 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α2 + α3 ≤ α1 + α2 + α3,

α3 ≤ α2 + α3 ≤ α1 + α2 + α3

A3: •α1

•α2

•α3

Φ+ = {α1, α2, α3, α1 + α2, α2 + α3, α1 + α2 + α3}

α1 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α2 + α3 ≤ α1 + α2 + α3,

α3 ≤ α2 + α3 ≤ α1 + α2 + α3

A3: •α1

•α2

•α3

Φ+ = {α1, α2, α3, α1 + α2, α2 + α3, α1 + α2 + α3}

α1 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α1 + α2 ≤ α1 + α2 + α3,

α2 ≤ α2 + α3 ≤ α1 + α2 + α3,

α3 ≤ α2 + α3 ≤ α1 + α2 + α3

Free Arrangements and their Exponents

A: an arrangement of hyperplanes in an ℓ-dimensional vectorspace V

αH ∈ V∗: ker(αH) = H for H ∈ AS := S(V∗): the symmetric algebra of the dual space V∗

Define a graded S-module

D(A) := {θ | θ is an R-linear derivation with

θ(αH) ∈ αHS for all H ∈ A}.

A is said to be a free arrangement if D(A) is a free S-module.

When A is free, then ∃θ1, θ2, . . . , θℓ: homogeneous basis withdegθi = di. The nonnegative integers d1,d2, . . . , dℓ are calledthe exponents of A.

αH ∈ V∗: ker(αH) = H for H ∈ A

S := S(V∗): the symmetric algebra of the dual space V∗

Every Weyl arrangement is free

.Theorem..

......

(K. Saito 1976 et al.) The Weyl arrangement A is a freearrangement. The exponents of the Weyl arrangement A coincidewith the exponents of the corresponding root system.

Example. (Weyl arrangement of type B2)Φ+ := {α1 := x1 − x2, α2 := x2, α1 + α2 = x1, α1 + 2α2 = x1 + x2}The S-module D(AG) is a free module with a basis

θ1 = x1(∂/∂x1) + x2(∂/∂x2), θ2 = x31(∂/∂x1) + x3

2(∂/∂x2),

The exponents are

d1 = degθ1 = 1, d2 = degθ2 = 3.

.Theorem..

......

θ1 = x1(∂/∂x1) + x2(∂/∂x2), θ2 = x31(∂/∂x1) + x3

2(∂/∂x2),

The exponents are

d1 = degθ1 = 1, d2 = degθ2 = 3.

.Theorem..

......

Example. (Weyl arrangement of type B2)Φ+ := {α1 := x1 − x2, α2 := x2, α1 + α2 = x1, α1 + 2α2 = x1 + x2}

The S-module D(AG) is a free module with a basis

θ1 = x1(∂/∂x1) + x2(∂/∂x2), θ2 = x31(∂/∂x1) + x3

2(∂/∂x2),

The exponents are

d1 = degθ1 = 1, d2 = degθ2 = 3.

.Theorem..

......

θ1 = x1(∂/∂x1) + x2(∂/∂x2), θ2 = x31(∂/∂x1) + x3

2(∂/∂x2),

The exponents are

d1 = degθ1 = 1, d2 = degθ2 = 3.

.Theorem..

......

θ1 = x1(∂/∂x1) + x2(∂/∂x2), θ2 = x31(∂/∂x1) + x3

2(∂/∂x2),

The exponents are

d1 = degθ1 = 1, d2 = degθ2 = 3.

Factorization Theorem

.Theorem..

......

(H. T.(1981)) Assume that A is a free arrangement in the complexspace V = Cℓ with exponents (d1, . . . ,dℓ). Define the complementof A by

M(A) := V \∪H∈A

Then the Poincare polynomial (with its coefficients equal to the Bettinumbers ) of the topological space M(A) splits as

Poin(M(A), t) =ℓ∏

(1+ dit).

Factorization Theorem

.Theorem..

......

(H. T.(1981)) Assume that A is a free arrangement in the complexspace V = Cℓ with exponents (d1, . . . ,dℓ). Define the complementof A by

M(A) := V \∪H∈A

Then the Poincare polynomial (with its coefficients equal to the Bettinumbers ) of the topological space M(A) splits as

Poin(M(A), t) =ℓ∏

(1+ dit).

Exponents

Dynkin diagrams (root systems) and exponents

Aℓ: •α1

•α2

· · · •αℓ−1

•αℓ

(1, 2, . . . , ℓ)

Bℓ: •α1

•α2

· · · •αℓ−1

αℓ(1, 3,5, . . . , 2ℓ − 1)

Cℓ: •α1

•α2

· · · •αℓ−1

αℓ(1,3,5, . . . , 2ℓ − 1)

Dℓ: •α1

•α2

· · · •αℓ−2

•αℓ−1

•αℓ

(1,3,5, . . . ,2ℓ − 3, ℓ − 1)

E6: •α1

•α3

•α4

•α2

•α5

•α6

(1,4,5, 7,8,11)

Exponents

Aℓ: •α1

•α2

· · · •αℓ−1

•αℓ

(1, 2, . . . , ℓ)

Bℓ: •α1

•α2

· · · •αℓ−1

αℓ(1, 3,5, . . . , 2ℓ − 1)

Cℓ: •α1

•α2

· · · •αℓ−1

αℓ(1,3,5, . . . , 2ℓ − 1)

Dℓ: •α1

•α2

· · · •αℓ−2

•αℓ−1

•αℓ

(1,3,5, . . . ,2ℓ − 3, ℓ − 1)

E6: •α1

•α3

•α4

•α2

•α5

•α6

(1,4,5, 7,8,11)

Exponents

Aℓ: •α1

•α2

· · · •αℓ−1

•αℓ

(1, 2, . . . , ℓ)

Bℓ: •α1

•α2

· · · •αℓ−1

αℓ(1, 3,5, . . . , 2ℓ − 1)

Cℓ: •α1

•α2

· · · •αℓ−1

αℓ(1,3,5, . . . , 2ℓ − 1)

Dℓ: •α1

•α2

· · · •αℓ−2

•αℓ−1

•αℓ

(1,3,5, . . . ,2ℓ − 3, ℓ − 1)

E6: •α1

•α3

•α4

•α2

•α5

•α6

(1,4,5, 7,8,11)

Exponents

E7: (1,5,7,9,11,13,17) •α1

•α3

•α4

•α2

•α5

•α6

•α7

E8: (1,7,11,13, 17,19,23, 29)

•α1

•α3

•α4

•α2

•α5

•α6

•α7

•α8

F4: •α1

•α2

•//α3

•α4

(1,5,7, 11)

G2: •α1

•ooα2

(from http://www.ms.u-tokyo.ac.jp/ abenori/tex/tex7.html )H. Terao (Hokkaido University) 2014.07.28 10/ 29

Main Theorem

.Theorem..

......

Φ : an irreducible root system of rank ℓ

I : an ideal of Φ+

B : the corresponding ideal subarrangement to I

Then(1) B is free, and (2) the exponents of B and the heightdistribution of the positive roots in I satisfy thedual-partition formula.

This positively settles a conjecture by Sommers-Tymoczko (2006).

Main Theorem

.Theorem..

......

I : an ideal of Φ+

Main Theorem

.Theorem..

......

I : an ideal of Φ+

Main Theorem

.Theorem..

......

I : an ideal of Φ+

Main Theorem

.Theorem..

......

I : an ideal of Φ+

(1) B is free, and (2) the exponents of B and the heightdistribution of the positive roots in I satisfy thedual-partition formula.

Main Theorem

.Theorem..

......

I : an ideal of Φ+

Then(1) B is free, and

(2) the exponents of B and the heightdistribution of the positive roots in I satisfy thedual-partition formula.

Main Theorem

.Theorem..

......

I : an ideal of Φ+

Main Theorem

.Theorem..

......

I : an ideal of Φ+

the dual-partition formula

In particular, when the ideal I is equal to the entire Φ+, our maintheorem yields:

.Corollary..

......

(The dual-partition formula by Shapiro, Steinberg, Kostant,Macdonald )The exponents of the entire Φ and the height distribution of theentire positive roots satisfy the dual-partition formula.

.Corollary..

......

.Corollary..

......

(The dual-partition formula by Shapiro, Steinberg, Kostant,Macdonald )

The exponents of the entire Φ and the height distribution of theentire positive roots satisfy the dual-partition formula.

.Corollary..

......

Height of positive roots

∆ = {α1, . . . , αℓ} : a simple system of Φ

Φ+: the set of positive roots

ht(α) :=∑ℓ

i=1 ci (height) for a positive rootα =∑ℓ

i=1 ciαi (ci ∈ Z≥0)

The height distribution in Φ+ is a sequenceof positive integers (i1, i2, . . . , im), wherei j := |{α ∈ Φ+ | ht(α) = j}| (1 ≤ j ≤ m)

ht(α) :=∑ℓ

Height of positive roots (E6)

E6: •α1

•α3

•α4

•α2

•α5

•α6

Exponents: (1,4,5,7,8,11)

List of positive roots:height 1 : α1, α2, α3, α4, α5, α6

height 2 : α1 + α3, α2 + α4, α3 + α4, α4 + α5, α5 + α6

height 3 : α1 + α3 + α4, α2 + α3 + α4, . . .....

height 11: α = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 (the highest root)

E6: •α1

•α3

•α4

•α2

•α5

•α6

Exponents: (1,4,5,7,8,11)

List of positive roots:

height 1 : α1, α2, α3, α4, α5, α6

height 3 : α1 + α3 + α4, α2 + α3 + α4, . . .....

E6: •α1

•α3

•α4

•α2

•α5

•α6

Exponents: (1,4,5,7,8,11)

height 3 : α1 + α3 + α4, α2 + α3 + α4, . . .

.height 11: α = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 (the highest root)

E6: •α1

•α3

•α4

•α2

•α5

•α6

Exponents: (1,4,5,7,8,11)

height 3 : α1 + α3 + α4, α2 + α3 + α4, . . .....

E6: •α1

•α3

•α4

•α2

•α5

•α6

Exponents: (1,4,5,7,8,11)

height 3 : α1 + α3 + α4, α2 + α3 + α4, . . .....

Height of positive roots (E6)he

sht=11 α

ht=10 •ht=9 •ht=8 • •ht=7 • • •ht=6 • • •ht=5 • • • •ht=4 • • • • •ht=3 • • • • •ht=2 α1 + α3 α2 + α4 α3 + α4 • •ht=1 α1 α2 α3 α4 α5 α6

α = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6, ht(α) = 11 (the highest root)H. Terao (Hokkaido University) 2014.07.28 15/ 29

Height Distribution ( E6 )

n1 •1 •1 •2 • •3 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

7 5 4 3 1 exponentsH. Terao (Hokkaido University) 2014.07.28 16/ 29

Exponents (E6)

n•••• •• • •• • •• • • •• • • • •• • • • •• • • • •• • • • • •

11 8 7 5 4 1 exponents

The Dual-Partition Formula ( E6)

n1 •1 •1 •2 • •3 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

11 8 7 5 4 1 exponentsH. Terao (Hokkaido University) 2014.07.28 18/ 29

History of the Dual-Partition Formula

. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g wasdiscovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers producedby this procedure agree with the exponents . . . . . . the important question of proving that this“agreement” is more than just a coincidence remained open.

(1959) A. Shapiro (empirical proof using the classification)

(1959) R. Steinberg (empirical proof using the classification)

(1959) B. Kostant (1st proof without using the classification)

(1972) I. G. Macdonald (2nd proof: generating functions)

(2014?) ABCHT (for ideal subarr.: using free arrangements)

MAT (Multiple Addition Theorem - key to our proof -)

.Theorem..

......

(ABCHT(2014?)) Let A′ be a free arrangement with exponents(d1, . . . ,dℓ) (d1 ≤ · · · ≤ dℓ) and 1 ≤ p ≤ ℓ the multiplicity of thehighest exponent d. Let H1, . . . ,Hq be hyperplanes with Hi < A′ fori = 1, . . . , q. Define A′′j := {H ∩ Hj | H ∈ A′} (j = 1, . . . , q). Assumethat the following three conditions are satisfied:(1) X := H1 ∩ · · · ∩ Hq is q-codimensional,(2) X ⊈ H (∀H ∈ A′),(3) |A′| − |A′′j | = d (1 ≤ j ≤ q).Then q ≤ p and A := A′ ∪ {H1, . . . ,Hq} is free withexponents (d1, . . . ,dℓ−q, (d + 1)q).

MAT (Multiple Addition Theorem - key to our proof -)

.Theorem..

......

Main Theorem (again)

.Theorem..

......

A : the Weyl arrangement (= the collection ofhyperplanes orthogonal to the positive roots of Φ)

...1 any ideal subarrangement B of A is free,

...2 its exponents and the height distribution of thepositive roots satisfy the dual-partition formula.

We prove the Main Theorem applying MAT inductively.

Proof of Main Theorem (just an outline)

Proof.

Let B be an ideal subarrangement of the Weyl arrangementA of a root system Φ.

For k ∈ Z>0, define B≤k := {H ∈ B | ht(αH) ≤ k}.We may easily verify B≤1 is a free arrangement with exponents(0,0, . . . ,0,1,1, . . . ,1).

We may apply MAT for A′ := B≤k and A := B≤k+1 .

To verify the three assumtions of MAT, we verify thecorresponding combinatorial and geometric properties of theroot system Φ. (A key Lemma is in the next page.)

Proof.

For k ∈ Z>0, define B≤k := {H ∈ B | ht(αH) ≤ k}.

We may easily verify B≤1 is a free arrangement with exponents(0,0, . . . ,0,1,1, . . . ,1).

Proof.

To verify the three assumtions of MAT, we verify thecorresponding combinatorial and geometric properties of theroot system Φ.

(A key Lemma is in the next page.)

Proof.

Local-global formula for heights (A key Lemma)

For X ∈ L(A), let ΦX := Φ ∩ X⊥. Then ΦX is a rootsystem of rank codimX.

The height of α in ΦX is called the local height and isdenoted by htX α.

For α ∈ Φ+, let

Aα := AHα = {K ∩ Hα | K ∈ A \ {Hα}}..Lemma (Local-global formula for heights)..

......

For α ∈ Φ+, we have

htΦ α − 1 =∑

X∈Aα(htX α − 1).

For α ∈ Φ+, let

......

htΦ α − 1 =∑

For α ∈ Φ+, let

Aα := AHα = {K ∩ Hα | K ∈ A \ {Hα}}.

.Lemma (Local-global formula for heights)..

......

htΦ α − 1 =∑

For α ∈ Φ+, let

......

htΦ α − 1 =∑

MAT (Revisited)

.Theorem..

......

MAT (Revisited)

.Theorem..

......

Inductive use of MAT (E6) : I = Φ+0

00000000000

0 0 0 0 0 0 exponents

n00000000000

n00000000006 • • • • • •

n0000000005 • • • • •6 • • • • • •

n000000005 • • • • •5 • • • • •6 • • • • • •

n00000005 • • • • •5 • • • • •5 • • • • •6 • • • • • •

n0000004 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

n000003 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

n00003 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

n0002 • •3 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

n001 •2 • •3 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

n01 •1 •2 • •3 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

The Dual-Partition Formula ( E6) (again)

n1 •1 •1 •2 • •3 • • •3 • • •4 • • • •5 • • • • •5 • • • • •5 • • • • •6 • • • • • •

Summary

...1 The celebrated dual-partition formula for a rootsystem by Shapiro-Steinberg-Kostant-Macdonald isgeneralized to the class of ideals.

...2 The theory of free arrangements (the multipleaddition theorem ( MAT)) provides a base of ourproof.

...3 Our proof needs combinatorial and geometricproperties concerning the height of positive roots.

Summary

A Natural Question

Is there any nice characterization of the set

F := {F ⊆ Φ+ | A(F) is free},

whereA(F) is the corresponding subarrangement toF?

Remark. At least we know

F ⊇ {wI | w ∈W, I is an ideal}.

A Natural Question

F := {F ⊆ Φ+ | A(F) is free},

A Natural Question

F := {F ⊆ Φ+ | A(F) is free},

A related theorem by W. Slofstra

For w ∈W, define its inverse set I(w) := {α ∈ Φ+ | −wα ∈ Φ+}.Consider the inverse subarrangement

A(w) := A(I(w)) := {ker(α) | α ∈ I(w)}

of the Weyl arrangement A = A(Φ+).

.Theorem..

......

(W. Slofstra (2014)) The element w is rationally smooth (≒ theSchubert cell X(w) is smooth) if and only if the inversesubarrangement A(w) is a free arrangement and its number ofchambers is equal to the size of the Bruhat interval [1,w].

For w ∈W, define its inverse set I(w) := {α ∈ Φ+ | −wα ∈ Φ+}.

Consider the inverse subarrangement

A(w) := A(I(w)) := {ker(α) | α ∈ I(w)}

.Theorem..

......

A(w) := A(I(w)) := {ker(α) | α ∈ I(w)}

.Theorem..

......

A(w) := A(I(w)) := {ker(α) | α ∈ I(w)}

.Theorem..

......

I stop here.

Thanks for your attention!

I stop here.

Thanks for your attention!

On ideal subarrangements of Weyl arrangements @let@tokenterao/20140728SaoCarlosBrazil.ver.5.p… ·...

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