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Most computer algebra packages for Weyl groups use generators and relations and the Weyl group elements are expressed as reduced words in the generators. This representation is not unique and leads to computational problems. In [HHR06], the authors introduce the representation of Weyl group elements uniquely as signed permutations. This representation is useful for the study of symmetric spaces and their representations.
A computer algebra package enabling one to do computations related to symmetric spaces would be an important tool for researchers in many areas of mathematics, including representation theory, Harish Chandra modules, singularity theory, differential and algebraic geometry, mathematical physics, character sheaves, Lie theory, etc. In this paper, we use the representation of Weyl group elements as signed permutations to improve the algorithms of [DH05]. These algorithms compute the fine structure of symmetric spaces and nice bases for local symmetric spaces.