Quadratic Capelli operators and Okounkov polynomials
Quadratic Capelli operators and Okounkov polynomials
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二次 Capelli 算子和 Okounkov 多项式
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Hadi Salmasian
中科院分区：
文献类型：
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作者：
S. Sahi;Hadi Salmasian
设Z是真实的除代数F上r \times r正定Hermite矩阵的对称锥.然后$Z$允许一个自然族不变微分算子-Capelli算子$C_\lambda$ -由长度至多为$r$的分区$\lambda$索引，其特征值由Knop-Sahi插值多项式的特殊化给出.
本文考虑一个双纤维化$Y \longleftarrow X \longrightarrow Z$，其中$Y$是$\mathbb F^n $的$r$维子空间的格拉斯曼算子，$n \geq 2 r $。使用这个，我们构建了一个家庭的不变微分算子$D_{\lambda，s}$上$Y$，我们称之为二次卡佩利算子。我们的主要结果表明，$D_{\lambda，s}$的特征值由Okounkov插值多项式的特殊化给出.
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.