LAUSR

Fix a bounded domain Omega subset of C-n and a positive definite kernel K on Omega, both invariant under G(n), the permutation group on n symbols. Let H subset of Hol(Omega) be the Hilbert module determined by K. We show that H splits into orthogonal direct sum of subspaces PpH indexed by the partitions p of n. We prove that each submodule PpH is a locally free Hilbert module of rank equal to square of the dimension chi(p) (1) of the irreducible representation corresponding to p. Given two partitions p and q, we show that if chi(p)(1) not equal chi(q)(1), then the sub-modules PpH and PqH are not unitarily equivalent. We prove that if H is a contractive analytic Hilbert module on Omega, then the `Taylor joint spectrum of the n-tuple of multiplication operators by elementary symmetric polynomials on PpH is clos (s(Omega)), where s : C-n -> C-n is the symmetrization map. It is then shown that this commuting tuple of operators defines a contractive homomorphism of the ring of symmetric polynomials Cz](Gn) in n variables, equipped with the sup norm on clos (s(Omega)). (C) 2018 Elsevier Inc. All rights reserved.