Plethysm for characters of relative operads and PROPs

TL;DR

This paper develops new algebraic operations on symmetric functions to study the representation theory of relative operads and props, with applications to automorphism groups and cohomology.

Contribution

It introduces the relative plethysm and box product, extending classical character maps to new algebraic structures in operad theory.

Findings

Defined relative plethysm and box product operations.

Computed characters for automorphism groups of free groups.

Applied to Albanese cohomology of IA-automorphism groups.

Abstract

We investigate the relationship between symmetric functions and the representation theory of operads, relative operads, and props. We extend the classical character map for symmetric sequences to relative bisymmetric sequences and symmetric bimodules. We introduce new operations on symmetric functions, the relative plethysm and the (connected) box product, which model via the character map the composition product of relative operads and the box product of prop(erad)s. As applications, we include the computation of characters for stable twisted cohomology of automorphism groups of free groups and the Albanese cohomology of the IA-automorphism group.