Combinatorial Interpretations of Primitivity in the Algebra of Symmetric Functions

Abstract

Given a Hopf algebra with distinguished bases indexed by combinatorial objects, along with a primitive generating set for this algebra, it is natural to consider how we can combinatorially interpret the primitivity of the elements in this generating set, by expanding these generators in terms of the distinguished bases of this algebra and then applying the comultiplication operation to these expansions and constructing sign-reversing involutions that determine the resultant cancellations that we obtain. In this article, we explore this idea, as applied to the power sum generators of the algebra of symmetric functions.