Orthogonality of super-Jack polynomials and a Hilbert space interpretation of deformed Calogero–Moser–Sutherland operators
Artikel i vetenskaplig tidskrift, 2019

We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials SPλ((z1, … , zn), (w1, … , wm); θ) with respect to a natural positive semi-definite, but degenerate, Hermitian product ‹·, ·›'n,m,θ. In case m = 0 (or n = 0), our product reduces to Macdonald's well-known inner product ‹·, ·›'n,θ, and we recover his corresponding orthogonality results for the Jack polynomials Pλ((z1, …, zn); θ). From our main results, we readily infer that the kernel of ‹·, ·›'n,m,θ is spanned by the super-Jack polynomials indexed by a partition λ not containing the m × n 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).

81R12 (secondary)

33C52 (primary)


Farrokh Atai

Kobe University

Martin Hallnäs

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

Edwin Langmann

Kungliga Tekniska Högskolan (KTH)

Bulletin of the London Mathematical Society

0024-6093 (ISSN) 1469-2120 (eISSN)

Vol. 51 2 1-18


Algebra och logik


Matematisk analys


Grundläggande vetenskaper



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