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Given a chain of groups $G_0 \le G_1 \le G_2 ... $, we may form thecorresponding chain of their representation rings, together with induction andrestriction operators. We may let $\textrm{Res}^l$ denote the operator whichrestricts down $l$ steps, and similarly for $\textrm{Ind}^l$. Observe then that$\textrm{Ind}^l \textrm{Res}^l$ is an operator from any particularrepresentation ring to itself. The central question that this paper addressesis: "What happens if the $\textrm{Ind}^l \textrm{Res}^l$ operator is apolynomial in the $\textrm{Ind} \textrm{Res}$ operator?". We show that chainsof wreath products $\{H^n \rtimes S_n\}_{n \in \mathbb{N}}$ have this property,and in particular, the polynomials that appear in the case of symmetric groupsare the falling factorial polynomials. An application of this fact gives aremarkable new way to compute characters of wreath products (in particularsymmetric groups) using matrix multiplication. We then consider arbitrarychains of groups, and find very rigid constraints that such a chain mustsatisfy in order for $\textrm{Ind}^l \textrm{Res}^l$ to be a polynomial in$\textrm{Ind} \textrm{Res}$. Our rigid constraints justify the intuition thatthis property is indeed a very rare and special property.

Polynomial relations between operators on chains of representation rings