NDLI: On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements

Content Provider	Springer Nature Link
Author	Athanasiadis, Christos A. Tzanaki, Eleni
Copyright Year	2006
Abstract	Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice $$\check{Q}$$ , spanning a Euclidean space V. Let m be a positive integer and $${\mathcal A}^{m}_{\Phi}$$ be the arrangement of hyperplanes in V of the form $$(\alpha, x) = k$$ for $$\alpha \in \Phi$$ and $$k = 0, 1,\dots,m$$ . It is known that the number $$N^+ (\Phi, m)$$ of bounded dominant regions of $${\mathcal A}^{m}_{\Phi}$$ is equal to the number of facets of the positive part $$\Delta^m_+ (\Phi)$$ of the generalized cluster complex associated to the pair $$(\Phi, m)$$ by S. Fomin and N. Reading.We define a statistic on the set of bounded dominant regions of $${\mathcal A}^{m}_{\Phi}$$ and conjecture that the corresponding refinement of $$N^+ (\Phi, m)$$ coincides with the $h$-vector of $$\Delta^m_+ (\Phi)$$ . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ, orbits of the action of W on the quotient $$\check{Q} / \, (mh-1) \, \check{Q}$$ and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of $${\mathcal A}^{m}_{\Phi}$$ . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1.
Starting Page	355
Ending Page	375
Page Count	21
File Format	PDF
ISSN	09259899
Journal	Journal of Algebraic Combinatorics
Volume Number	23
Issue Number	4
e-ISSN	15729192
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2006-01-01
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Catalan arrangement Bounded region Generalized cluster complex Positive part h-vector Computer Science Group Theory and Generalizations Order, Lattices, Ordered Algebraic Structures Combinatorics Convex and Discrete Geometry
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Algebra and Number Theory

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