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Register a new userCapelli operators for spherical superharmonics and the Dougall-Ramanujan identity
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Let $(V,omega)$ be an orthosympectic $mathbb Z_2$-graded vector space and let $mathfrak g:=mathfrak{gosp}(V,omega)$ denote the Lie superalgebra of similitudes of $(V,omega)$. When the space $mathscr P(V)$ of superpolynomials on $V$ is emph{not} a completely reducible $mathfrak g$-module, we construct a natural basis $D_lambda$ of Capelli operators for the algebra of $mathfrak g$-invariant superpolynomial superdifferential operators on $V$, where the index set $mathcal P$ is the set of integer partitions of length at most two. We compute the action of the operators $D_lambda$ on maximal indecomposable components of $mathscr P(V)$ explicitly, in terms of Knop-Sahi interpolation polynomials. Our results show that, unlike the cases where $mathscr P(V)$ is completely reducible, the eigenvalues of a subfamily of the $D_lambda$ are emph{not} given by specializing the Knop-Sahi polynomials. Rather, the formulas for these eigenvalues involve suitably regularized forms of these polynomials. In addition, we demonstrate a close relationship between our eigenvalue formulas for this subfamily of Capelli operators and the Dougall-Ramanujan hypergeometric identity. We also transcend our results on the eigenvalues of Capelli operators to the Deligne category $mathsf{Rep}(O_t)$. More precisely, we define categorical Capelli operators ${mathbf D_{t,lambda}}_{lambdainmathcal P}^{}$ that induce morphisms of indecomposable components of symmetric powers of $mathsf V_t$, where $mathsf V_t$ is the generating object of $mathsf{Rep}(O_t)$. We obtain formulas for the eigenvalue polynomials associated to the $left{mathbf D_{t,lambda}right}_{lambdainmathcal P}$ that are analogous to our results for the operators ${D_lambda}_{lambdainmathcal P}^{}$.
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Let $Z$ be the symmetric cone of $r times r$ positive definite Hermitian matrices over a real division algebra $mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_lambda$ -- indexed by partitions $lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y longleftarrow X longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $mathbb F^n $ with $n geq 2r$. Using this we construct a family of invariant differential operators $D_{lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{lambda,s}$ are given by specializations of Okounkov interpolation polynomials.
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We give a simple crystal theoretic interpretation of the Lascouxs expansion of a non-symmetric Cauchy kernel $prod_{i+ jleq n+1}(1-x_iy_j)^{-1}$, which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.
For a finite dimensional unital complex simple Jordan superalgebra $J$, the Tits-Kantor-Koecher construction yields a 3-graded Lie superalgebra $mathfrak g_flatcong mathfrak g_flat(-1)oplusmathfrak g_flat(0)oplusmathfrak g_flat(1)$, such that $mathfrak g_flat(-1)cong J$. 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 $mathcal P(V):=bigoplus_{lambdainOmega}V_lambda$, where $left(V_lambdaright)_{lambdainOmega}$ is a family of irreducible $mathfrak g$-modules parametrized by a set of partitions $Omega$. In these cases, one can define a natural basis $left(D_lambdaright)_{lambdainOmega}$ of Capelli operators for the algebra $mathcal{PD}(V)^{mathfrak g}$. In this paper we complete the solution to the Capelli eigenvalue problem, which is to determine the scalar $c_mu(lambda)$ by which $D_mu$ acts on $V_lambda$. We associate a restricted root system $mathit{Sigma}$ to the symmetric pair $(mathfrak g,mathfrak k)$ that corresponds to $J$, which is either a deformed root system of type $mathsf{A}(m,n)$ or a root system of type $mathsf{Q}(n)$. We prove a necessary and sufficient condition on the structure of $mathit{Sigma}$ for $mathcal{P}(V)$ to be completely reducible and multiplicity-free. When $mathit{Sigma}$ satisfies the latter condition we obtain an explicit formula for the eigenvalue $c_mu(lambda)$, in terms of Sergeev-Veselovs shifted super Jack polynomials when $mathit{Sigma}$ is of type $mathsf{A}(m,n)$, and Okounkov-Ivanovs factorial Schur $Q$-polynomials when $mathit{Sigma}$ is of type $mathsf{Q}(n)$.
We introduce the category of finite strings and study its basic properties. The category is closely related to the augmented simplex category, and it models categories of linear representations. Each lattice of non-crossing partitions arises naturally as a lattice of subobjects.
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