On irreducible representations of quandles

Abstract: We consider irreducible representations of finite quandles over $\mathbb{C}$. For $Q$ a finite quandle whose inner automorphism group $Inn(Q)$ have trivial Schur multipliers, we prove that the irreducible representations of $Q$ can be constructed out of what we call characters of $Q$ and irreducible linear represenations of the group $Inn(Q)$. For $G$ a finite groiup having trivial Schur multiplier or being a Schur cover of another group, we show that the irreducible representations of the conjugacy quandle $Conj(G)$ can be constructed out of characters of $Conj(G)$ and irreducible linear representations of the group $G$. In both cases, the finite unitary irreducible representations can be determined from the results. For instance, these results allow to solve the problem of constucting irreducible represenations of the conjugacy quandles of dihedral groups and generalised quaternion groups. In general, we relate the irreducible representations of a finite quandle $Q$ to irreducible projective representations of $Inn(Q)$ and prove that the irreducible representations of $Q$ can be in theory constructed out of characters of $Q$ and irreducible representations of a finite quotient of the enveloping group $G(Q)$. The quotient is a stem extensions of $Inn(Q)$ with nucleus a finite subgroup of the center of $G(Q)$. This allows, using a result from the litterature, to show that the irreducible quandle representations of $Conj(S_n)$ ($S_n$ the symmetric group) can be constructed out of characters of the corresponding quandle and irreducible linear group representations of the symmetric group.

• The paper establishes a categorical equivalence showing every finite quandle irreducible representation arises from an irreducible representation of its enveloping group via pullback.
• It utilizes Schur multiplier conditions to precisely classify representations, especially in conjugacy quandles from symmetric, dihedral, and quaternion groups.
• The study connects quandle structure with group cohomology by demonstrating how characters and projective representations lift through finite central extensions.

Irreducible Representations of Finite Quandles: Structural Analysis and Classification

Overview

This paper addresses the classification and construction of irreducible representations of finite quandles over $\mathbb{C}$. It establishes deep structural results that relate such representations to characters and to irreducible representations and projective representations of the inner automorphism group, enveloping group, and related central extensions. The analysis is particularly sharp for cases where the Schur multipliers are trivial or when the groups in question are Schur covers. The methodology yields both a conceptual understanding of the interplay between quandle, group, and cohomological data and concrete classification results for families of quandles, including conjugacy quandles associated to classical finite groups.

Definitions and Construction Principles

A quandle is defined as a set $Q$ endowed with a binary operation $\triangleright$ satisfying idempotency, right distributivity, and right invertibility, granting a close relationship with the conjugacy operation in group theory. The canonical construction associates to every finite quandle $Q$ an enveloping group $G(Q)$, generated by $Q$ subject to the relations $xyx^{-1} = x \triangleright y$. This group, together with the associated inner automorphism group $Inn(Q)$, serves as the algebraic base for developing a theory of representations parallel to that for groups.

A (linear) quandle representation is a quandle morphism $Q \rightarrow Conj(GL(V))$. The focus is on irreducible representations, i.e., those with no proper invariant subspaces under the action of $\rho(Q)$.

Irreducible Representations and the Role of Characters

The core result is a categorical equivalence between representations of $Q$0 and representations of $Q$1, meaning every irreducible quandle representation arises from an irreducible group representation of the enveloping group via pullback along the universal morphism $Q$2. Characters of a quandle, defined as quandle morphisms to $Q$3, are shown to correspond to multiplicative characters of $Q$4.

A significant structural theorem is established: If $Q$5 has trivial Schur multiplier, then every irreducible representation of $Q$6 is of the form $Q$7, where $Q$8 is an irreducible linear representation of $Q$9 and $\triangleright$0 is a character of $\triangleright$1. This leads to a full classification in favorable cases.

Conjugacy Quandles and Schur Cover Analysis

When $\triangleright$2 is a conjugacy quandle $\triangleright$3 for a finite group $\triangleright$4, the analysis is sharpened. If $\triangleright$5 is a Schur cover or has trivial Schur multiplier, all irreducible quandle representations are parametrized by characters and irreducible linear representations of $\triangleright$6. The same holds when $\triangleright$7 is a Schur cover and for the group $\triangleright$8.

These results allow explicit constructions for quandles arising from dihedral and generalized quaternion groups, with classification governed by the group-theoretic properties (notably the Schur multipliers).

Projective Representations, Central Extensions, and Infinitesimal Structure

For general finite quandles, irreducible quandle representations induce irreducible projective representations of $\triangleright$9, classified by the subgroup $Q$0 of the Schur multiplier corresponding to classes that can be lifted. A structural theorem asserts: There is a finite central extension $Q$1 of $Q$2 (a stem extension with nucleus given by the torsion subgroup $Q$3 of the center of $Q$4), such that irreducible quandle representations are obtained via pullback from irreducible representations of $Q$5 and characters.

This cements the correspondence between quandle representation theory and the obstruction-theoretic data arising in group cohomology: the ability of a projective representation to lift through $Q$6 detects which projective representations of $Q$7 can be realized by quandle representations.

Explicit Calculations and Case Studies

The classification results are rendered explicit for several families:

• For $Q$8, with $Q$9 the symmetric group, the result from the literature that $G(Q)$0 has center of trivial torsion implies $G(Q)$1, even when $G(Q)$2 for $G(Q)$3. Consequently, all irreducible quandle representations of $G(Q)$4 arise from linear representations and characters.
• For dihedral and generalized quaternion groups ($G(Q)$5, $G(Q)$6), the torsion and Schur multiplier computations provide a classification regime: when the Schur multiplier is trivial, every irreducible quandle representation is realized by characters and group representations; when nontrivial, $G(Q)$7 detects precisely which projective classes can be lifted.

A table summarizes these results for the cases $G(Q)$8 and $G(Q)$9 as $Q$0 varies, detailing the structure of $Q$1, its Schur multiplier, the subgroup $Q$2, and the torsion in the center of the enveloping group.

Implications and Future Directions

These structural results elucidate the precise relationship between elementary quandle-theoretic data and the finer aspects of group extension and cohomology theory. The practical upshot is a complete recipe for constructing and enumerating irreducible quandle representations in regimes where the Schur multiplier is controlled or fully understood. The analysis foregrounds the central role of characters and of lifting projective representations, suggesting natural directions in the study of more general algebraic structures, including racks and their associated cohomological constructions.

From a theoretical perspective, the results set a groundwork for further extensions—for instance, examining representations over more general fields, or exploring connections with topological and categorical quantum invariants, particularly where quandles function as algebraic encodings of symmetry via knot and braid theoretical applications.

Conclusion

The paper offers a rigorous, detailed bridge from the combinatorial structure of finite quandles to the intricacies of group and projective representation theory, anchored via enveloping groups, central extensions, and Schur multipliers. It provides a precise and exhaustive classification in key families, highlighting the interplay between internal quandle operations, the associated automorphism groups, and the cohomological obstructions that mediate between projective and linear representation theories. This foundational approach furnishes both specific tools for computation and broad conceptual insight for subsequent algebraic and topological explorations.