Singularities of Schubert varieties within a right cell

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We describe an algorithm which takes as input any pair of
permutations and gives as output two permutations lying in the same
Kazhdan-Lusztig right cell. There is an isomorphism between the
Richardson varieties corresponding to the two pairs of permutations
which preserves the singularity type. This fact has applications in the
study of W-graphs for symmetric groups, as well as in finding examples
of reducible associated varieties of sln-highest weight modules, and
comparing various bases of irreducible representations of the symmetric
group or its Hecke algebra. This is joint work with Peter McNamara.