Jack superpolynomials: physical and combinatorial definitions

Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland. They are orthogonal with respect to the scalar product, dubbed physical, that is naturally induced by this quantum-mechanical problem. But Jack superpolynomials can also be defined more combinatorially, starting from the multiplicative bases of symmetric superpolynomials, enforcing orthogonality with respect to a one-parameter deformation of the combinatorial scalar product. Both constructions turns out to be equivalent. This provides strong support for the correctness of the various underlying constructions and for the pivotal role of Jack superpolynomials in the theory of symmetric superpolynomials. 1 [email protected] [email protected] [email protected] To appear in the proceedings of the XIII International Colloquium on Integrable Systems and Quantum Groups, Czech. J . Phys., June 17-19 2004, Doppler Institute, Czech Technical University, ed. by C. Burdik 1 Jack superpolynomials as eigenfunctions of the stCMS model 2 The aim of this contribution is to highlight some aspects of our work [1, 2], focussing on the equivalence between two totally different approaches to the construction of Jack superpolynomials: a physical one, in terms of an eigenvalue problem in supersymmetric quantum mechanics [1] and a more mathematical definition linked to algebraic combinatorics [2], along the lines of [3]. Most references to original works are omitted here and can be found in these quoted articles. Moreover, the presentation is kept at a rather informal level. 1 Jack superpolynomials as eigenfunctions of the stCMS model Jack superpolynomials are eigenfunctions of the supersymmetric extension of the quantum trigonometric Calogero-Moser-Sutherland (stCMS) Hamiltonian, without its ground state contribution. This model describes the interaction of N particles on a unit radius circle, with canonical variables xi and pi = −i∂/∂xi (with xj subsequently replaced by zj = exp(ixj)), together with their fermionic partners, the grassmannian variables θi (with θiθj = −θjθi) and θ † i = ∂/∂θi: H̄ = ∑