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.