On the algebra of the M\"obius crown

TL;DR

This paper explores how certain algebraic structures over fields relate to representations of categories of finite sets and surjective maps, revealing non-isomorphic algebras can produce identical representations on restricted subcategories.

Contribution

It introduces two non-isomorphic algebras that induce isomorphic representations when restricted to sets of size at most r, highlighting subtle distinctions in algebraic representation theory.

Findings

Two distinct algebras yield the same restricted representation.

Representation restrictions can obscure algebraic differences.

The study advances understanding of algebra-category relationships.

Abstract

A commutative algebra over a field gives rise to a representation of the category of finite sets and surjective maps. We consider the restriction of this representation to the subcategory of sets of cardinality at most $r$. For each $r$, we present two non-isomorphic algebras that give rise to isomorphic representations of this subcategory.