REVIEW 4 minor 1 references
Commutators, matrices and an identity of Copeland
T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Every power $(ba)^n$ in a noncommutative ring is a row-vector times matrix power times column-vector product.
desk verdict A rigorous, original generalization of Copeland's identity to noncommutative rings; the central proof is sound, with two small erratum-level errors. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing identity is Lemma 3.3, $a^i b = \sum_{j=0}^i \binom{i}{j} \operatorname{ad}_a^{i-j}(b) a^j$, which is the noncommutative binomial expansion obtained by writing $\operatorname{ad}_a = L_a - R_a$. On top of it, the proof defines the congruence $A \equiv_k B$ (agreement on the first $m-k+1$ rows) and proves that multiplying by the lower-triangular $U_b$ preserves $k$ while multiplying by $S$ raises it by one; Lemma 3.13 iterates this to get $(U_bS)^n H_1 \equiv_{n+1} H_{(ba)^n}$, and the condition $n<m$ puts row 0 inside the agreed block.
What would settle it
In a ring where $b^2 a^2\neq 0$, take $m=2$ and $n=2$; a direct computation gives $e_0^T (U_b S)^2 H_1 = b(ab-ba)a = baba - bbaa$, whereas $(ba)^2 = baba$, so the identity would force $bbaa=0$, which is not true in a free algebra.
Extended reading notes
Core claim
Theorem 2.7 states: for $n,m$ with $n<m$ and elements $a,b$ of an arbitrary associative unital ring $L$, $$(ba)^n = e_0^T (U_b S)^n H_1,$$ where $S$ is the shift matrix $S_{i,j}=[j=i+1]$, $H_1=(1,a,a^2,\ldots,a^{m-1})^T$, and $U_b$ is the lower-triangular matrix with entries $\binom{i}{j} \operatorname{ad}_a^{i-j}(b)$ for $i\ge j$ and $0$ otherwise. The proof does not require commutativity: it uses the binomial-type identity $a^i b = \sum_j \binom{i}{j} \operatorname{ad}_a^{i-j}(b) a^j$, then tracks how the shift $S$ and the multiplication by $U_b$ act on columns $H_c$. The infinite-matrix case $m=\infty$ is the clean case, where $S H_c = H_{ac}$ and $U_b H_c = H_{bc}$ hold exactly; for finite $m$, these identities fail at the last row, and the proof shows the error moves up one row per multiplication by $S$, so it has not reached the first row before the $n$-th power (since $n<m$).
Load-bearing premise
The proof depends on the power $n$ being strictly smaller than the matrix size $m$: only then has the shift-relation boundary error, which travels one row upward per multiplication, failed to reach the first row.
Editorial extensions
If this is right
- In the Weyl-algebra case $[a,x]=h$ with $[h,a]=[h,x]=0$, the theorem gives $(g(x)a)^n = e_0^T (V_g S)^n H_1$ with $V_g$ built from derivatives $g^{(i-j)}(x) h^{i-j}$, a direct consequence of identifying $U_{g(x)}$ with $V_g$.
- For a fixed pair $a,b$ and a desired exponent $n$, taking $m=n+1$ (or $m=\infty$) always works, so the identity supplies an exact finite matrix computation of $(ba)^n$ without expanding the word.
- No commutativity assumption on $a$ and $b$ is needed, so the identity applies to differential operators, endomorphism rings, and matrix algebras alike.
- The $n<m$ condition is not an artifact of the proof: at $n=m$ the top row can be corrupted, as the example $m=2,n=2$ shows in the falsifier.
Reading between the lines
- The boundary-error mechanism suggests that extending the identity to $n\ge m$ is possible by adding correction terms that compensate the last-row corruption, perhaps by including a $U_b$-dependent boundary operator.
- One can test the identity as a normal-ordering tool: expanding $e_0^T (U_b S)^n H_1$ gives $(ba)^n$ plus terms where the top-row error has not yet arrived; reversing the direction might yield an algorithm for rewriting powers of sums of words.
- A similar construction could apply to longer words like $(abc)^n$ by replacing the single shift $S$ with a multi-shift tensor and building $U$ from iterated commutators of the letters, though the paper does not address this.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves an identity expressing powers of products in noncommutative rings: for any associative unital ring L, elements a,b ∈ L, and integers n < m (where m may be finite or infinite), it shows that (ba)^n = e_0^T (U_b S)^n H_1. Here U_b is a lower-triangular matrix whose entries are binomial coefficients times iterated adjoint actions ad_a^{i-j}(b), S is the shift matrix, and H_1 is the column vector of powers a^i. The proof proceeds by relating the rows of the matrices via a carefully tailored notion of quasi-lower-triangular matrices and a series of equivalence relations ≡_k that control how the finite-size boundary effects propagate. The paper also derives a Weyl-algebraic specialization that recovers a formula of Copeland for differential operators. The main theorem is stated in full generality and verified by a concrete example.
Significance. The paper gives a self-contained, first-principles proof of a clean algebraic identity that generalizes Copeland's formula from a commutative/differential-operator setting to arbitrary noncommutative rings. The proof is detailed and does not assume the identity; it derives it as a special case. The infinite-matrix framework is handled with a precise quasi-lower-triangularity condition, and the finite case is treated by tracking the propagation of boundary effects one row at a time. The result is modest but solid, and the exposition is unusually careful about noncommutative ordering issues. The paper provides a useful reference for matrix representations of iterated commutators and for the algebraic core of Copeland's identity.
minor comments (4)
- [Section 2.3, Proposition 2.3] Proposition 2.3 is false as stated: it claims that any k×ℓ matrix is (ℓ−1)-lower-triangular, but the 1×2 matrix [1 0] is not 1-lower-triangular because its (0,0) entry is nonzero while 0 < 0 + (2−1) = 1. The true statement is that any k×ℓ matrix is k-lower-triangular (with k the number of rows), so the conclusion that finite matrices are quasi-lower-triangular remains correct. Since this proposition is not used in the proof of Theorem 2.7, the main result is unaffected, but the statement should be corrected.
- [Section 2.5, Example 2.8] The displayed square (U_b S)^2 is computed incorrectly for noncommutative rings. In particular, its (1,2) entry should be ad_a(b) b + 2 b ad_a(b), not 3 b ad_a(b); its (2,1) entry should be ad_a^2(b) ad_a(b) + 2 ad_a(b) ad_a^2(b), not 3 ad_a(b) ad_a^2(b); and its (2,2) entry should be ad_a^2(b) b + 4 (ad_a(b))^2, not 4 (ad_a(b))^2 + b ad_a^2(b). The first row is correct, so the subsequent verification of Theorem 2.7 in the example is unaffected, but the displayed matrix should be fixed or explicitly identified as computed in the commutative case.
- [Abstract] The abstract says the matrix is "the n-th power of a matrix with entries binom(i,j) ad_a^{i-j}(b)", but the theorem uses (U_b S)^n, not U_b^n. This wording is imprecise and should be changed to reflect the actual product, e.g., "the n-th power of a matrix built from U_b and the shift S" or by stating the identity explicitly.
- [Section 3.7, proof of Theorem 2.7] The sentence "since n and m are integers" near the end is slightly inaccurate when m = ∞; in that case n+1 ≤ m holds vacuously by the conventions of Section 2.3. The argument is correct, but the wording could be adjusted to say "since n+1 < m+1" or "since n < m implies n+1 ≤ m under the stated conventions."
Circularity Check
No circularity: Theorem 2.7 is proved from the definitions; Copeland's identity is a derived corollary, not an input.
full rationale
The paper's central claim, Theorem 2.7, is derived self-containedly. The proof uses only the definitions of S, U_b, and H_c (equations (2)-(5)), the elementary commutator identity Lemma 3.3, the shift properties in Proposition 3.7, the multiplication rule U_b H_c = H_{bc} in Proposition 3.8, and the congruence propagation Lemmas 3.10-3.12, together with a straightforward induction in Lemma 3.13. No parameter is fitted, no target identity is assumed, and no load-bearing external result is imported. Copeland's formula is not used as an input: Theorem 4.2 is explicitly derived as a special case of Theorem 2.7 after proving U_{g(x)} = V_g in Proposition 4.3. The only citation, [MO337766], is acknowledged as the source of the original formula being generalized, and it plays no role in the proof. Two non-central errata appear: Proposition 2.3 is false as stated (a 1x2 row matrix (1 0) with k=1 and ell=2 is not (ell-1)-lower-triangular), and the displayed square in Example 2.8 has incorrect noncommutative entries off the first row; neither affects the proof of Theorem 2.7, which uses only row 0. Thus there is no significant circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption L is an associative ring with unity, not necessarily commutative.
- standard math Binomial theorem for commuting elements: (x+y)^n = sum (n choose k) x^k y^{n-k} for any commuting x,y in a ring.
- standard math Multiplication of quasi-lower-triangular matrices is well-defined and associative even for infinite matrices.
- domain assumption For the Weyl application (Section 4): [a,x]=h, [h,a]=0, [h,x]=0.
Cite this review
Pith. "Pith review of Commutators, matrices and an identity of Copeland." pith.science (2026). https://pith.science/paper/QKFHQYIO
@misc{pith2026190809179,
author = {Pith},
title = {Pith review of: Commutators, matrices and an identity of Copeland},
year = {2026},
howpublished = {\url{https://pith.science/paper/QKFHQYIO}},
note = {Machine review of arXiv:1908.09179}
}
abstract
Given two elements $a$ and $b$ of a noncommutative ring, we express $\left( ba\right)^n$ as a "row vector times matrix times column vector" product, where the matrix is the $n$-th power of a matrix with entries $\dbinom{i}{j}\operatorname{ad}_a^{i-j}\left( b\right)$. This generalizes a formula by Tom Copeland used in the study of Pascal-style matrices.
Reference graph
Works this paper leans on
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[1]
://mathoverflow.net/questions/337766/
Tom Copeland, MathOverflow question \#337766 ( Expansions of iterated, or nested, derivatives, or vectors -- conjectured matrix computation ). ://mathoverflow.net/questions/337766/
Reviewed August 14, 2026 · model on record in the stance chip above.
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