Involve: A Journal of Mathematics

  • Involve
  • Volume 12, Number 8 (2019), 1379-1397.

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

David Beers and Matt Szczesny

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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.

Article information

Involve, Volume 12, Number 8 (2019), 1379-1397.

Received: 9 May 2019
Revised: 18 September 2019
Accepted: 20 September 2019
First available in Project Euclid: 12 December 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E10: Combinatorial aspects of representation theory [See also 20C30] 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80] 16W22: Actions of groups and semigroups; invariant theory 18F30: Grothendieck groups [See also 13D15, 16E20, 19Axx]

field of one element combinatorics rooted trees skew shapes Grothendieck rings


Beers, David; Szczesny, Matt. Split Grothendieck rings of rooted trees and skew shapes via monoid representations. Involve 12 (2019), no. 8, 1379--1397. doi:10.2140/involve.2019.12.1379.

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