Descent Algebras , Hyperplane Arrangements , and Shuffling Cards
نویسنده
چکیده
Abstract Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon’s descent algebra and another using random walk on chambers of hyperplane arrangements. These definitions coincide for types A,B,H3, and rank two groups. Both notions satisfy a convolution property and have the same simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when W is crystallographic and the relevant parameter is a good prime. The hyperplane viewpoint suggests deep connections with Lie theory and leads to a notion of riffle shuffling for arbitrary real hyperplane arrangements and oriented matroids. 1991 AMS Subject Classification: 20F55, 20G40
منابع مشابه
Descent algebras, hyperplane arrangements, and shuffling cards. To appear
This note establishes a connection between Solomon’s descent algebras and the theory of hyperplane arrangements. It is shown that card-shuffling measures on Coxeter groups, originally defined in terms of descent algebras, have an elegant combinatorial description in terms of random walk on the chambers of hyperplane arrangements. As a corollary, a positivity conjecture of Fulman is proved.
متن کاملv 4 [ m at h . C O ] 1 5 Ju l 1 99 9 Descent Algebras , Hyperplane Arrangements , and Shuffling Cards
Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon’s descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types A,B,C, H3, and rank two groups. Both notions have the same, simple eigenvalues. The hyperplane definition is especially natural and satisfies a positivity property when W is crystallographic and the...
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This note establishes a connection between Solomon's descent algebras and the theory of hyperplane arrangements. It is shown that card-shu ing measures on Coxeter groups, originally de ned in terms of descent algebras, have an elegant combinatorial description in terms of randomwalk on the chambers of hyperplane arrangements. As a corollary, a positivity conjecture of Fulman is proved. 2
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