Parabolic Hilbert schemes via the Dunkl-Opdam subalgebra

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In this note we give an alternative presentation of the rational
Cherednik algebra H_c corresponding to the permutation representation of
S_n. As an application, we give an explicit combinatorial basis for all
standard and simple modules if the denominator of c is at least n, and
describe the action of H_c in this basis. We also give a basis for the
irreducible quotient of the polynomial representation and compare it to
the basis of fixed points in the homology of the parabolic Hilbert
scheme of points on the plane curve singularity {x^n=y^m}. This is a
joint work with José Simental and Monica Vazirani.