Polynomial relations between operators on chains of representation rings

Abstract

Given a chain of groups G 0 ≤ G 1 ≤ G 2 ..., we may form the corresponding chain of their representation rings, together with induction and restriction operators. We may let Res l denote the operator which restricts down l steps, and similarly for Ind l . Observe then that Ind l Res l is an operator from any particular representation ring to itself. The central question that this paper addresses is: "What happens if the Ind l Res l operator is a polynomial in the Ind Res operator?". We show that chains of wreath products {H n S n } n∈N have this property, and in particular, the polynomials that appear in the case of symmetric groups are the falling factorial polynomials. An application of this fact gives a remarkable new way to compute characters of wreath products (in particular symmetric groups) using matrix multiplication. We then consider arbitrary chains of groups, and find very rigid constraints that such a chain must satisfy in order for Ind l Res l to be a polynomial in Ind Res. Our rigid constraints justify the intuition that this property is indeed a very rare and special property.