.
.
On ideal subarrangements
of
Weyl arrangements (The Fourth Lecture)
Hiroaki Terao
(Hokkaido University, Sapporo, Japan)
at
Geometry and Topology of Singular Varieties
Theory and Applications
Hanoi, VietNam
2013.12.06
H. Terao (Hokkaido University)
2013.12.06
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Credit
H. Terao (Hokkaido University)
2013.12.06
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Credit
with
H. Terao (Hokkaido University)
2013.12.06
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Credit
with
Takuro Abe (Kyoto University)
H. Terao (Hokkaido University)
2013.12.06
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Credit
with
Takuro Abe (Kyoto University)
Mohamed Barakat (Universität Kaiserlautern)
H. Terao (Hokkaido University)
2013.12.06
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Credit
with
Takuro Abe (Kyoto University)
Mohamed Barakat (Universität Kaiserlautern)
Michael Cuntz (Leibniz Universität Hannover )
H. Terao (Hokkaido University)
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Credit
with
Takuro Abe (Kyoto University)
Mohamed Barakat (Universität Kaiserlautern)
Michael Cuntz (Leibniz Universität Hannover )
Torsten Hoge (Ruhr Universität Bochum)
H. Terao (Hokkaido University)
2013.12.06
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Credit
with
Takuro Abe (Kyoto University)
Mohamed Barakat (Universität Kaiserlautern)
Michael Cuntz (Leibniz Universität Hannover )
Torsten Hoge (Ruhr Universität Bochum)
arXiv:1304.8033
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Weyl arrangements
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Weyl arrangements
A reflection arrangement A is called a Weyl
arrangement if there exists a set Φ of roots such that
2(α, β)/(β, β) ∈ Z
for any α, β ∈ Φ.
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Weyl arrangements
A reflection arrangement A is called a Weyl
arrangement 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 group
W(A).
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Weyl arrangements
A reflection arrangement A is called a Weyl
arrangement 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 group
W(A).
Φ+ : the set of positive roots
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Weyl arrangements
A reflection arrangement A is called a Weyl
arrangement 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 group
W(A).
Φ+ : the set of positive roots
∆ = {α1 , α2 , . . . , αℓ } : the simple system (=the set of
simple roots)
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the root poset and ideals
.
Definition
.
.
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the root poset and ideals
.
Definition
.
Introduce a partial order ≥ into the set Φ+ of positive roots by
.
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the root poset and ideals
.
Definition
.
Introduce a partial order ≥ into the set Φ+ of positive roots by
β1 ≥ β2 ⇐⇒ β1 − β2 ∈
ℓ
∑
Z≥0 αi .
i=1
The poset is called the (positive) root poset.
.
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the root poset and ideals
.
Definition
.
Introduce a partial order ≥ into the set Φ+ of positive roots by
β1 ≥ β2 ⇐⇒ β1 − β2 ∈
ℓ
∑
Z≥0 αi .
i=1
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.
.
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the root poset and ideals
.
Definition
.
Introduce a partial order ≥ into the set Φ+ of positive roots by
β1 ≥ β2 ⇐⇒ β1 − β2 ∈
ℓ
∑
Z≥0 αi .
i=1
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) := {ker α | α ∈ I} is
called
an ideal subarrangement of A.
.
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Examples of ideals/non-ideals of the root poset of A3
A3 : •
α1
•
α2
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•
α3
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Examples of ideals/non-ideals of the root poset of A3
A3 : •
α1
•
α2
•
α3
Φ+ = {α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 }
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Examples of ideals/non-ideals of the root poset of A3
A3 : •
α1
•
α2
•
α3
Φ+ = {α1 , α2 , α3 , α1 + α2 , α2 + α3 , α1 + α2 + α3 }
α1 ≤ α1 + α2 ≤ α1 + α2 + α3 ,
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Examples of ideals/non-ideals of the root poset of A3
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 ,
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Examples of ideals/non-ideals of the root poset of A3
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 ,
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Examples of ideals/non-ideals of the root poset of A3
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
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Examples of ideals/non-ideals of the root poset of A3
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.
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Examples of ideals/non-ideals of the root poset of A3
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.
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Free Arrangements and their Exponents
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Free Arrangements and their Exponents
A: an arrangement of hyperplanes in an ℓ-dimensional vector
space V
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Free Arrangements and their Exponents
A: an arrangement of hyperplanes in an ℓ-dimensional vector
space V
αH ∈ V ∗ : ker(αH ) = H for H ∈ A
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Free Arrangements and their Exponents
A: an arrangement of hyperplanes in an ℓ-dimensional vector
space V
αH ∈ V ∗ : ker(αH ) = H for H ∈ A
S := S(V ∗ ): the symmetric algebra of the dual space V ∗
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Free Arrangements and their Exponents
A: an arrangement of hyperplanes in an ℓ-dimensional vector
space V
αH ∈ V ∗ : ker(αH ) = H for H ∈ A
S := 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 ) ∈ αH S for all H ∈ A}.
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Free Arrangements and their Exponents
A: an arrangement of hyperplanes in an ℓ-dimensional vector
space V
αH ∈ V ∗ : ker(αH ) = H for H ∈ A
S := 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 ) ∈ αH S for all H ∈ A}.
A is said to be a free arrangement if D(A) is a free S-module.
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Free Arrangements and their Exponents
A: an arrangement of hyperplanes in an ℓ-dimensional vector
space V
αH ∈ V ∗ : ker(αH ) = H for H ∈ A
S := 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 ) ∈ αH S 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 with
deg θi = di . The nonnegative integers d1 , d2 , . . . , dℓ are called
the exponents of A.
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Every Weyl arrangement is free
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Every Weyl arrangement is free
.
Theorem
.
(K. Saito 1976 et al.) The Weyl arrangement A is a free
arrangement. The exponents of the Weyl arrangement A coincide
.with the exponents of the corresponding root system.
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Every Weyl arrangement is free
.
Theorem
.
(K. Saito 1976 et al.) The Weyl arrangement A is a free
arrangement. The exponents of the Weyl arrangement A coincide
.with 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 }
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Every Weyl arrangement is free
.
Theorem
.
(K. Saito 1976 et al.) The Weyl arrangement A is a free
arrangement. The exponents of the Weyl arrangement A coincide
.with 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 = x13 (∂/∂x1 ) + x23 (∂/∂x2 ),
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Every Weyl arrangement is free
.
Theorem
.
(K. Saito 1976 et al.) The Weyl arrangement A is a free
arrangement. The exponents of the Weyl arrangement A coincide
.with 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 = x13 (∂/∂x1 ) + x23 (∂/∂x2 ),
The exponents are
d1 = deg θ1 = 1, d2 = deg θ2 = 3.
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Factorization Theorem
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Factorization Theorem
.
Theorem
.
(H. T.(1981)) Assume that A is a free arrangement in the complex
space V = Cℓ with exponents (d1 , . . . , dℓ ). Define the complement
of A by
∪
M(A) := V \
H.
H∈A
Then the Poincaré polynomial (with its coefficients equal to the Betti
numbers ) of the topological space M(A) splits as
Poin(M(A), t) =
.
ℓ
∏
(1 + di t).
i=1
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Exponents
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Exponents
Dynkin diagrams (root systems) and exponents
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Exponents
Dynkin diagrams (root systems) and exponents
Aℓ : •
α1
•
α2
H. Terao (Hokkaido University)
···
•
αℓ−1
• (1, 2, . . . , ℓ)
αℓ
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Exponents
Dynkin diagrams (root systems) and exponents
Aℓ : •
α1
Bℓ : •
α1
•
α2
•
α2
H. Terao (Hokkaido University)
···
···
•
αℓ−1
• /
αℓ−1
• (1, 2, . . . , ℓ)
αℓ
• (1, 3, 5, . . . , 2ℓ − 1)
αℓ
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Exponents
Dynkin diagrams (root systems) and exponents
Aℓ : •
α1
Bℓ : •
α1
Cℓ : •
α1
•
α2
•
α2
•
α2
H. Terao (Hokkaido University)
···
···
···
• (1, 2, . . . , ℓ)
•
αℓ−1
αℓ
• / • (1, 3, 5, . . . , 2ℓ − 1)
αℓ−1
αℓ
o
•
• (1, 3, 5, . . . , 2ℓ − 1)
αℓ−1
αℓ
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Exponents
Dynkin diagrams (root systems) and exponents
Aℓ : •
α1
Bℓ : •
α1
Cℓ : •
α1
•
α2
•
α2
•
α2
···
Dℓ : •
α1
•
α2
···
H. Terao (Hokkaido University)
···
···
• (1, 2, . . . , ℓ)
•
αℓ−1
αℓ
• / • (1, 3, 5, . . . , 2ℓ − 1)
αℓ−1
αℓ
o
•
• (1, 3, 5, . . . , 2ℓ − 1)
αℓ−1
αℓ
•
αℓ−1 (1, 3, 5, . . . , 2ℓ − 3, ℓ − 1)
•
αℓ−2
•
αℓ
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Exponents
Dynkin diagrams (root systems) and exponents
Aℓ : •
α1
Bℓ : •
α1
Cℓ : •
α1
•
α2
•
α2
•
α2
···
Dℓ : •
α1
•
α2
···
E6 : •
α1
•
α3
•
···
···
α4
• (1, 2, . . . , ℓ)
•
αℓ−1
αℓ
• / • (1, 3, 5, . . . , 2ℓ − 1)
αℓ−1
αℓ
o
•
• (1, 3, 5, . . . , 2ℓ − 1)
αℓ−1
αℓ
•
αℓ−1 (1, 3, 5, . . . , 2ℓ − 3, ℓ − 1)
•
αℓ−2
•
αℓ
•
• (1, 4, 5, 7, 8, 11)
α5
α6
• α2
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Exponents
Dynkin diagrams (root systems) and exponents
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Exponents
Dynkin diagrams (root systems) and exponents
E7 : (1, 5, 7, 9, 11, 13, 17) •
α1
•
α3
•
α4
•
α5
•
α6
•
α7
• α2
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Exponents
Dynkin diagrams (root systems) and exponents
E7 : (1, 5, 7, 9, 11, 13, 17) •
α1
•
α3
•
α4
•
α5
•
α6
•
α7
• α2
E8 : (1, 7, 11, 13, 17, 19, 23, 29)
•
•
•
•
•
α3
α6
α1
α4 α5
•
α7
•
α8
• α2
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Exponents
Dynkin diagrams (root systems) and exponents
E7 : (1, 5, 7, 9, 11, 13, 17) •
α1
•
α3
•
α4
•
α5
•
α6
•
α7
• α2
E8 : (1, 7, 11, 13, 17, 19, 23, 29)
•
•
•
•
•
α3
α6
α1
α4 α5
•
α7
•
α8
• α2
F4 : •
α1
•
α2
/
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•
α3
• (1, 5, 7, 11)
α4
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Exponents
Dynkin diagrams (root systems) and exponents
E7 : (1, 5, 7, 9, 11, 13, 17) •
α1
•
α3
•
α4
•
α5
•
α6
•
α7
• α2
E8 : (1, 7, 11, 13, 17, 19, 23, 29)
•
•
•
•
•
α3
α6
α1
α4 α5
•
α7
•
α8
• α2
F4 : •
α1
G2 : •
α1
o
•
α2
/
•
α3
• (1, 5, 7, 11)
α4
• (1, 5)
α2
(from http://www.ms.u-tokyo.ac.jp/ abenori/tex/tex7.html )
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Main Theorem
.
Theorem
.
.
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Main Theorem
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
.
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Main Theorem
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
I : an ideal of Φ+
.
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Main Theorem
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
I : an ideal of Φ+
B : the corresponding ideal subarrangement to I
.
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Main Theorem
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
I : an ideal of Φ+
B : the corresponding ideal subarrangement to I
Then
.
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Main Theorem
.
Theorem
.
If
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
.
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Main Theorem
.
Theorem
.
If
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 height
distribution of the positive roots in I satisfy the
dual-partition
formula.
.
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Main Theorem
.
Theorem
.
If
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 height
distribution of the positive roots in I satisfy the
dual-partition
formula.
.
This positively settles a conjecture by Sommers-Tymoczko (2006).
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the dual-partition formula
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the dual-partition formula
In particular, when the ideal I is equal to the entire Φ+ ,
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the dual-partition formula
In particular, when the ideal I is equal to the entire Φ+ , our main
theorem yields the dual-partition formula by Shapiro, Steinberg,
Kostant, and Macdonald:
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the dual-partition formula
In particular, when the ideal I is equal to the entire Φ+ , our main
theorem yields the dual-partition formula by Shapiro, Steinberg,
Kostant, and Macdonald:
.
Corollary
.
The exponents of Φ and the height distribution of positive roots
.satisfy the dual-partition formula.
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Height of positive roots
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Height of positive roots
Φ : an irreducible root system of rank ℓ
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Height of positive roots
Φ : an irreducible root system of rank ℓ
∆ = {α1, . . . , αℓ } : a simple system of Φ
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Height of positive roots
Φ : an irreducible root system of rank ℓ
∆ = {α1, . . . , αℓ } : a simple system of Φ
Φ+: the set of positive roots
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Height of positive roots
Φ : an irreducible root system of rank ℓ
∆ = {α1, . . . , αℓ } : a simple system of Φ
Φ+: the set of positive roots
∑
ht(α) := ℓi=1 ci (height) for a positive root
∑
α = ℓi=1 ciαi (ci ∈ Z≥0)
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Height of positive roots
Φ : an irreducible root system of rank ℓ
∆ = {α1, . . . , αℓ } : a simple system of Φ
Φ+: the set of positive roots
∑
ht(α) := ℓi=1 ci (height) for a positive root
∑
α = ℓi=1 ciαi (ci ∈ Z≥0)
The height distribution in Φ+ is a sequence
of positive integers (i1 , i2 , . . . , im ), where
ij := |{α ∈ Φ+ | ht(α) = j}| (1 ≤ j ≤ m)
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Height of positive roots (E6 )
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
Exponents: (1, 4, 5, 7, 8, 11)
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
Exponents: (1, 4, 5, 7, 8, 11)
List of positive roots:
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
Exponents: (1, 4, 5, 7, 8, 11)
List of positive roots:
height 1 : α1 , α2 , α3 , α4 , α5 , α6
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
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
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
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 , . . .
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
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 , . . .
.
.
.
.
H. Terao (Hokkaido University)
2013.12.06
14 / 32
Height of positive roots (E6 )
E6 : •
α1
•
α3
•
α4
•
α5
•
α6
• α2
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: α̃ = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 (the highest root)
H. Terao (Hokkaido University)
2013.12.06
14 / 32
heights
Height of positive roots (E6 )
ht=11
ht=10
ht=9
ht=8
ht=7
ht=6
ht=5
ht=4
ht=3
ht=2
ht=1
α̃
•
•
•
•
•
•
•
•
α1 + α3
α1
•
•
•
•
•
•
α2 + α4
α2
α̃ = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 ,
H. Terao (Hokkaido University)
•
•
•
•
•
α3 + α4
α3
•
•
•
•
α4
•
•
•
α5
α6
ht(α̃) = 11 (the highest root)
2013.12.06
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height distribution
Height Distribution (E6 )
1
1
1
2
3
3
4
5
5
5
6
H. Terao (Hokkaido University)
•
•
•
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•
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•
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•
•
•
7
5
4
3
1
•
exponents
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height distribution
Exponents (E6 )
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•
•
•
•
•
•
•
•
•
•
•
11
8
7
5
4
1
H. Terao (Hokkaido University)
exponents
2013.12.06
17 / 32
height distribution
The Dual-Partition Formula (E6 )
•
•
•
•
•
•
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•
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•
•
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•
•
•
•
•
•
•
•
•
•
11
8
7
5
4
1
1
1
1
2
3
3
4
5
5
5
6
H. Terao (Hokkaido University)
exponents
2013.12.06
18 / 32
History of the Dual-Partition Formula
H. Terao (Hokkaido University)
2013.12.06
19 / 32
History of the Dual-Partition Formula
. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g was
discovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers produced
by this procedure agree with the exponents . . . . . . the important question of proving that this
“agreement” is more than just a coincidence remained open.
H. Terao (Hokkaido University)
2013.12.06
19 / 32
History of the Dual-Partition Formula
. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g was
discovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers produced
by 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)
H. Terao (Hokkaido University)
2013.12.06
19 / 32
History of the Dual-Partition Formula
. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g was
discovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers produced
by 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)
H. Terao (Hokkaido University)
2013.12.06
19 / 32
History of the Dual-Partition Formula
. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g was
discovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers produced
by 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)
H. Terao (Hokkaido University)
2013.12.06
19 / 32
History of the Dual-Partition Formula
. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g was
discovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers produced
by 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)
H. Terao (Hokkaido University)
2013.12.06
19 / 32
History of the Dual-Partition Formula
. . . . . . we shall presently describe, of “reading off” the exponents from the root structure of g was
discovered by Arnold Shapiro. . . . . . . However, even though one verifies that the numbers produced
by 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)
(2013) ABCHT (for ideal subarr.: using free arrangements)
H. Terao (Hokkaido University)
2013.12.06
19 / 32
MAT (Multiple Addition Theorem - key to our proof -)
H. Terao (Hokkaido University)
2013.12.06
20 / 32
MAT (Multiple Addition Theorem - key to our proof -)
.
Theorem
.
(ABCHT(2013)) Let A′ be a free arrangement with exponents
(d1 , . . . , dℓ ) (d1 ≤ · · · ≤ dℓ ) and 1 ≤ p ≤ ℓ the multiplicity of the
highest exponent d. Let H1 , . . . , Hq be hyperplanes with Hi < A′ for
i = 1, . . . , q. Define A′′j := {H ∩ Hj | H ∈ A′ } (j = 1, . . . , q). Assume
that the following three conditions are satisfied:
(1) X := H1 ∩ · · · ∩ Hq is q-codimensional,
∪
(2) X ⊈ H∈A′ H ,
(3) |A′ | − |A′′j | = d (1 ≤ j ≤ q).
Then q ≤ p and A := A′ ∪ {H1 , . . . , Hq } is free with
exponents
(d1 , . . . , dℓ−q , (d + 1)q ).
.
H. Terao (Hokkaido University)
2013.12.06
20 / 32
Main Theorem (again)
.
Theorem
.
.
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Main Theorem (again)
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
.
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Main Theorem (again)
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
A : the Weyl arrangement (= the collection of
hyperplanes orthogonal to the positive roots of Φ)
.
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Main Theorem (again)
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
A : the Weyl arrangement (= the collection of
hyperplanes orthogonal to the positive roots of Φ)
Then
.
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Main Theorem (again)
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
A : the Weyl arrangement (= the collection of
hyperplanes orthogonal to the positive roots of Φ)
Then
1. any ideal subarrangement B of A is free,
.
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Main Theorem (again)
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
A : the Weyl arrangement (= the collection of
hyperplanes orthogonal to the positive roots of Φ)
Then
1. any ideal subarrangement B of A is free,
. its exponents and the height distribution of the
positive roots satisfy the dual-partition formula.
.
2
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Main Theorem (again)
.
Theorem
.
If
Φ : an irreducible root system of rank ℓ
A : the Weyl arrangement (= the collection of
hyperplanes orthogonal to the positive roots of Φ)
Then
1. any ideal subarrangement B of A is free,
. its exponents and the height distribution of the
positive roots satisfy the dual-partition formula.
.
2
We prove the Main Theorem applying MAT inductively.
H. Terao (Hokkaido University)
2013.12.06
21 / 32
Proof of Main Theorem (just an outline)
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Proof of Main Theorem (just an outline)
Proof.
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Proof of Main Theorem (just an outline)
Proof.
Let B be an ideal subarrangement of the Weyl arrangement
A of a root system Φ.
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Proof of Main Theorem (just an outline)
Proof.
Let B be an ideal subarrangement of the Weyl arrangement
A of a root system Φ.
For k ∈ Z>0 , define B≤k := {H ∈ B | ht(αH ) ≤ k}.
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Proof of Main Theorem (just an outline)
Proof.
Let B be an ideal subarrangement of the Weyl arrangement
A 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
(1, . . . , 1).
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Proof of Main Theorem (just an outline)
Proof.
Let B be an ideal subarrangement of the Weyl arrangement
A 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
(1, . . . , 1).
We may apply MAT for A′ := B≤k and A := B≤k+1 .
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Proof of Main Theorem (just an outline)
Proof.
Let B be an ideal subarrangement of the Weyl arrangement
A 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
(1, . . . , 1).
We may apply MAT for A′ := B≤k and A := B≤k+1 .
To verify the three assumtions of MAT, we verify the
corresponding combinatorial and geometric properties of the
root system Φ. (next page)
H. Terao (Hokkaido University)
2013.12.06
22 / 32
Verifying the three assumptions of MAT (when A = B)
H. Terao (Hokkaido University)
2013.12.06
23 / 32
Verifying the three assumptions of MAT (when A = B)
Let β1 , . . . , βq be positive roots of the same height k ≥ 1.
H. Terao (Hokkaido University)
2013.12.06
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Verifying the three assumptions of MAT (when A = B)
Let β1 , . . . , βq be positive roots of the same height k ≥ 1.
1. X := {β
1 = · · · = βq = 0} is q-codimensional.
H. Terao (Hokkaido University)
2013.12.06
23 / 32
Verifying the three assumptions of MAT (when A = B)
Let β1 , . . . , βq be positive roots of the same height k ≥ 1.
1. X := {β
1 = · · · = βq = 0} is q-codimensional.
2. The q-codimensional subspace X is not entirely
contained in a hyperplane {β = 0} if β ∈ Φ+ with
ht(β) < k.
H. Terao (Hokkaido University)
2013.12.06
23 / 32
Verifying the three assumptions of MAT (when A = B)
Let β1 , . . . , βq be positive roots of the same height k ≥ 1.
1. X := {β
1 = · · · = βq = 0} is q-codimensional.
2. The q-codimensional subspace X is not entirely
contained in a hyperplane {β = 0} if β ∈ Φ+ with
ht(β) < k.
3. Note that
Φ+k := {β ∈ Φ+ | ht(β) ≤ k}
is an ideal which we call the k-th height ideal . If a
positive root β0 saifies ht(β0 ) = k + 1 > 1, then
|Φ+k | − |{⟨β, β0 ⟩ | β ∈ Φ+k }| = k.
H. Terao (Hokkaido University)
2013.12.06
23 / 32
Verifying the three assumptions of MAT (when A = B)
Let β1 , . . . , βq be positive roots of the same height k ≥ 1.
1. X := {β
1 = · · · = βq = 0} is q-codimensional.
2. The q-codimensional subspace X is not entirely
contained in a hyperplane {β = 0} if β ∈ Φ+ with
ht(β) < k.
3. Note that
Φ+k := {β ∈ Φ+ | ht(β) ≤ k}
is an ideal which we call the k-th height ideal . If a
positive root β0 saifies ht(β0 ) = k + 1 > 1, then
|Φ+k | − |{⟨β, β0 ⟩ | β ∈ Φ+k }| = k.
No classification is needed for this verification.
H. Terao (Hokkaido University)
2013.12.06
23 / 32
To verify the third assumption we need the coheights
H. Terao (Hokkaido University)
2013.12.06
24 / 32
To verify the third assumption we need the coheights
Define the coheight of α by
cohtΦ α := h − 1 − ht(α),
where h is the Coxeter number of Φ.
H. Terao (Hokkaido University)
2013.12.06
24 / 32
To verify the third assumption we need the coheights
Define the coheight of α by
cohtΦ α := h − 1 − ht(α),
where h is the Coxeter number of Φ.
This coheight of α is called the global coheight.
H. Terao (Hokkaido University)
2013.12.06
24 / 32
Local-global formula for coheights
For X ∈ L(A), let ΦX := Φ ∩ X ⊥ . Then ΦX is a root
system of rank codim X .
H. Terao (Hokkaido University)
2013.12.06
25 / 32
Local-global formula for coheights
For X ∈ L(A), let ΦX := Φ ∩ X ⊥ . Then ΦX is a root
system of rank codim X .
The coheight of α in ΦX is called the local coheight.
H. Terao (Hokkaido University)
2013.12.06
25 / 32
Local-global formula for coheights
For X ∈ L(A), let ΦX := Φ ∩ X ⊥ . Then ΦX is a root
system of rank codim X .
The coheight of α in ΦX is called the local coheight.
For α ∈ Φ+ , let
Aα := AHα = {K ∩ Hα | K ∈ A \ {Hα }}.
H. Terao (Hokkaido University)
2013.12.06
25 / 32
Local-global formula for coheights
For X ∈ L(A), let ΦX := Φ ∩ X ⊥ . Then ΦX is a root
system of rank codim X .
The coheight of α in ΦX is called the local coheight.
For α ∈ Φ+ , let
Aα := AHα = {K ∩ Hα | K ∈ A \ {Hα }}.
.
Lemma (Local-global formula for coheights)
.
For α ∈ Φ+ , we have
cohtΦ α =
∑
cohtX α.
X∈Aα
.
H. Terao (Hokkaido University)
2013.12.06
25 / 32
Inductive use of MAT (E6 ) : I = Φ+0
H. Terao (Hokkaido University)
2013.12.06
26 / 32
height distribution
Inductive use of MAT (E6 ) : I = Φ+0
0
0
0
0
0
0
0
0
0
0
0
0
H. Terao (Hokkaido University)
0
0
0
0
0
exponents
2013.12.06
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height distribution
Inductive use of MAT (E6 ) : I = Φ+1
0
0
0
0
0
0
0
0
0
0
6
H. Terao (Hokkaido University)
•
•
•
•
•
•
1
1
1
1
1
1
exponents
2013.12.06
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height distribution
Inductive use of MAT (E6 ) : I = Φ+2
0
0
0
0
0
0
0
0
0
5
6
H. Terao (Hokkaido University)
•
•
•
•
•
•
•
•
•
•
•
2
2
2
2
2
1
exponents
2013.12.06
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height distribution
Inductive use of MAT (E6 ) : I = Φ+3
0
0
0
0
0
0
0
0
5
5
6
H. Terao (Hokkaido University)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
3
3
3
3
3
1
exponents
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height distribution
Inductive use of MAT (E6 ) : I = Φ+4
0
0
0
0
0
0
0
5
5
5
6
H. Terao (Hokkaido University)
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4
4
4
4
4
1
exponents
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height distribution
Inductive use of MAT (E6 ) : I = Φ+5
0
0
0
0
0
0
4
5
5
5
6
H. Terao (Hokkaido University)
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5
5
5
5
4
1
exponents
2013.12.06
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height distribution
Inductive use of MAT (E6 ) : I = Φ+6
0
0
0
0
0
3
4
5
5
5
6
H. Terao (Hokkaido University)
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6
6
6
5
4
1
exponents
2013.12.06
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height distribution
Inductive use of MAT (E6 ) : I = Φ+7
0
0
0
0
3
3
4
5
5
5
6
H. Terao (Hokkaido University)
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•
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7
7
7
5
4
1
exponents
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height distribution
Inductive use of MAT (E6 ) : I = Φ+8
0
0
0
2
3
3
4
5
5
5
6
H. Terao (Hokkaido University)
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8
8
7
5
4
1
exponents
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height distribution
Inductive use of MAT (E6 ) : I = Φ+9
0
0
1
2
3
3
4
5
5
5
6
H. Terao (Hokkaido University)
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9
8
7
5
4
1
exponents
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height distribution
Inductive use of MAT (E6 ) : I = Φ+10
0
1
1
2
3
3
4
5
5
5
6
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•
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•
•
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•
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•
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•
•
•
•
•
•
•
10
8
7
5
4
1
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exponents
2013.12.06
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height distribution
The Dual-Partition Formula (E6 ) (again)
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H. Terao (Hokkaido University)
exponents
2013.12.06
26 / 32
Summary
H. Terao (Hokkaido University)
2013.12.06
27 / 32
Summary
. The celebrated dual-partition formula for a root
system by Shapiro-Steinberg-Kostant-Macdonald is
generalized to the class of ideals.
1
H. Terao (Hokkaido University)
2013.12.06
27 / 32
Summary
. The celebrated dual-partition formula for a root
system by Shapiro-Steinberg-Kostant-Macdonald is
generalized to the class of ideals.
2. The theory of free arrangements (the multiple
1
addition theorem ( MAT)) provides a base of our
proof.
H. Terao (Hokkaido University)
2013.12.06
27 / 32
Summary
. The celebrated dual-partition formula for a root
system by Shapiro-Steinberg-Kostant-Macdonald is
generalized to the class of ideals.
2. The theory of free arrangements (the multiple
1
addition theorem ( MAT)) provides a base of our
proof.
. Our proof needs combinatorial and geometric
properties concerning the height of positive roots.
3
H. Terao (Hokkaido University)
2013.12.06
27 / 32
height distribution
The Dual-Partition Formula (E6 ) (3rd time)
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H. Terao (Hokkaido University)
exponents
2013.12.06
28 / 32
height distribution
The Dual-Partition Formula for the ideal Φ+5 of E6
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H. Terao (Hokkaido University)
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exponents
2013.12.06
29 / 32
height distribution
The Dual-Partition Formula for an ideal I with
Φ+5 ⊂ I ⊂ Φ+6 of E6
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H. Terao (Hokkaido University)
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exponents
2013.12.06
30 / 32
A Natural Question
H. Terao (Hokkaido University)
2013.12.06
31 / 32
A Natural Question
Is there any nice characterization of the set
F := {F ⊆ Φ+ | A(F) is free},
where A(F) is the corresponding subarrangement to
F?
H. Terao (Hokkaido University)
2013.12.06
31 / 32
A Natural Question
Is there any nice characterization of the set
F := {F ⊆ Φ+ | A(F) is free},
where A(F) is the corresponding subarrangement to
F?
Remark. At least we know
F ⊇ {wI | w ∈ W, I is an ideal}.
H. Terao (Hokkaido University)
2013.12.06
31 / 32
I stop here.
H. Terao (Hokkaido University)
2013.12.06
32 / 32
I stop here.
Thanks for your attention!
H. Terao (Hokkaido University)
2013.12.06
32 / 32