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

Beers, David; Szczesny, Matt

doi:10.2140/involve.2019.12.1379

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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

DOI: 10.2140/involve.2019.12.1379

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 $F_{1}$ — 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.

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

Vol. 12, No. 8, 2019