2019/12/12 by Siddhartha Sahi, Sahi, Siddhartha, Hadi Salmasian +3
Mathematics · #05E05 #17B10 #33C20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1912.06301
openalex publication_date 2019/12/12 · openalex created_date 2022/07/26 · openalex updated_date 2026/08/01
Let (V,\ω) be an orthosympectic mathbb Z2-graded vector space and\nlet mathfrak g:= mathfrakgosp(V,\ω) denote the Lie superalgebra of\nsimilitudes of (V,\ω). When the space mathscr P(V) of superpolynomials\non V is \not a completely reducible mathfrak g-module, we construct\na natural basis D_\λ of Capelli operators for the algebra of mathfrak\ng-invariant superpolynomial superdifferential operators on V, where the\nindex set mathcal P is the set of integer partitions of length at most two.\nWe compute the action of the operators D_\λ on maximal indecomposable\ncomponents of mathscr P(V) explicitly, in terms of Knop-Sahi interpolation\npolynomials. Our results show that, unlike the cases where mathscr P(V) is\ncompletely reducible, the eigenvalues of a subfamily of the D_\λ are\n\not given by specializing the Knop-Sahi polynomials. Rather, the\nformulas for these eigenvalues involve suitably regularized forms of these\npolynomials. In addition, we demonstrate a close relationship between our\neigenvalue formulas for this subfamily of Capelli operators and the\nDougall-Ramanujan hypergeometric identity.\n We also transcend our results on the eigenvalues of Capelli operators to the\nDeligne category \Rep(Ot). More precisely, we define categorical\nCapelli operators mathbf Dt,\λ \λ\∈ mathcal P that\ninduce morphisms of indecomposable components of symmetric powers of mathsf\nVt, where mathsf Vt is the generating object of \Rep(Ot). We\nobtain formulas for the eigenvalue polynomials associated to the\n \ mathbf Dt,\λ \\λ\∈ mathcal P that are\nanalogous to our results for the operators D_\λ \λ\∈ mathcal\nP.\n