2018/07/19 by Siddhartha Sahi, Sahi, Siddhartha, Hadi Salmasian +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1807.07340
openalex publication_date 2018/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a finite dimensional unital complex simple Jordan superalgebra J, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra \mathfrak g_\flat≅ \mathfrak g_\flat(-1)⊕\mathfrak g_\flat(0)⊕\mathfrak g_\flat(1), such that \mathfrak g_\flat(-1)≅ J. Set V:=\mathfrak g_\flat(-1)^* and \mathfrak g:=\mathfrak g_\flat(0). In most cases, the space \mathcal P(V) of superpolynomials on V is a completely reducible and multiplicity-free representation of \mathfrak g, with a decomposition \mathcal P(V):=\bigoplusλ∈ΩVλ, where (Vλ)λ∈Ω is a family of irreducible \mathfrak g-modules parametrized by a set of partitions Ω. In these cases, one can define a natural basis (Dλ)λ∈Ω of "Capelli operators" for the algebra PD(V)\mathfrak g. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar cμ(λ) by which Dμ acts on Vλ. We associate a restricted root system \mathitΣ to the symmetric pair (\mathfrak g,\mathfrak k) that corresponds to J, which is either a deformed root system of type A(m,n) or a root system of type Q(n). We prove a necessary and sufficient condition on the structure of \mathitΣ for P(V) to be completely reducible and multiplicity-free. When \mathitΣ satisfies the latter condition we obtain an explicit formula for the eigenvalue cμ(λ), in terms of Sergeev-Veselov's shifted super Jack polynomials when \mathitΣ is of type A(m,n), and Okounkov-Ivanov's factorial Schur Q-polynomials when \mathitΣ is of type Q(n).