Hayashi property for conjugation quandles over finite Coxeter groups

Abstract: In this article, we show that for all finite irreducible Coxeter groups $G$ with a conjugation closed subset $C$, the conjugation quandle $\rm{Conj}(G,C)$ satisfies the Hayashi property.

• The paper demonstrates that conjugation quandles over finite Coxeter groups satisfy the Hayashi property, ensuring that every right translation map is a regular permutation.
• It employs both combinatorial arguments and computational group theory (using GAP) to verify the property across classical families and exceptional types.
• The study extends the analysis to unions of conjugacy classes, revealing deeper structural implications and prompting further group-theoretic classifications.

Hayashi Property for Conjugation Quandles over Finite Coxeter Groups

Introduction and Motivation

This paper investigates the Hayashi property for conjugation quandles constructed from finite irreducible Coxeter groups. Quandles, introduced to encode algebraic invariants of knots and links, form a class of idempotent, right self-distributive, right quasigroups whose automorphisms capture essential symmetries in various algebraic and geometric settings. The profile of a quandle, defined via the cycle types of its right translation maps, is a combinatorial invariant closely tied to both group structure and applications in the study of solutions to the Yang-Baxter equations and invariants in Hopf algebra and knot theory.

The Hayashi property generalizes the Hayashi conjecture, stating that for finite quandles, every right translation map is a regular permutation—meaning all cycle lengths in its decomposition divide the length of the largest cycle. The focus here is on conjugation quandles $\mathrm{Conj}(G, C)$, where $G$ is a finite irreducible Coxeter group and $C$ is a conjugation-closed subset. Establishing the Hayashi property across all such cases resolves an important structural question at the intersection of algebra, combinatorics, and group theory.

Preliminaries: Quandles and the Hayashi Property

Formally, a quandle $(X, \triangleright)$ is a set $X$ with a binary operation satisfying idempotence and the property that each right translation map $R_x: y \mapsto y \triangleright x$ is an automorphism of the structure. The main construction of interest is the conjugation quandle $\mathrm{Conj}(G, C)$, with $C$ a conjugation-closed subset of the group $G$, and operation $x \triangleright y = y x y^{-1}$. The profile of a quandle is the multiset of cycle lengths in the right translation map $G$0 (for $G$1), which for connected quandles becomes uniform across all $G$2.

The Hayashi property asserts that for every right translation map $G$3, all cycle lengths divide the maximal cycle length (i.e., $G$4 is a regular permutation). This property can be characterized combinatorially via group-theoretic intersections: for $G$5, requiring $G$6 suffices for $G$7 to be regular in $G$8.

Results for Finite Coxeter Groups

The core of the paper is devoted to a systematic verification of the Hayashi property for conjugation quandles over all finite irreducible Coxeter groups, including both the classical families ($G$9, $C$0, $C$1, $C$2, $C$3) and the exceptional types ($C$4, $C$5, $C$6, $C$7, $C$8, $C$9).

Symmetric Groups ($(X, \triangleright)$0) and Dihedral Groups ($(X, \triangleright)$1)

For $(X, \triangleright)$2, explicit combinatorial arguments show that for any conjugacy class $(X, \triangleright)$3, there exist $(X, \triangleright)$4 such that $(X, \triangleright)$5 is trivial or central, guaranteeing the regularity of $(X, \triangleright)$6 in $(X, \triangleright)$7. Similarly, dihedral groups $(X, \triangleright)$8, being well-understood, are handled by explicit computations distinguishing between rotational and reflectional conjugacy classes.

Hyperoctahedral and Related Types ($(X, \triangleright)$9, $X$0, $X$1)

Types $X$2 and $X$3 correspond to the wreath product $X$4, and the analysis hinges on the combinatorial data of signed partitions. The order of a conjugation element and the structure of its centralizer in $X$5 is characterized using the cycle product formalism. The key technical results (Lemmas \ref{lc1}, \ref{lc2}, \ref{lc3}) give, for each possible cycle type, explicit witnesses $X$6 certifying the Hayashi property as demanded.

Type $X$7 is addressed by reduction from $X$8 using the subgroup relation and properties of index two subgroups. In all these cases, the arguments are robust for $X$9 (with small values checked computationally).

Exceptional Coxeter Groups ($R_x: y \mapsto y \triangleright x$0)

For the exceptional types, the paper employs computational group theory (using GAP) to conduct exhaustive searches for suitable $R_x: y \mapsto y \triangleright x$1 as witnesses for the property $R_x: y \mapsto y \triangleright x$2. For all cases except $R_x: y \mapsto y \triangleright x$3, this search is computationally trivial; for $R_x: y \mapsto y \triangleright x$4, a more sophisticated subset search suffices. In all cases considered, witnesses are found, confirming the Hayashi property for all exceptional types and all conjugation-closed subsets.

Unions of Conjugacy Classes and Structural Properties

A significant aspect of the work is the extension from single conjugacy classes to arbitrary unions of conjugacy classes $R_x: y \mapsto y \triangleright x$5. The analysis makes crucial use of properties of centralizers and centers of normal subgroups generated by these unions. Lemmas demonstrate that if $R_x: y \mapsto y \triangleright x$6 is regular for each $R_x: y \mapsto y \triangleright x$7 and $R_x: y \mapsto y \triangleright x$8, regularity extends to $R_x: y \mapsto y \triangleright x$9 acting on $\mathrm{Conj}(G, C)$0. The property is further generalized for unions of $\mathrm{Conj}(G, C)$1-conjugate classes.

The paper also catalogs the normal subgroups generated by conjugacy classes in $\mathrm{Conj}(G, C)$2, identifying precisely when the center inclusion needed for regularity holds.

Implications and Directions for Future Research

The principal implication is that all finite Coxeter groups are "good" in the sense of Filip. This sharply delineates the structural class of groups (at least, all direct products of finite irreducible Coxeter groups) where the Hayashi property holds robustly.

The existence of finite groups (such as $\mathrm{Conj}(G, C)$3 with certain conjugation-closed subsets) failing the property asserts that goodness is nontrivial and warrants further group-theoretic and combinatorial classification. The posed open problem is the classification of all finite groups $\mathrm{Conj}(G, C)$4 such that $\mathrm{Conj}(G, C)$5 satisfies the property for all conjugation-closed $\mathrm{Conj}(G, C)$6. This challenges group theorists to uncover new algebraic characterizations, possibly connecting to broader theory around permutation group profiles and quandle automorphisms.

From a practical perspective, the results ensure that algebraic invariants derived from conjugation quandles associated to Coxeter groups retain regular profile properties, facilitating further work in topological, combinatorial, and representation-theoretic applications involving Coxeter symmetries.

Conclusion

The paper establishes that, for all finite irreducible Coxeter groups and all conjugation-closed subsets, the corresponding conjugation quandles satisfy the Hayashi property. The argument combines explicit combinatorial reasoning, group-theoretic classification, and computational verifications, covering both classical and exceptional Coxeter groups, and generalizes to unions of conjugacy classes under verifiable center conditions. This comprehensive structural result solidifies the connection between the algebraic profiles of quandles and the deep symmetries encoded by finite Coxeter groups, with broad implications for algebraic combinatorics and knot theory. Further classification of "good" groups remains an open line of inquiry.