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Authors |
F. Atai Martin Hallnäs E. Langmann |
---|---|
Published in | Bulletin of the London Mathematical Society |
Volume | 51 |
Issue | 2 |
Pages | 353-370 |
ISSN | 0024-6093 |
Publication year | 2019 |
Published at |
Department of Mathematical Sciences |
Pages | 353-370 |
Language | en |
Links |
dx.doi.org/10.1112/blms.12234 |
Keywords | integrability, Mathematics |
Subject categories | Geometry |
We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials SP lambda((z1, horizontal ellipsis ,zn),(w1, horizontal ellipsis ,wm);theta) with respect to a natural positive semi-definite, but degenerate, Hermitian product ⟨center dot,center dot⟩n,m,theta '. In case m=0 (or n=0), our product reduces to Macdonald's well-known inner product ⟨center dot,center dot⟩n,theta ', and we recover his corresponding orthogonality results for the Jack polynomials P lambda((z1, horizontal ellipsis ,zn);theta). From our main results, we readily infer that the kernel of ⟨center dot,center dot⟩n,m,theta ' is spanned by the super-Jack polynomials indexed by a partition lambda not containing the mxn rectangle (mn). As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero-Moser-Sutherland operators of type A(n-1,m-1).