2018/02/06 by Farrokh Atai, Martin Hallnäs, Edwin Langmann
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Hermitian matrix #Hilbert space #Inner product space #Kernel (algebra) #Mathematical functions and polynomials #Orthogonal polynomials #Orthogonality #Partition (number theory) #Product (mathematics) #Rectangle #math-ph #math.MP #math.QA
paper · pdf · doi:10.1112/blms.12234
published as Bull. Lond. Math. Soc. 51 (2019), no. 2, 353-370 · 19 pages, 1 figure
arxiv created 2018/02/06 · openalex created_date 2018/02/23 · openalex publication_date 2019/02/03 · arxiv updated 2021/01/20 · openalex updated_date 2026/08/05
We prove orthogonality and compute explicitly the (quadratic) norms for super-Jack polynomials S P λ ( ( z 1 , … , z n ) , ( w 1 , … , w m ) ; 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 P λ ( ( z 1 , … , z n ) ; 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 ( m n ) . As an application, we provide a Hilbert space interpretation of the deformed trigonometric Calogero–Moser–Sutherland operators of type A ( n − 1 , m − 1 ) .