Simple-root bases for Shi arrangements

Abe, Takuro; Terao, Hiroaki

doi:10.1016/j.jalgebra.2014.09.011

Cited by 8 publications

(15 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…3. Those among (4), (5) and (6) follow from the deletion-restriction theorem 2.2. Since χ 0 (A; t) = χ 0 (A ′ ; t) − χ 0 (A H ; t), that between (7) and (8) is easy.…”

Section: Example 37mentioning

confidence: 91%

See 1 more Smart Citation

Divisionally free arrangements of hyperplanes

Abe

2015

Invent. math.

Self Cite

102

View full text Add to dashboard Cite

Let (A, A ′ , A H ) be the triple of hyperplane arrangements. We show that the freeness of A H and the division of χ(A; t) by χ(A H ; t) imply the freeness of A. This "division theorem" improves the famous addition-deletion theorem, and it has several applications, which include a definition of "divisionally free arrangements". It is a strictly larger class of free arrangements than the classical important class of inductively free arrangements. Also, whether an arrangement is divisionally free or not is determined by the combinatorics. Moreover, we show that a lot of recursively free arrangements, to which almost all known free arrangements are belonging, are divisionally free. Main resultsLet V be an ℓ-dimensional vector space over an arbitrary field K with ℓ ≥ 1, S = Sym(V * ) = K[x 1 , . . . , x ℓ ] its coordinate ring and Der S := ⊕ ℓ i=1 S∂ x i the module of K-linear S-derivations. A hyperplane arrangement A is a finite set of hyperplanes in V . We say that A is central if every hyperplane is linear. In this article every arrangement is central unless otherwise specified. In the central cases, we fix a linear form α H ∈ V * such that ker(α H ) = H for each H ∈ A. An ℓ-arrangement is an arrangement in an ℓ-dimensional vector space. Let L(A) := {∩ H∈B H | B ⊂ A} be an intersection lattice. L(A) has a partial order by reverse inclusion, which equips L(A) with a poset structure. For X ∈ L(A), define the localization A X of A at X by A X := {H ∈ A | H ⊃ X}, which is a subarrangement of A. Also, the restriction A X of A onto X is defined by A X := {H ∩ X | H ∈ A \ A X }, which is an *

show abstract

“…3. Those among (4), (5) and (6) follow from the deletion-restriction theorem 2.2. Since χ 0 (A; t) = χ 0 (A ′ ; t) − χ 0 (A H ; t), that between (7) and (8) is easy.…”

Section: Example 37mentioning

confidence: 91%

“…Also, we can show that a lot of recursively free arrangements are divisionally free in Theorem 5. 5. Moreover, as in Theorem 1.4, applications to those related to root systems are also given.…”

Section: Definition 15 (Divisionally Free Arrangements)mentioning

confidence: 96%

Divisionally free arrangements of hyperplanes

Abe

2015

Invent. math.

Self Cite

102

View full text Add to dashboard Cite

show abstract

“…Then, the Ziegler restriction of S k ±Σ onto H z is equal to (A, 2k ± m). [2] for example). Note that Ψ := Φ ∩ Y ⊥ is a (not necessarily irreducible) root system of rank two.…”

Section: Casementioning

confidence: 99%

Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements

Abe

Terao

2015

J Algebr Comb

Self Cite

View full text Add to dashboard Cite

In this article, we prove that the ideal-Shi arrangements are free central arrangements of hyperplanes satisfying the dual partition formula. Then, it immediately follows that there exists a saturated free filtration of the cone of any affine Weyl arrangement such that each subarrangement of the filtration satisfies the dual partition formula. This generalizes the main result in Abe et al. (J. Eur. Math. Soc., to appear) which affirmatively settled a conjecture by Sommers and Tymoczko (Trans. Am. Math. Soc. 358:3493-3509, 2006)

show abstract

“…Let S D S.V / be the symmetric algebra of V and Der K .S/ be the module of derivations A is called free if D.A/ is free. There are a lot of works on the freeness of central arrangements, especially on Coxeter arrangements and the cones over Catalan and Shi arrangements [1][2][3][4][5][6]. For proving the freeness of arrangements, Terao's Addition Theorem [7] provides a standard tool and this theorem leads to the notion of inductively freeness.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we assert that the inductively free arrangement with the required induction table is supersolvable. Finally, for the Coxeter arrangements of type A` 1 and B`, we analyze the supersolvable orders and construct the triangular bases for the logarithmic derivation modules of them.…”

Section: Introductionmentioning

confidence: 99%