Algebras of symmetric holomorphic functions of several complex variables

Abstract

Given a proper holomorphic mapping \(g:\varOmega \subseteq {\mathbb {C}}^{n}\longrightarrow \varOmega ' \subseteq {\mathbb {C}}^{n}\) and an algebra of holomorphic functions \({\mathcal {B}}\) (e.g. \({\mathscr {P}}(K)\) where \(K\subset \varOmega \) is a compact set, \({\mathcal {H}}(U)\), A(U) or \({\mathcal {H}}^{\infty }(U)\) where U is an open and bounded set with \(\overline{U}\subset \varOmega \)), we study the subalgebra \({\mathcal {B}}_{g}\) of all functions compatible with the equivalence relation defined by the proper mapping g. We provide alternative representations of these algebras and describe the fibers in their spectra. Among other examples we relate the algebras of functions that are invariant under permutations and the algebras of functions defined on the symmetrized polydisk.