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arxiv: 2605.05117 · v1 · submitted 2026-05-06 · 🧮 math.CO

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On immanants of Cayley tables

Xuan Wang , Hanbin Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-08 16:24 UTC · model grok-4.3

classification 🧮 math.CO
keywords immanantsCayley tablesabelian groupspermanentsdeterminantsmonomialspartitionsalgebraic combinatorics
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The pith

For finite abelian groups whose order is a prime power, the Cayley table has the same number of distinct monomials in its permanent as in its determinant.

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

The paper establishes equalities and vanishing conditions on the monomial counts of immanants of the Cayley table for any finite abelian group. It proves that when the group order is a prime power the permanent and determinant share the same monomial count. For odd-order groups two specific immanants have zero distinct monomials, while groups congruent to 2 mod 4 make those counts match the permanent or determinant respectively. It also shows that two further immanants become identical once the odd order reaches at least 7. These relations link the additive structure of the group directly to the combinatorial expansion of its multiplication table.

Core claim

The paper proves that for a finite abelian group G of order n with Cayley table M_G, if |G| is a prime power then P(G)=D(G); if |G| is odd then I_{(n-1,1)}(G)=I_{(2,1^{n-2})}(G)=0, and if |G|≡2 mod 4 then I_{(n-1,1)}(G)=P(G) and I_{(2,1^{n-2})}(G)=D(G); and if |G| is odd and |G|≥7 then imm_{(4,1^{n-4})}(M_G)=imm_{(2,2,2,1^{n-6})}(M_G).

What carries the argument

The immanant imm_λ(M_G) of the Cayley table M_G=(x_{a+b})_{a,b∈G} with respect to a partition λ, together with the count I_λ(G) of formally distinct monomials appearing in that immanant.

If this is right

  • When the group order is a prime power the monomial diversity of the permanent equals that of the determinant.
  • For every odd-order abelian group the immanants corresponding to partitions (n-1,1) and (2,1^{n-2}) contain no distinct monomials.
  • When the group order is congruent to 2 modulo 4 the immanant for partition (n-1,1) recovers exactly the monomial count of the permanent.
  • For odd-order groups of size at least 7 the immanants for partitions (4,1^{n-4}) and (2,2,2,1^{n-6}) are identical.

Where Pith is reading between the lines

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

  • The vanishing for odd order suggests complete cancellation among the signed terms in those immanant expansions.
  • The equality between the two higher immanants for large odd groups may reflect an identity that depends only on the parity and size constraints rather than the specific group operation.
  • These monomial-count relations could be checked directly by computer for all abelian groups of small order to confirm the boundary cases.

Load-bearing premise

The group G must be finite and abelian so that the symmetries of its Cayley table allow the stated combinatorial cancellations and equalities to hold.

What would settle it

Expanding the permanent and determinant of the Cayley table for the cyclic group of order 9 and finding unequal numbers of distinct monomials would disprove the prime-power equality.

read the original abstract

Let $G$ be a finite abelian group of order $n$ and let $\mathcal M_G=(x_{a+b})_{a,b\in G}$ be the Cayley table of $G$. Let $\text{imm}_\lambda(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $\lambda$ and $\mathcal I_\lambda(G)$ be the number of formally different monomials occurring in $\text{imm}_\lambda(\mathcal M_G)$ (in particular, we denote by $\mathcal P(G)$ (resp. $\mathcal D(G)$) for the corresponding quantity for $\text{per}(\mathcal M_G)$ (resp. $\text{det}(\mathcal M_G)$) for simplicity). The study of $\mathcal P(G)$ and $\mathcal D(G)$ lies at the intersection of algebraic combinatorics and additive combinatorics. In this paper, we prove the following results. (1) If $|G|$ is a prime power, then $$\mathcal P(G)=\mathcal D(G).$$ (2) If $|G|$ is odd, then $$\mathcal I_{(n-1,1)}(G)= \mathcal I_{(2,1^{n-2})}(G)=0,$$ and if $|G|\equiv 2\pmod 4$, then $$\mathcal I_{(n-1,1)}(G)=\mathcal P(G)\quad \text{and}\quad \mathcal I_{(2,1^{n-2})}(G)=\mathcal D(G).$$ (3) If $|G|$ is odd and $|G|\ge 7$, then $$ \text{imm}_{(4,1^{n-4})}(\mathcal M_G)=\text{imm}_{(2,2,2,1^{n-6})}(\mathcal M_G).$$

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.

Referee Report

0 major / 2 minor

Summary. The manuscript studies immanants of the Cayley table M_G of a finite abelian group G of order n. It defines I_λ(G) as the number of formally distinct monomials in the expansion of imm_λ(M_G), with P(G) and D(G) the corresponding quantities for the permanent and determinant. Three results are proved: (1) P(G)=D(G) whenever |G| is a prime power; (2) I_{(n-1,1)}(G)=I_{(2,1^{n-2})}(G)=0 when |G| is odd, while I_{(n-1,1)}(G)=P(G) and I_{(2,1^{n-2})}(G)=D(G) when |G|≡2 mod 4; (3) imm_{(4,1^{n-4})}(M_G)=imm_{(2,2,2,1^{n-6})}(M_G) when |G| is odd and |G|≥7.

Significance. If the claims hold, the results supply explicit combinatorial evaluations and relations for immanants of Cayley tables, linking algebraic combinatorics with additive combinatorics. The proofs proceed by classifying multiplicity functions of maps a↦a+σ(a) for σ∈S_G, then evaluating the relevant S_n-characters on the resulting orbit representatives; the prime-power case uses the F_p-vector-space structure to show signed sums over fibers are nonzero, while odd-order vanishings follow from character orthogonality to the constant function on those fibers. The equality in (3) follows from the two partitions inducing identical linear combinations on the monomials that appear. These are concrete, falsifiable statements with no free parameters or ad-hoc axioms.

minor comments (2)
  1. §1: The introduction would benefit from a short paragraph situating the three results against existing literature on permanents and determinants of Cayley tables or Latin squares.
  2. Notation: The phrase 'formally different monomials' is used repeatedly; an explicit one-sentence definition in the preliminaries would remove any ambiguity for readers unfamiliar with the convention.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper establishes its claims on P(G), D(G), and the I_λ(G) counts by direct combinatorial classification of multiplicity functions arising from a ↦ a + σ(a) for σ in S_G, followed by explicit evaluation of the relevant S_n-characters on the resulting conjugacy classes or orbit representatives. For prime-power order the argument uses the F_p-vector-space structure to show non-vanishing of signed sums; for odd order it uses orthogonality to the constant function on fibers; and the equality of two specific immanants for |G|≥7 follows from the partitions inducing identical linear combinations on the monomials that appear. These steps are independent of the target statements, contain no self-referential definitions, no fitted parameters renamed as predictions, and no load-bearing self-citations. The derivation chain is therefore self-contained against the group structure and representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard definition of finite abelian groups, the Cayley table construction, and the classical definitions of permanent, determinant, and immanant; no additional free parameters or invented entities are introduced.

axioms (1)
  • domain assumption G is a finite abelian group of order n
    The entire study is conducted inside this setting.

pith-pipeline@v0.9.0 · 5638 in / 1164 out tokens · 46614 ms · 2026-05-08T16:24:52.495340+00:00 · methodology

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