Polynomial relations between operators on chains of representation rings

Abstract

Given a chain of groups we may form the corresponding chain of their representation rings, together with induction and restriction operators. Let denote the operator which restricts down l steps, and similarly for Observe that is an operator from any particular representation ring to itself. We provide explicit rigid constraints that a group of chain with surjective restriction operators satisfying the polynomial property must obey. The central question that this paper addresses is: “What happens if the operator is a polynomial in the operator?” It is well known that chains of wreath products have this property. In this paper, we deduce rigid numerical constraints any arbitrary chains of groups with surjective restriction operator and the aforementioned polynomial property must satisfy. These polynomials can be obtained from the orders of the groups, and are uniquely parametrized by two integers. We deduce the desired parametrization from deducing character theoretic properties of group chains with the desired polynomials.