Symmetric Functions and Rings of Multinumbers Associated with Finite Groups

In this paper, we introduce ωn-symmetric polynomials associated with the finite group ωn, which consists of roots of unity, and groups of permutations acting on the Cartesian product of Banach spaces ℓ1. These polynomials extend the classical notions of symmetric and supersymmetric polynomials on ℓ1. We explore algebraic bases in the algebra of ωn-symmetric polynomials and derive corresponding generating functions. Building on this foundation, we construct rings of multisets (multinumbers), defined as equivalence classes on the underlying space under the action of ωn-symmetric polynomials, and investigate their fundamental properties. Furthermore, we examine the ring of integer multinumbers associated with the group ωn, proving that it forms an integral domain when n is prime or n=4.