Irreducible representations of conjugacy quandles

Abstract: For $G$ a finite group, we show that the irreducible quandle representations of the conjugacy quandle $Conj(G)$ are the products of what we call characters of $Conj(G)$ by irreducible linear group representations of $G$. The unitary quandle representations can be distinguished. We apply this to the study of the enveloping group of $Conj(G)$ showing that it injects into $G \times \mathbb{Z}<sup>{c_G}$, where $c_G$ is the number of conjugacy classes of $G$. The injection is an isomorphism for $G$ perfect.

• The paper establishes that every irreducible quandle representation is expressed as a product of a quandle character and an irreducible linear group representation.
• It leverages cohomological analysis to prove symmetric 2-cocycles are trivial, eliminating the emergence of exotic projective representations.
• The study details explicit embeddings of enveloping groups into group products, elucidating central exact sequences and implications for unitary representations.

Irreducible Representations of Conjugacy Quandles: Structural Classification and Group Embeddings

Overview

This paper ("Irreducible representations of conjugacy quandles" (2604.22078)) establishes a comprehensive classification of irreducible quandle representations for conjugacy quandles associated to finite groups. Specifically, the study demonstrates that irreducible quandle representations of $Conj(G)$ over $\mathbb{C}$ are always of the form $\chi \cdot \rho$, where $\chi$ is a quandle character and $\rho$ is an irreducible linear representation of $G$. The work further explores the implications for enveloping groups of these conjugacy quandles, providing injective group homomorphisms and characterizing exact sequences, particularly in the case of perfect groups.

Fundamental Definitions and Structural Background

A quandle is a set $Q$ equipped with a binary operation $\triangleright$ satisfying three axioms: idempotency, invertibility, and self-distributivity. The conjugacy quandle $Conj(G)$ associated to a group $G$ is defined by $\mathbb{C}$0, recasting conjugation as a quandle operation.

Quandle representations are morphisms $\mathbb{C}$1 for a vector space $\mathbb{C}$2, where irreducibility is defined analogously to linear representations: the only preserved subspaces are $\mathbb{C}$3 and $\mathbb{C}$4. The group-theoretic notion extends to quandle characters, morphisms from $\mathbb{C}$5 to $\mathbb{C}$6 with the property $\mathbb{C}$7.

The enveloping group $\mathbb{C}$8 of a quandle $\mathbb{C}$9 is formed by adjoining generators corresponding to elements in $\chi \cdot \rho$0 and imposing relations derived from the quandle operation, creating a nontrivial central extension.

Symmetric 2-Cocycles and Their Cohomological Triviality

A significant technical result centers on symmetric 2-cocycles $\chi \cdot \rho$1 for finite groups. The paper shows that every such symmetric cocycle is necessarily a coboundary, using explicit calculations involving extensions and abelianization arguments. This result nullifies the possibility of nontrivial Schur multiplier classes arising from symmetric cocycles, which bears on projective representation lifting.

Classification of Irreducible Quandle Representations

Leveraging the cohomological triviality of symmetric 2-cocycles, the paper demonstrates the following:

• Every irreducible quandle representation $\chi \cdot \rho$2 of $\chi \cdot \rho$3 corresponds to a projective representation $\chi \cdot \rho$4 whose class in $\chi \cdot \rho$5 is represented by a symmetric cocycle.
• Since symmetric cocycles are coboundaries, all such classes are trivial, and hence every irreducible quandle representation lifts to an irreducible linear group representation.
• The paper proves that $\chi \cdot \rho$6, where $\chi \cdot \rho$7 is an irreducible linear representation and $\chi \cdot \rho$8 is a quandle character. The set of irreducible quandle representations is thus parameterized by the product of irreducible linear representations of $\chi \cdot \rho$9 and quandle characters.

This result formalizes the link between the representation theory of the quandle and that of the group, showing that the quandle structure does not introduce new irreducible objects beyond products with characters.

Unitary and Characteristic Representations

The paper notes that the set of unitary quandle representations are precisely those for which both the underlying group representation and the quandle character take values in the appropriate unitary subgroups. This distinction allows precise identification within the class of irreducible quandle representations.

Enveloping Groups of Conjugacy Quandles

The enveloping group $\chi$0 is studied with respect to its central extensions and abelianization:

• $\chi$1 admits a central exact sequence $\chi$2, where $\chi$3 is the free abelian group on conjugacy classes.
• The map $\chi$4 is injective; for perfect groups, it is an isomorphism.
• The torsion subgroup $\chi$5 equals $\chi$6 (the derived subgroup), giving a tight correspondence between group and quandle-theoretic commutator structures.

This embedding demonstrates that enveloping groups of conjugacy quandles retain much of the algebraic structure of $\chi$7, augmented by a free abelian component indexed by conjugacy class count.

Implications and Future Directions

The results yield several practically and theoretically significant implications:

• Representation Theory: The classification confirms that quandle representation theory in the context of conjugacy quandles is fully controlled by group representation theory, after accounting for quandle characters. This closes the possibility for exotic irreducible structures in $\chi$8.
• Cohomology and Extensions: The proof that all symmetric $\chi$9-cocycles for finite groups are coboundaries removes potential obstructions to lifting projective representations, with broader implications for lifting problems in similar algebraic frameworks.
• Group Embeddings: The explicit embedding of enveloping groups into $\rho$0 for finite groups, with isomorphism in the perfect case, clarifies the fundamental algebraic nature of structure groups associated with quandles.
• Unitary Representation Rigidity: The ability to distinguish unitary representations within the framework provides a basis for further study of unitary quandle representations and their applications.

Potential future directions include extending the classification to infinite groups, exploring implications for knot-theoretic invariants derived from quandle representations, and analyzing automorphism groups of conjugacy quandles and their enveloping groups in more depth.

Conclusion

The paper delivers a complete structural description of irreducible quandle representations for conjugacy quandles arising from finite groups, proving their equivalence to the product of quandle characters and irreducible linear group representations. The study’s analysis of symmetric 2-cocycles and enveloping group embeddings resolves foundational representation and extension questions, positioning quandle representation theory in close alignment with classical group representation frameworks. The results lay a robust platform for further algebraic exploration and potential application in related areas.