Martina LANINI, Univ. Rome, ``Right cells and singularities of type A Schubert varieties

We describe an algorithm which takes as input any pair of permutations and gives as output two permutations lying in the same right cell. The main feature of such an algorithm is that the Kazhdan-Lusztig variety corresponding to the input pair is isomorphic to the Kazhdan-Lusztig variety of the output pair, hence preserves the singularity type. This implies a negative answer to a question by Borho-Brylinki and Joseph from the Eighties. A negative answer had been already obtained by Williamson and relied on a computer calculation. Our algorithm provides an elementary way to recover Williamson's example and allows to show methodically that there is an infinite family of permutations
for which (a variant of) the property expected by Borho-Brylinki and Joseph fails. This is joint work with P. McNamara.