The freeness of ideal subarrangements of Weyl arrangements

Abstract

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. Our proof of the main theorem heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.

Changes are in Section 4. Results are unchanged

References in corpus (1)

• Exponents for B-stable ideals

Cited by in corpus (23)

• Hessenberg varieties and hyperplane arrangements
• Volumes of orthogonal groups and unitary groups
• Signed graphs and the freeness of the Weyl subarrangements of type $B_{\ell }$
• Average four-genus of two-bridge knots
• Root system chip-firing I: Interval-firing
• Plus-one generated and next to free arrangements of hyperplanes
• MAT-free reflection arrangements
• Characteristic quasi-polynomials of ideals and signed graphs of classical root systems
• A survey of recent developments on Hessenberg varieties
• Worpitzky-compatible subarrangements of braid arrangements and cocomparability graphs
• Uniform bases for ideal arrangements
• Multiple addition, deletion and restriction theorems for hyperplane arrangements
• On $A_1^2$ restrictions of Weyl arrangements
• Vines and MAT-labeled graphs
• Inductive and divisional posets
• Solomon-Terao algebra of hyperplane arrangements
• Arrangements of ideal type
• Projective dimension of weakly chordal graphic arrangements
• Hyperpolygonal arrangements
• The largest coefficient of the highest root and the second smallest exponent
• V-systems, holonomy Lie algebras and logarithmic vector fields
• Accurate Arrangements
• The height distribution in root systems