Cardinalities in finite monoids of $G$-equivariant functions
Ramón H. Ruiz-Medina
TL;DR
The paper addresses counting problems for the monoid of $G$-equivariant self-maps $End_G(X)$ on a finite $G$-set $X$ by expressing cardinalities in terms of group-action data. It develops a combinatorial framework based on orbit-stabilizer structure, conjugacy classes of subgroups, and a box decomposition indexed by stabilizer types, leading to explicit formulas for $|End_G(X)|$ and its group of units $|Aut_G(X)|$. A key concept is the generating set of $End_G(X)$ given by $Aut_G(X)$ together with fixing elementary collapsings, with precise counting formulas for the number and types of these collapsings. The results enable systematic enumeration of all $G$-equivariant maps and automorphisms using only the stabilizer and conjugacy data, facilitating applications in areas like equivariant topology and combinatorial dynamics. Overall, the work provides concrete, computable expressions that connect the algebraic structure of $End_G(X)$ to the orbit-stabilizer geometry of the $G$-action.
Abstract
A set with a group action is referred to as a $G$-set, and the set of functions that commute with this action forms a monoid under function composition. This paper examines the case where the $G$-set is finite, which implies that the monoid of $G$-equivariant functions is also finite. The document provides formulas for calculating the cardinality of this monoid, its group of units, and explores special cases of $G$-equivariant functions, known as fixing elementary collapsings. All of these results are expressed in terms of specific properties of the $G$-set, including the number of orbits and certain indices of the subgroups acting as stabilizers.