We study commutative ring structures on the integral span of rooted trees and
-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
— the
“field” of one element. We also study the base-change homomorphism from
-modules to
-modules
for a field
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
