The harmonic transvector algebra in two vector variables

2015/10/22 by De Bie, Hendrik, Eelbode, David, Roels, Matthias · 2 citations
#17B10 #30G35 #42B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1510.06566

Abstract

The decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the Howe duality theorem and leads to the study of spherical harmonics. The aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components. To do so, a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical Howe duality. The results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated Stiefel manifold.

Cited by

• Dirac reduction algebra
• Exact Solutions to the Klein--Gordon Equation via Reduction Algebras

Related

• Separation of variables in the semistable range
• The Weyl Algebra, Spherical Harmonics, and Hahn Polynomials
• A Lie algebra of Grassmannian Dirac operators and vector variables
• Harmonics on the Quantum Euclidean Space Related to the Quantum Orthogonal Group
• Orthogonal polynomials through the invariant theory of binary forms