#### Vol. 12, No. 8, 2019

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

### David Beers and Matt Szczesny

Vol. 12 (2019), No. 8, 1379–1397
##### 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 $〈t〉$-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.

##### Keywords
field of one element, combinatorics, rooted trees, skew shapes, Grothendieck rings
##### Mathematical Subject Classification 2010
Primary: 05E10, 05E15, 16W22, 18F30
##### Milestones
Revised: 18 September 2019
Accepted: 20 September 2019
Published: 25 October 2019

Communicated by Ravi Vakil
##### Authors
 David Beers Department of Mathematics and Statistics Boston University Boston, MA United States Matt Szczesny Department of Mathematics and Statistics Boston University Boston, MA United States