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 F1 — the “field” of one element. We also study the base-change homomorphism from t-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.

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Keywords
field of one element, combinatorics, rooted trees, skew shapes, Grothendieck rings
Mathematical Subject Classification 2010
Primary: 05E10, 05E15, 16W22, 18F30
Milestones
Received: 9 May 2019
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