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Characterizing Transfer Systems for Non-Abelian Groups

Sarah Klanderman, Chloe Lewis, Harlea Monson, Koki Shibata, Danika Van Niel

TL;DR

This work studies $G$-transfer systems for finite non-abelian groups, focusing on width as a combinatorial invariant and providing new width formulas and Hasse diagrams for dihedral, quaternion, dicyclic, and Frobenius groups. The authors employ meet-irreducible subgroups (via the RainbowMRC framework) to determine $w(G)$ and derive explicit expressions: $w(D_{2^m p_1^{i_1} igr ext{...} p_k^{i_k}}) = 2m + 1 + extstyleigl( extstyle ext{sum of } i_ulletigr)$ and $w( ext{Dic}_{2^m p_1^{i_1} igr ext{...} p_k^{i_k}}) = 2m + 2 + extstyleigl( extstyle ext{sum of } i_ulletigr)$, with $w(Q_{2^{m+2}}) = 2m + 2$. The paper also reports initial Frobenius-group results and outlines conjectures on lattice structure and path properties, providing a resource for homotopical combinatorialists to test conjectures and generate new transfer-system lattices for non-abelian groups.

Abstract

For a finite group $G$, the notion of a $G$-transfer system provides homotopy theorists with a combinatorial way to study equivariant objects. In this paper, we focus on the properties of transfer systems for non-abelian groups. We explicitly describe the width of all dihedral groups, quaternion groups, and dicyclic groups. For a given $G$, the set of all $G$-transfer systems forms a poset lattice under inclusion; these are a useful resource to homotopical combinatorialists for detecting patterns and checking conjectures. We expand the suite of known transfer system lattices for non-abelian groups including those which are dihedral, dicyclic, Frobenius, and alternating.

Characterizing Transfer Systems for Non-Abelian Groups

TL;DR

This work studies -transfer systems for finite non-abelian groups, focusing on width as a combinatorial invariant and providing new width formulas and Hasse diagrams for dihedral, quaternion, dicyclic, and Frobenius groups. The authors employ meet-irreducible subgroups (via the RainbowMRC framework) to determine and derive explicit expressions: and , with . The paper also reports initial Frobenius-group results and outlines conjectures on lattice structure and path properties, providing a resource for homotopical combinatorialists to test conjectures and generate new transfer-system lattices for non-abelian groups.

Abstract

For a finite group , the notion of a -transfer system provides homotopy theorists with a combinatorial way to study equivariant objects. In this paper, we focus on the properties of transfer systems for non-abelian groups. We explicitly describe the width of all dihedral groups, quaternion groups, and dicyclic groups. For a given , the set of all -transfer systems forms a poset lattice under inclusion; these are a useful resource to homotopical combinatorialists for detecting patterns and checking conjectures. We expand the suite of known transfer system lattices for non-abelian groups including those which are dihedral, dicyclic, Frobenius, and alternating.

Paper Structure

This paper contains 12 sections, 7 theorems, 21 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgments
  4. Background
  5. Transfer Systems
  6. Conjugacy Axiom Examples
  7. Properties of Transfer Systems
  8. Width of Some Non-Abelian Groups
  9. Dihedral and Quaternion Groups
  10. Dicyclic groups
  11. Frobenius groups
  12. Hasse Diagrams and Future Work

Key Result

Proposition 2.6

Lossless Let $G$ be a lossless group, and $T$ a transfer system on $\mathop{\mathrm{Sub}}\nolimits(G)/G$. The transfer system $T$ lifts to a $G$-transfer system if and only if for all edges $[K] \to [H]$ and any $K' \leq H$ with $[K'] = [K]$ we have the edge $[K \cap K'] \to [H]$ in $T$.

Figures (22)

  • Figure 1: The pentagon $N_5$.
  • Figure 2: Restrictions on the non-modular lattice $\mathop{\mathrm{Sub}}\nolimits(A_4)/A_4$.
  • Figure 3: An example of a $C_{pq}$-transfer system, a non-example of a $D_{p}$-transfer system, and a $D_p$-transfer system.
  • Figure 4: A $F_5$-transfer system with no DTC restrictions.
  • Figure 5: The lattices $\mathop{\mathrm{Sub}}\nolimits(D_{p^n})/D_{p^n}$ for $n=2,3$ and $n$ arbitrary.
  • ...and 17 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 25 more