tailieunhanh - Báo cáo toán học: "Computation in Coxeter Groups—I. Multiplication"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Computation in Coxeter Groups—I. Multiplication. | Computation in Coxeter Groups I. Multiplication Bill Casselman Mathematics Department University of British Columbia Canada cass@ Abstract. An efficient and purely combinatorial algorithm for calculating products in arbitrary Coxeter groups is presented which combines ideas of Fokko du Cloux and myself. Proofs are largely based on geometry. The algorithm has been implemented in practical Java programs and runs surprisingly quickly. It seems to be good enough in many interesting cases to build the minimal root reflection table of Brink and Howlett which can be used for a more efficient multiplication routine. MR subject classifications 20H15 20-04 Submitted March 28 2001 accepted August 25 2001. A Coxeter group is a pair W S where W is a group generated by elements from its subset S subject to relations st ms t 1 for all s and t in S where a the exponent ms s 1 for each s in S and b for all s t the exponent ms t is either a non-negative integer or 1 indicating no relation . Although there some interesting cases where S is infinite in this paper no harm will be done by assuming S to be finite. Since ms s 1 each s in S is an involution s2 1 for all s 2 S. If we apply this to the other relations we deduce the braid relations st. ts . ms t terms on each side . The array ms t indexed by pairs of elements of S is called a Coxeter matrix. A pair of distinct elements s and t will commute if and only if ms t 2. The labeled graph whose nodes are elements of S with an edge linking non-commuting s and t labeled by ms t is called the associated Coxeter graph. For ms t 3 the labels are often omitted. Coxeter groups are ubiquitous. The symmetry group of a regular geometric figure for example any of the five Platonic solids is a Coxeter group and so is the Weyl group of any Kac-Moody Lie algebra and in particular any finite-dimensional semi-simple Lie algebra . The Weyl groups of finite-dimensional semi-simple Lie algebras are those associated to the finite root systems