Super-Jack polynomials: orthogonality and a physical interpretation

Martin Hallnäs, Chalmers University of Technology and University of Gothenburg.

The celebrated Jack symmetric polynomials, depending on a partition and an additional (real) parameter, were introduced some fifty years ago by Henry Jack to interpolate between Schur polynomials and zonal polynomials. We consider a particular generalisation, namely the super-Jack polynomials, introduced by Kerov, Okounkov and Olshanski in 1998, which have appeared in the context of discrete potential theory, (deformed) Calogero-Moser-Sutherland systems as well as beta-ensembles of random matrices. I will present a natural, albeit somewhat nonstandard, generalisation of Macdonald’s well-known orthogonality relations and norms formulae for the Jack polynomials to the super case. Time permitting, I will also sketch how the super-Jack polynomials can be used to construct orthogonal eigenfunctions of a second quantised (deformed) Calogero-Moser-Sutherland operator.

The talk is based on joint work with Farrokh Atai and Edwin Langmann.