On irreducible representations of quandles

Mohamad Maassarani

arxiv: 2604.12550 · v1 · submitted 2026-04-14 · 🧮 math.RT

On irreducible representations of quandles

Mohamad Maassarani This is my paper

Pith reviewed 2026-05-10 14:31 UTC · model grok-4.3

classification 🧮 math.RT

keywords quandlesirreducible representationsSchur multiplierinner automorphism groupconjugacy quandleprojective representationsenveloping group

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The pith

Irreducible representations of finite quandles are constructed from quandle characters and linear representations of their inner automorphism groups when the Schur multiplier is trivial.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for a finite quandle Q whose inner automorphism group Inn(Q) has trivial Schur multiplier, the irreducible representations of Q over the complex numbers arise by combining characters of Q with the ordinary irreducible linear representations of Inn(Q). It gives a parallel construction for the conjugacy quandle of a finite group G under the same multiplier condition or when G is a Schur cover. In the general case without the multiplier hypothesis, the representations of Q correspond to projective representations of Inn(Q) and can still be built from characters of Q together with linear representations of a suitable finite quotient of the enveloping group G(Q). These results supply explicit constructions for the conjugacy quandles of dihedral groups, generalized quaternion groups, and symmetric groups.

Core claim

For Q a finite quandle whose inner automorphism group Inn(Q) has trivial Schur multipliers, the irreducible representations of Q can be constructed out of characters of Q and irreducible linear representations of the group Inn(Q). For G a finite group having trivial Schur multiplier or being a Schur cover, the irreducible representations of the conjugacy quandle Conj(G) can be constructed out of characters of Conj(G) and irreducible linear representations of G. In general, the irreducible representations of a finite quandle Q relate to irreducible projective representations of Inn(Q) and can be constructed from characters of Q and irreducible representations of a finite quotient of the envol

What carries the argument

Characters of the quandle combined with irreducible linear representations of its inner automorphism group Inn(Q), which generate the quandle representations precisely when Inn(Q) has trivial Schur multiplier.

If this is right

The finite unitary irreducible representations of such quandles can be determined explicitly.
Irreducible representations of conjugacy quandles of dihedral groups and generalized quaternion groups can be constructed in full.
Irreducible representations of the conjugacy quandle of the symmetric group S_n can be constructed from characters of the quandle and linear representations of S_n.
In general, irreducible quandle representations correspond to projective irreducible representations of the inner automorphism group.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The stem-extension quotient of the enveloping group offers a route to linearize representations even when the Schur multiplier is non-trivial.
The same character-plus-linear-rep method may apply to other families of finite quandles once their inner automorphism groups are shown to satisfy the multiplier condition.
Explicit lists obtained this way could be used to compare representation-theoretic invariants across different quandle structures.

Load-bearing premise

The inner automorphism group of the quandle has a trivial Schur multiplier.

What would settle it

Explicit computation of all irreducible representations of the conjugacy quandle of a small dihedral group, followed by direct comparison against the list obtained by combining the quandle characters with the linear irreps of its inner automorphism group.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reduces quandle irreps to quandle characters plus linear irreps of Inn(Q) when the Schur multiplier is trivial, with explicit applications to Conj(S_n) and dihedral groups.

read the letter

The main takeaway is that for finite quandles where Inn(Q) has trivial Schur multiplier, the irreducible representations over C can be built from a new notion of quandle characters together with ordinary linear representations of Inn(Q). The same reduction applies to conjugacy quandles Conj(G) when G itself satisfies the condition or is a Schur cover. This gives concrete constructions for families like dihedral groups, generalized quaternion groups, and Conj(S_n) that were previously open problems in the literature the abstract cites. The general case without the multiplier hypothesis is handled by moving to projective representations of Inn(Q) and a stem extension of the enveloping group G(Q), which matches standard facts on central extensions. That part is consistent and does not introduce circularity or fitting parameters. The work is useful because it turns abstract existence questions into explicit recipes once the characters are computed. The definitions of quandle characters and the handling of unitary representations look workable on the surface, though the proofs will need checking for any hidden assumptions about finiteness or the enveloping group. The Schur multiplier hypothesis is a real limitation rather than a minor caveat, but the paper states it clearly and provides the projective fallback. This is for readers already working on quandle representations or knot invariants that rely on them. Someone computing concrete examples will get direct value; a generalist in representation theory might skim the applications. The paper deserves a serious referee because the reductions are new in this framing and the applications are falsifiable. I would send it to peer review with requests to expand the character definitions and add a small computational check for one of the families.

Referee Report

2 major / 2 minor

Summary. The paper studies irreducible representations of finite quandles over ℂ. For a finite quandle Q whose inner automorphism group Inn(Q) has trivial Schur multiplier, the irreducible representations of Q are constructed from characters of Q together with irreducible linear representations of Inn(Q). Analogous results are given for conjugacy quandles Conj(G) when G has trivial Schur multiplier or is a Schur cover. In general, quandle irreps are related to projective representations of Inn(Q) and constructed from quandle characters and linear representations of a stem extension (quotient of the enveloping group G(Q)) whose nucleus is a finite central subgroup. Concrete applications are given for dihedral groups, generalized quaternion groups, and Conj(S_n).

Significance. If the derivations hold, the results supply an explicit bridge between quandle representation theory and ordinary group representation theory via Schur multipliers and stem extensions. The conditional statements on trivial Schur multipliers allow direct use of linear representations, while the general case reduces the problem to a finite central extension whose representations are in principle accessible. The applications to Conj(D_n), Conj(Q_{2^n}), and Conj(S_n) are concrete and falsifiable.

major comments (2)

[Definition of characters of Q (prior to main theorems)] The definition and basic properties of the newly introduced 'characters of Q' are load-bearing for all stated theorems; the manuscript must contain a self-contained section proving that these characters classify the relevant 1-dimensional representations or cocycles before they are used to parametrize the irreps of Q.
[General reduction to stem extensions] In the general case, the precise stem extension of Inn(Q) (the quotient of G(Q) with nucleus in Z(G(Q))) must be constructed explicitly or shown to be the universal central extension; without this, the claim that all irreps arise from linear representations of this finite quotient remains formal.

minor comments (2)

[Abstract] Abstract contains multiple typographical errors ('groiup', 'represenations', 'constucting', 'litterature') that should be corrected.
[Introduction and notation] Notation for the enveloping group G(Q) and its relation to Inn(Q) should be introduced once and used consistently; cross-references to standard facts on Schur multipliers would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their thorough review and constructive feedback on our manuscript. We respond to each major comment below, specifying the changes we will implement in the revised version.

read point-by-point responses

Referee: [Definition of characters of Q (prior to main theorems)] The definition and basic properties of the newly introduced 'characters of Q' are load-bearing for all stated theorems; the manuscript must contain a self-contained section proving that these characters classify the relevant 1-dimensional representations or cocycles before they are used to parametrize the irreps of Q.

Authors: We agree that the characters of Q require a self-contained treatment to support the main results. In the revised manuscript, we will introduce a new section dedicated to the definition and properties of quandle characters. This section will prove that the characters of Q classify the 1-dimensional representations of the quandle and the associated cocycles on the inner automorphism group, prior to their application in the theorems on irreducible representations. revision: yes
Referee: [General reduction to stem extensions] In the general case, the precise stem extension of Inn(Q) (the quotient of G(Q) with nucleus in Z(G(Q))) must be constructed explicitly or shown to be the universal central extension; without this, the claim that all irreps arise from linear representations of this finite quotient remains formal.

Authors: We acknowledge that the general reduction requires more explicit detail to avoid being formal. We will revise the relevant section to provide an explicit construction of the stem extension as the quotient of the enveloping group G(Q) by its commutator subgroup intersected with the center. We will further show that this extension is the universal central extension of Inn(Q), ensuring that the irreducible representations of Q are indeed obtained from the linear representations of this finite quotient in a concrete manner. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external representation theory

full rationale

The paper presents conditional mathematical proofs: under the hypothesis that Inn(Q) has trivial Schur multiplier, irreducible quandle representations are constructed from characters of Q and linear irreps of Inn(Q) via standard facts on central extensions, projective representations, and stem extensions of groups. The general case reduces to irreps of a quotient of the enveloping group G(Q), again using established theory rather than internal fitting or self-definition. Specific applications (e.g., Conj(S_n)) invoke an external literature result. No equations or steps in the derivation chain reduce the claimed outputs to quantities defined by construction from the paper's own inputs or self-citations; the argument is self-contained against external benchmarks in group representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims depend on the standard axioms of representation theory over ℂ, the definition of quandle characters introduced in the paper, and the domain assumption of trivial Schur multipliers; no free parameters or new entities with external falsifiable handles are introduced.

axioms (2)

standard math Representation theory of finite groups and algebras over ℂ is well-defined and semisimple when applicable.
Invoked for the existence and classification of irreducible representations.
domain assumption Inn(Q) has trivial Schur multiplier.
Explicit hypothesis required for the direct construction from linear representations.

invented entities (1)

Characters of Q no independent evidence
purpose: Auxiliary objects used to construct the irreducible representations of the quandle.
Introduced in the paper as part of the construction; no independent external evidence supplied.

pith-pipeline@v0.9.0 · 5585 in / 1360 out tokens · 21901 ms · 2026-05-10T14:31:09.056227+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Clarendon Press, 1987

Karpilovsky, Gregory, The Schur Multiplier. Clarendon Press, 1987

work page 1987
[2]

Fundamenta Mathematicae, vol

Eisermann, Michael, Quandle Coverings and Their Galois Correspondence. Fundamenta Mathematicae, vol. 225, no. 1, pp. 103–67, 2014

work page 2014
[3]

and Moutuou, E

Elhamdadi, M. and Moutuou, E. kaïoum M.. Finitely stable racks and rack representations. Communications in Algebra, 46(11), 4787–4802, 2018

work page 2018
[4]

arxiv, 2022

Jacques Darné, Nilpotent quandles. arxiv, 2022

work page 2022
[5]

Lebed Victoria, Conjugation groups and structure groups of quandles, arxiv

work page
[6]

, Journal of Algebra and Its Applications, 2026

Rodr\' i guez-Nieto, Jos\' e Gregorio and Salazar-D\' i az, Olga Patricia and Vallejos-Cifuentes, Ricardo Esteban and Vel\' a squez, Ra\' u l, Rack representations and connections with groups representations. , Journal of Algebra and Its Applications, 2026

work page 2026
[7]

arxiv, 2026

Maassarani Mohamad, Groups and quandles. arxiv, 2026

work page 2026

[1] [1]

Clarendon Press, 1987

Karpilovsky, Gregory, The Schur Multiplier. Clarendon Press, 1987

work page 1987

[2] [2]

Fundamenta Mathematicae, vol

Eisermann, Michael, Quandle Coverings and Their Galois Correspondence. Fundamenta Mathematicae, vol. 225, no. 1, pp. 103–67, 2014

work page 2014

[3] [3]

and Moutuou, E

Elhamdadi, M. and Moutuou, E. kaïoum M.. Finitely stable racks and rack representations. Communications in Algebra, 46(11), 4787–4802, 2018

work page 2018

[4] [4]

arxiv, 2022

Jacques Darné, Nilpotent quandles. arxiv, 2022

work page 2022

[5] [5]

Lebed Victoria, Conjugation groups and structure groups of quandles, arxiv

work page

[6] [6]

, Journal of Algebra and Its Applications, 2026

Rodr\' i guez-Nieto, Jos\' e Gregorio and Salazar-D\' i az, Olga Patricia and Vallejos-Cifuentes, Ricardo Esteban and Vel\' a squez, Ra\' u l, Rack representations and connections with groups representations. , Journal of Algebra and Its Applications, 2026

work page 2026

[7] [7]

arxiv, 2026

Maassarani Mohamad, Groups and quandles. arxiv, 2026

work page 2026