Annals of Combinatorics· 2026Q2
Extendability of the $$B_2$$-Arrangement
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- Q2SCImago
- 2026year
Short summary
Researchers found that for the Coxeter arrangement of type B2, infinitely many multiplicities exist for which no free extension is possible, unlike the A2 type.
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Key points
- The study focuses on free extensions of multiarrangements, specifically the Coxeter arrangement of type B2.
- Unlike the A2 arrangement where free extensions always exist, B2 arrangements have infinite multiplicities without free extensions.
- This result impacts the general understanding of free extensions in higher-rank arrangements.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract Let $$({\mathscr {A}},m)$$ ( A , m ) be a free multiarrangement, and let $${\mathscr {E}}$$ E be an extension of $$({\mathscr {A}},m)$$ ( A , m ) . It is well known that if $${\mathscr {A}}$$ A is the Coxeter arrangement of type $$A_2$$ A 2 , then a free extension of $$({\mathscr {A}},m)$$ ( A , m ) always exists. In this work, we demonstrate that if $${\mathscr {A}}$$ A is the Coxeter arrangement of type $$B_2$$ B 2 , there exist infinitely many multiplicities for which no free extension of $$({\mathscr {A}},m)$$ ( A , m ) exists. This result has immediate consequences for the existence of free extensions in higher rank.
The authors' abstract, as published at the source. Annals of Combinatorics, 2026 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics