On minimal bases in homotopical combinatorics

TL;DR

This paper introduces minimal bases for $N_$ operads in homotopical combinatorics, using closure operators and combinatorial invariants to advance computational methods for finite groups.

Contribution

It establishes the existence of minimal bases for $N_$ operads and develops new combinatorial invariants for finite groups.

Findings

Existence of minimal bases for $N_$ operads.

Introduction of novel combinatorial invariants for finite groups.

Enhanced computational framework for $N_$ operads.

Abstract

We present a development in the computational suite for the study of $N_\infty$ operads for a finite group $G$. This progress is achieved using the simple yet powerful observation that Rubin's generation algorithm can be interpreted as a closure operator. Leveraging this perspective, we establish the existence of minimal bases for $N_\infty$ operads. By investigating these bases for certain families of groups we are led to introduce and analyze several novel combinatorial invariants for finite groups.