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

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 ${\mathbb{F}}_1$ — the “field” of one element. We also study the base-change homomorphism from $\langle t\rangle$-modules to $k\left[t\right]$-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.