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arxiv: 2604.12619 · v1 · submitted 2026-04-14 · 🧮 math.CO

Noncommutative Abel-like identities

Darij Grinberg This is my paper

Pith reviewed 2026-05-10 15:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords noncommutative ringsAbel-Hurwitz identitiessubset sumsordered tuplescentral elementscombinatorial identitiesnoncommutative algebra
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The pith

In a noncommutative ring with X plus Y central, three Abel-like summation identities equate subset power expressions to sums over ordered distinct tuples of the x elements times powers of X plus Y.

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

This paper generalizes the Abel-Hurwitz identities to noncommutative rings. For a finite set V of size n, a noncommutative ring L, elements x_s in L, and X, Y in L with X plus Y in the center of L, it proves three summation identities. The first equates the sum over all subsets S of (X plus x of S) to the power |S| times (Y minus x of S) to the power n minus |S| with a sum over all distinct ordered k-tuples of the x elements times (X plus Y) to the power n minus k. The second and third insert an extra X factor on the left and simplify to (X plus Y) to the n and (X plus Y minus x of V) times (X plus Y) to n minus 1. A reader would care because these identities let one manipulate sums in algebras where multiplication order matters, such as matrices or operators, without losing the combinatorial structure of the classical versions.

Core claim

Let V be a finite set of size n, and let L be any noncommutative ring. For each s in V, let x_s in L. Set x(S) to be the sum of x_s over s in S. Let X and Y be elements of L such that X plus Y lies in the center of L. Then the sum over subsets S of (X plus x(S)) to |S| times (Y minus x(S)) to n minus |S| equals the sum over distinct ordered tuples i1 to ik of (X plus Y) to n minus k times the product x_i1 to x_ik, and the two variants that multiply the left side by an extra X equal (X plus Y) to the n and (X plus Y minus x(V)) times (X plus Y) to n minus 1.

What carries the argument

The three summation identities that equate subset-based power sums to ordered-tuple products of the x elements, made possible by the centrality of X plus Y.

If this is right

  • The identities hold in every ring satisfying the centrality condition on X plus Y.
  • When the ring is commutative the identities reduce directly to the classical Abel-Hurwitz formulas.
  • The second identity shows that one modified subset sum always equals the pure power (X plus Y) to the n.
  • The third identity gives an exact factorization for a slightly adjusted sum that isolates the factor (X plus Y minus x of V).
  • Any negative powers appearing in the statements are cancelled by positive powers from other factors in the same term.

Where Pith is reading between the lines

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

  • The same technique might produce identities for other generating functions built from noncommuting variables.
  • The identities could be checked directly in matrix algebras or quaternion algebras to see how they behave in concrete noncommutative examples.
  • They suggest a route to noncommutative versions of other classical binomial or multinomial identities by replacing ordinary powers with suitably ordered expressions.

Load-bearing premise

X plus Y must lie in the center of the ring so that it commutes with every x_s and allows terms to rearrange freely in the power expansions.

What would settle it

Pick the ring of 2 by 2 real matrices, choose concrete X, Y, and x_s such that X plus Y commutes with all elements, compute both sides of the first identity for n equals 3, and check whether the numerical values match.

read the original abstract

We generalize the Abel--Hurwitz identities to an almost entirely noncommutative setting. Namely, let $V$ be a finite set of size $n$, and let $\mathbb{L}$ be any noncommutative ring. For each $s\in V$, let $x_{s}\in\mathbb{L}$. Set $x\left( S\right) :=\sum_{s\in S}x_{s}$ for any $S\subseteq V$. Let $X$ and $Y$ be two elements of $\mathbb{L}$ such that $X+Y$ lies in the center of $\mathbb{L}$. Then, we show that% \begin{align*} & \sum_{S\subseteq V}\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert }\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\sum_{\substack{i_{1},i_{2},\ldots,i_{k}\in V\text{ distinct}% }}\left( X+Y\right) ^{n-k}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}};\\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert }=\left( X+Y\right) ^{n};\\ & \sum_{S\subseteq V}X\left( X+x\left( S\right) \right) ^{\left\vert S\right\vert -1}\left( Y-x\left( S\right) \right) ^{n-\left\vert S\right\vert -1}\left( Y-x\left( V\right) \right) =\left( X+Y-x\left( V\right) \right) \left( X+Y\right) ^{n-1}. \end{align*} (Negative powers are understood to be cancelled by other factors.)

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 / 3 minor

Summary. The manuscript generalizes the classical Abel-Hurwitz identities to noncommutative rings. For a finite set V of cardinality n, a noncommutative ring L, elements x_s in L, and elements X, Y in L such that X+Y is central, it establishes three summation identities. The first equates the sum over all subsets S of (X + x(S))^{|S|} (Y - x(S))^{n-|S|} to a sum over distinct ordered tuples of the x_i's multiplied by (X+Y)^{n-k}. The second, with an inserted factor of X and exponent |S|-1 on the first term, equals (X+Y)^n. The third, with an additional (Y - x(V)) factor, equals (X+Y - x(V))(X+Y)^{n-1}. Negative exponents are understood to cancel against other factors in each term.

Significance. If the identities hold, the work supplies a direct algebraic extension of a classical combinatorial identity to the noncommutative setting, with the centrality hypothesis serving as the sole mechanism for rearranging terms. The derivation proceeds from the ring axioms without free parameters, fitted constants, or self-referential definitions, and the right-hand sides are expressed explicitly in the input elements. This constitutes a clean, falsifiable algebraic statement that may support further noncommutative generalizations.

minor comments (3)
  1. [Abstract] Abstract, displayed equations: the summation index on the right-hand side of the first identity is described only as 'distinct ordered tuples'; an explicit notation (e.g., sum over injections or over 1 ≤ k ≤ n and distinct i1,...,ik) would remove any ambiguity about whether the tuples are ordered or the sum includes the empty product for k=0.
  2. [Abstract] Abstract: a one-sentence recall of the classical (commutative) Abel-Hurwitz identity would help readers situate the noncommutative generalization.
  3. [Abstract] The manuscript should state explicitly whether the identities are intended to hold in rings with or without identity, since the cancellation of negative powers implicitly relies on the existence of a multiplicative identity when |S|=0.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful summary of our results and for the positive assessment of their significance as a direct algebraic extension of the classical identities. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves the three summation identities by direct algebraic manipulation in a noncommutative ring, relying only on the ring axioms, the definition of subset sums x(S), and the centrality of X+Y to allow free rearrangement of factors. The proofs (likely by induction on n or by expanding products and collecting terms) do not invoke self-citations for uniqueness, do not rename fitted parameters as predictions, and do not define any quantity in terms of the claimed result. The negative-power convention is explicitly reduced to ordinary multiplication by cancellation in each term. The derivation is therefore self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard axioms of associative rings together with the domain assumption that X+Y is central; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math L is an associative ring (possibly noncommutative) with the usual addition and multiplication operations.
    Invoked throughout the statement of the identities and the subset sums.
  • domain assumption X + Y lies in the center of L, so (X + Y) z = z (X + Y) for every z in L.
    Explicitly required in the abstract for the identities to hold; this cancels noncommuting terms during expansion.

pith-pipeline@v0.9.0 · 5658 in / 1474 out tokens · 69799 ms · 2026-05-10T15:14:37.400626+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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