Polynomial relations between operators on chains of representation rings

TL;DR

This paper investigates when the operator formed by induction and restriction along chains of groups can be expressed as a polynomial in a simpler operator, revealing special properties of wreath product chains and providing new character formulas.

Contribution

It characterizes chains of groups, especially wreath products, where induction-restriction operators are polynomial functions of a basic operator, introducing new computational methods for characters.

Findings

Wreath product chains have induction-restriction operators as polynomials, specifically falling factorials for symmetric groups.

A new method for computing characters of wreath products using matrix multiplication.

Rigid constraints on group chains where the operator is polynomial, indicating this is a rare property.

Abstract

Given a chain of groups $G_0 \le G_1 \le G_2 ... $, we may form the corresponding chain of their representation rings, together with induction and restriction operators. We may let $\textrm{Res}^l$ denote the operator which restricts down $l$ steps, and similarly for $\textrm{Ind}^l$. Observe then that $\textrm{Ind}^l \textrm{Res}^l$ is an operator from any particular representation ring to itself. The central question that this paper addresses is: "What happens if the $\textrm{Ind}^l \textrm{Res}^l$ operator is a polynomial in the $\textrm{Ind} \textrm{Res}$ operator?". We show that chains of 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 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 $\textrm{Ind}^l \textrm{Res}^l$ to be a polynomial in $\textrm{Ind} \textrm{Res}$. Our rigid constraints justify the intuition that this property is indeed a very rare and special property.