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- W1659185503 abstract "AbstractOur main theorem asserts that the ideal-Shi arrangements are freecentral arrangements of hyperplanes. In the special case of the conesof the Shi (or Catalan) arrangements, our main theorem had beenconjectured by Edelman and Reiner in [5] before it was proved byYoshinaga in [11]. As a corollary, we prove that there exists a linearorder of hyperplanes in the cone of any affine Weyl arrangement suchthat every initial segment makes a free arrangement satisfying thedual-partition theorem. Thisgeneralizes the main theorem in [1] whichaffirmatively settled a conjecture by Sommers and Tymoczko [9]. 1 Introduction Let Φ be an irreducible crystallographic root system of rank l over the realnumber field R. Let ∆ = {α 1 ,...,α l } be a simple system of Φ and Φ + the corresponding positive system. An ideal I ⊆ Φ + is a set such that ifα ∈ I, β ∈ Φ + with α − β ∈P li=1 Z ≥0 α i , then β ∈ I. For any subset Σ ofΦ + , define A(Σ) := {H α | α ∈ Σ}, where H α is the hyperplane perpendicularto α in the l-dimensional Euclidean space.Let V be the (l + 1)-dimensional Euclidean space with a basis ∆ ∪ {z},where z is a unit vector perpendicular to each α" @default.
- W1659185503 created "2016-06-24" @default.
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- W1659185503 date "2014-05-25" @default.
- W1659185503 modified "2023-09-27" @default.
- W1659185503 title "The freeness of ideal-Shi arrangements and free paths in affine Weyl arrangements" @default.
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