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Gröbner-basis-free proofs of freeness of hyperplane arrangements

Mohamed Barakat

University of Kaiserslautern

Algebra, Algorithms, and Algebraic Analysis Abdij Rolduc, 6. September 2013. joint work with

• T. Abe, M. Cuntz, T. Hoge, and H. Terao

Mohamed Barakat Gröbner-basis-free proofs of freeness

Freeness of divisors

• k a field and S = k[x1, . . . , xℓ]
• Aℓ = Spec S
• Der = DerAℓ = ℓ

i=1 S∂xi ∼

= Sℓ is the free S-module of polynomial derivations or polynomial vector fields. Definition Let D = {f = 0} ⊂ Aℓ be a divisor:

• The submodule of logarithmic derivations is defined as

Der(− log D) = ker

• Der

df

− → S → S/f

• ≤ Der ∼

= Sℓ. These are all infinitesimal external symmetries of D in the sense of SOPHUS LIE.

• D is called free if Der(− log D) is a free S-module.

One can compute Der(− log D) and decide its freeness using GRÖBNER bases over S.

Mohamed Barakat Gröbner-basis-free proofs of freeness

The NEIL parabola

gap> LoadPackage( "Sheaves" );; gap> Q := HomalgFieldOfRationalsInSingular( ); Q gap> R := Q * "x,y"; Q[x,y] gap> AssignGeneratorVariables( R ); #I Assigned the global variables [ x, y ] gap> f := y^2-x^3;

• x^3+y^2

gap> D := Divisor( f ); <A divisor on <An affine space A^2 over Q>> gap> DerD := DerMinusLog( D ); <A torsion-free left submodule given by 2 generators> gap> IsFree( DerD ); true gap> DerD; <A free left submodule of rank 2 on free generators> gap> Display( DerD ); 2*x,3*y, 2*y,3*x^2 A left submodule generated by the 2 rows of the above matrix

Mohamed Barakat Gröbner-basis-free proofs of freeness

The NEIL parabola (continued)

Mohamed Barakat Gröbner-basis-free proofs of freeness

Hyperplane arrangements

• A = {H1, . . . , Hn} a hyperplane arrangement in Aℓ
• αH ∈ S a degree one defining equation of H ∈ A
• ∪A := {

H∈A αH = 0} is the associated divisor

Definition A is called free if D(A) := Der(− log ∪A) is free.

• If A is central then D(A) is a graded S-module.
• If A is central and free then D(A) = ℓ

i=1 S(−di).

• {d1 ≤ · · · ≤ dℓ} is called the multiset of exponents of A.

Root systems provide an interesting class of examples.

Mohamed Barakat Gröbner-basis-free proofs of freeness

Root systems

Let V be an ℓ-dimensional real vector space. A reflection along α ∈ V is an R-linear map s satisfying

• s(α) = −α,
• Hs := Fix(s) is a hyperplane.

Definition A root system Φ ⊂ V \ {0} is a finite generating system of V such that

1 for each α ∈ Φ there exists a reflection sα with sα(R) = R

(and hence exactly one),

2 for each α, β ∈ Φ: sα(β) ∈ β + Zα.

• Φ is called reduced if α = {α, −α}.
• Φ is called irreducible if it cannot be decomposed as a

direct sum of two root subsystems Φ1 ⊕ Φ2 ⊂ V1 ⊕ V2 = V .

Mohamed Barakat Gröbner-basis-free proofs of freeness

Classification of root systems

Example Let (e1, . . . , eℓ) denote the standard basis of Rℓ. The following subsets of V are reduced irreducible root systems

• Aℓ−1 := {ei − ej}i=j ⊂ Rℓ−1
• Dℓ := Aℓ−1 ∪ {±(ei + ej)}i=j ⊂ Rℓ
• Bℓ := Dℓ ∪ {±ei} ⊂ Rℓ
• Cℓ := Dℓ ∪ {±2ei} ⊂ Rℓ

Theorem The above example is an exhaustive list except for 5 sporadic root systems E6, E7, E8, F4, G2. From now on all root systems will be assumed reduced and irreducible.

Mohamed Barakat Gröbner-basis-free proofs of freeness

Simple system

Definition

• Fix a set Φ+ of positive roots, i.e., Φ+ ˙

∪ − Φ+ = Φ.

• The corresponding simple system

∆ := Φ+ \ {α + β | α, β ∈ Φ+} ⊂ Φ+ is a basis of V .

• ∆ is the only antichain in Φ+ with ℓ roots.
• The simple system ∆ := {α1, . . . , αℓ} turns Φ+ into a poset

α ≥ β :⇐ ⇒ α − β ∈

ℓ

• i=1

Z≥0αi

• An (order) ideal I in a poset is a subset closed under

passage to smaller elements.

Mohamed Barakat Gröbner-basis-free proofs of freeness

COXETER arrangements are inductively free

Set Hα = Hsα = Fix(sα) for α ∈ Φ+.

• For an ideal I ⊂ Φ+ we call

A(I) := {Hα | α ∈ I} the ideal subarrangement of I.

• A(Φ+) is called a WEYL arrangement.

Using invariant theory one can show that Theorem ([ORLIK, TERAO]) WEYL arrangements are free. The exponents can be described combinatorially. Theorem (–, CUNTZ) Crystallographic & COXETER arrangements are inductively free. Our proof is heavily computer assisted.

Mohamed Barakat Gröbner-basis-free proofs of freeness

Ideal subarrangements are free

Theorem (ABE, –, CUNTZ, HOGE, TERAO) Any ideal subarrangement A(I) ⊂ A(Φ+) is free. The exponents can be described combinatorially. Proof. Make an induction on the height ht

• α∈∆

λαα

• =
• α∈∆

λα and use the following multiple addition theorem to add the roots

• f largest height simultaneously.

Mohamed Barakat Gröbner-basis-free proofs of freeness

The multiple addition theorem (MAT)

• A′ be a free central arrangement
• H1, . . . , Hq new central hyperplanes (i.e., not in A′) with

X := H1 ∩ · · · ∩ Hq q-codimensional and ⊂ ∪A′

• Define the restrictions A′′

j := A′ ∩ Hj and assume that

|A′| − |A′′

j | = d

∀j = 1, . . . , q, (⋆) where d is the highest exponent of A Theorem (loc. cit.) Then A := A′ ∪ {H1, . . . , Hq} is free. The exponents of A are expressible in terms of those of A′.

Mohamed Barakat Gröbner-basis-free proofs of freeness

Idea of the proof.

• Let θ1, . . . , θℓ be a free basis of D(A′).
• Use (⋆) to show that θ1, . . . , θℓ−p ∈ D(A), where p is the

multiplicity of the highest exponent d.

• For z ∈ X ⊂ ∪A′

TV,z = θ1, . . . , θℓ−pz

• ⊂TX,z

⊕

p vectors

• θℓ−p+1, . . . , θℓz
• =

⇒ ∼ =TV,z/TX,z

• dim=q
• Hence p ≥ q and we may assume that the last q vectors

(θℓ−q+1, . . . , θℓ) form a basis of TV,z/TX,z.

• Again using (⋆) one can show that

(θ1, . . . , θℓ−q, αHqθℓ−q+1, . . . , αH1θℓ) is a free basis of D(A).

Mohamed Barakat Gröbner-basis-free proofs of freeness

Thank you

Mohamed Barakat Gröbner-basis-free proofs of freeness

Takuro Abe, Mohamed Barakat, Michael Cuntz, Torsten Hoge, and Hiroaki Terao, The freeness of ideal subarrangements of weyl arrangements, submitted (arXiv:1304.8033), 2013. Mohamed Barakat and Michael Cuntz, Coxeter and crystallographic arrangements are inductively free, Adv.

• Math. 229 (2012), no. 1, 691–709, (arXiv:1011.4228).

MR 2854188 (2012i:20048) Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR MR1217488 (94e:52014)

Mohamed Barakat Gröbner-basis-free proofs of freeness