A note on the lambda-structure on the Burnside ring

TL;DR

This paper derives an explicit formula for the lambda-operations on the Burnside ring of a finite group, enabling direct computation of these operations on G-sets.

Contribution

It provides the first explicit formula for lambda-operations on the Burnside ring, clarifying their structure and computational aspects.

Findings

Derived an explicit formula for lambda-operations on the Burnside ring

Expressed lambda-operations as linear combinations of G-set classes

Enhanced computational understanding of the Burnside ring's lambda-structure

Abstract

Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a lambda-ring. However, a priori the images of S under the lambda-operations can only be computed implicitly. In this paper we establish an explicit formula, expressing these images as linear combinations of classes of G-sets.