GENERALIZED CATALAN NUMBERS, WEYL GROUPS AND ARRANGEMENTS OF HYPERPLANES

Abstract

For an irreducible, crystallographic root system Φ in a Euclidean space V and a positive integer m, the arrangement of hyperplanes in V given by the affine equations (α, x) = k, for α ∈ Φ and k = 0, 1, …, m, is denoted here by A Φ m . The characteristic polynomial of A Φ m is related in the paper to that of the Coxeter arrangement AΦ (corresponding to m = 0), and the number of regions into which the fundamental chamber of AΦ is dissected by the hyperplanes of A Φ m is deduced to be equal to the product ∏ i = 1 l ( e i + m h + 1 ) / ( e i + 1 ) , where e1, e2, …, el are the exponents of Φ and h is the Coxeter number. A similar formula for the number of bounded regions follows. Applications to the enumeration of antichains in the root poset of Φ are included. 2000 Mathematics Subject Classification 20F55 (primary), 05A15, 52C35 (secondary).