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We will explain how some new algebraic solutions of the sixth Painlevé equation arise
from complex reflection groups, thereby extending some results of Hitchin and Dubrovin--
Mazzocco for real reflection groups. The problem of finding explicit formulae for these
solutions will be addressed elsewhere.
We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the -smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.
Here we prove a Poincaré - Verdier duality theorem for the o-minimal sheaf cohomology with definably compact supports of definably normal, definably locally compact spaces in an arbitrary o-minimal structure.
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