Rotational symmetries in polynomial rings (Q2082629)

The paper develops some basic results on spherical harmonics in a purely algebraic manner, working over a general base ring (instead of the field of real or complex numbers). More concretely, let \(R\) be a commutative ring with an identity (for certain results it is assumed that \(R\) contains the field of rational numbers). Let \(\mathcal{P}_N=R[X_1,\dots,X_N]\) denote the polynomial ring over \(R\) in \(N\ge 2\) indeterminates. The operators \(M_{jk}:=X_j\partial_k-X_k\partial_j\) (\(j\neq k\)) on \(\mathcal{P}_N\) are called \textit{rotation generators}; they span a Lie algebra over \(R\). An element of \(\mathcal{P}_N\) is \textit{harmonic} if it is annihilated by the Laplace operator \(\sum_{i=1}^N\partial_i^2\). Among other things, it is shown that the elements of \(\mathcal{P}_N\) annihilated by all the rotation generators are exactly the polynomials with coefficients in \(R\) of \(X\cdot X:=X_1^2+\cdots+X_N^2\). The \(R\)-linear derivations on \(\mathcal{P}_N\) annihilating \(X\cdot X\) are exactly the \(R[X_1,\dots,X_N]\)-linear combinations of the rotation generators \(M_{jk}\). Given any \(c\in R\), the \(R\)-module \(\mathcal{P}_N\) decomposes as the direct sum of the ideal generated by \(X\cdot X-c\) and the \(R\)-submodule of harmonic polynomials. An explicit \(R\)-module basis of the space of homogeneous harmonic polynomials of degree \(d\) is given. As a replacement of integration over spheres with center at the origin, \textit{spherical means} \(\lambda:\mathcal{P}_N\to R\) are defined algebraically and some explicit formulae are worked out for them, extending their analytic counterpart to the present context.