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{\bf Bill Casselman}
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{\bf Computation in Coxeter Groups---I. Multiplication}
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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.

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