# Split Grothendieck rings of rooted trees and skew shapes via monoid representations

## TL;DR

This paper explores the algebraic structures of rooted trees and skew shapes through monoid representations, revealing connections to Grothendieck rings and graph theory over the field with one element.

## Contribution

It introduces a novel interpretation of ring structures on rooted trees and skew shapes via monoid representations and their Grothendieck rings, linking to graph adjacency matrices.

## Key findings

- Ring structures derived from smash product on monoid representations.
- Interpretation of Grothendieck rings over the field with one element.
- Analysis of base-change homomorphisms and Jordan decompositions.

## Abstract

We study commutative ring structures on the integral span of rooted trees and $n$-dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over $\fun$ - the "field" of one element. We also study the base-change homomorphism from $\mt$-modules to $k[t]$-modules for a field $k$ containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs.