Pizzetti formula on the Grassmannian of 2-planes
type: Item
instance of: scholarly article
title: Pizzetti formula on the Grassmannian of 2-planes
zbMATH Open document ID: 1456.43005
author: Yanyan Li
published in: Annals of Global Analysis and Geometry
publication date: 2020-10-12T00:00:00Z
review text: The authors study the Higgs algebras \(H_3\) in the harmonic analysis of the Grassmann manifolds \(\mathrm{Gr}_0(m,2):=\mathrm{SO}(m)/(\mathrm{SO}(m-2)\times \mathrm{SO}(2))\). These algebras appear in the Howe duality where \(\mathrm{SO}(m)\) acts on \(\mathrm{Gr}_0(m,2)\). In particular, an orthogonal decomposition of the vector space of all poynomials under the joint action of \(\mathrm{SO}(m)\times H_3\) is given. This decomposition is then used to derive a Pizetti formula for integrals of the form \(\int_{\mathrm{Gr}_0(m,2)} f\> d\mu\) with the uniform distribution \(\mu\) on \(\mathrm{Gr}_0(m,2)\). The authors also present a connection to a Pizetti formula for Stiefel manifolds by K.~Coulembier and M.~Kieburg.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 33C55
Mathematics Subject Classification ID: 43A75
zbMATH DE Number: 7259251
zbMATH Keywords: harmonic polynomials
zbMATH Keywords: Grassmann manifolds
zbMATH Keywords: Pizetti formulas
reviewed by: Michael Voit
MaRDI profile type: Publication
full work available at URL: s10455-020-09731-8
OpenAlex ID: W3048324229
cites work: Volumes of compact symmetric spaces
cites work: Some Extensions of Lardner’s Relations between ${}_0 F_3 $ and Bessel Functions
cites work: Models for irreducible representations of Spin\((m)\)
cites work: Pizzetti formulae for Stiefel manifolds and applications
cites work: A higher rank Racah algebra and the $\mathbb{Z}_2^n$ Laplace–Dunkl operator
cites work: Spherical harmonics and integration in superspace
cites work: Higher spin generalisation of Fueter's theorem
cites work: Q3129746
cites work: The Racah algebra as a commutant and Howe duality
cites work: Q3995205
cites work: Pizzetti and Cauchy formulae for higher dimensional surfaces: a distributional approach
cites work: Q3331943
cites work: Dynamical symmetries in a spherical geometry. I
cites work: Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds
cites work: Transcending Classical Invariant Theory
cites work: Spherical harmonics on Grassmannians
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Systems of Singular Integral Operators on Spheres
cites work: Riesz potentials and integral geometry in the space of rectangular matrices
cites work: Invariant differential operators on the Grassmann manifold \(SG_{2,n- 1}(\mathbb{R})\)
cites work: The ``Higgs algebra'' as a ``quantum'' deformation of SU(2)
cites work: EXTREMAL PROJECTORS AND GENERALIZED MICKELSSON ALGEBRAS OVER REDUCTIVE LIE ALGEBRAS
DOI: 10.1007/S10455-020-09731-8
description: scientific article; zbMATH DE number 7259251