REVIEW 1 major objections 3 minor 12 references
An isoperimetric inequality for mean shadow
T0 review · 1 major / 3 minor · reviewed 2026-07-12 · grok-4.5
Pith's one-line read Spin and non-spin characters of the alternating double cover become proportional after reduction modulo 2 precisely when their labels are 4-stepped-and-semicongruent partitions of a rigid form.
desk verdict Clean Clifford reduction that finishes the p=2 mixed spin/non-spin classification for Ân, but the stated proportionality constant is off by a factor of 4 when α is even. 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
Clifford restriction from Ŝn to Ân together with the already-classified proportional pairs for the symmetric double cover; the argument reduces the alternating case to the symmetric case by comparing how self-conjugate partitions and even strict partitions split, then checks the remaining diagonal-hook and parity conditions that force r=s=0 when the characters are self-conjugate.
What would settle it
Exhibit a pair of partitions λ and α that do not satisfy the stated 4-stepped-and-semicongruent form yet whose corresponding characters of Ân still become proportional after reduction modulo 2, or verify that a pair claimed by the theorem fails to be proportional on some odd-order class.
Extended reading notes
Core claim
An irreducible non-spin character of Ân labelled by a partition λ is proportional to one (or both) of the irreducible spin characters labelled by a strict partition α if and only if λ equals the 2-core-and-quotient form [κ_a;(κ_r,κ_s)] and α equals the corresponding strict partition κ̄_a ⊔ 2(κ_r+κ_s), with either r eq s (in which case both characters are self-conjugate and the spin one is a concrete power of 2 times the non-spin one) or r=s=0 (in which case the two conjugate pairs coincide).
Load-bearing premise
The entire classification stands or falls with the earlier complete list of proportional pairs for the double cover of the symmetric group; if that list is incomplete, the alternating statement inherits the same gaps.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper classifies pairs consisting of an irreducible non-spin character φ of the double cover Ân (labelled by a partition λ) and an irreducible spin character ψ of Ân (labelled by a strict partition α) whose 2-modular reductions are proportional. Equivalently, it classifies when such characters are proportional on elements of odd order. Theorem 1.1 states that this occurs precisely when λ = [κ_a; (κ_r, κ_s)] and α = κ̄_a ⊔ 2(κ_r + κ_s) for nonnegative integers a, r, s, with either r ≠ s (in which case φ = φ^c and the common spin character equals a stated power of 2 times φ) or r = s = 0 (in which case the two conjugate pairs coincide). The argument reduces to the corresponding classification for Ŝn via restriction and Clifford theory, using two short lemmas that relate modular proportionality on Ŝn and Ân and that convert constants of proportionality between the two groups.
Significance. The result cleanly completes the p = 2 spin/non-spin case for the alternating double covers, complementing the author’s joint work with Fayers for the symmetric double covers and the earlier classifications of proportional modular reductions for pairs of spin characters or pairs of non-spin characters. The deduction is short, transparent, and correctly isolates the extra case analysis (self-conjugacy of λ, parity of α, and the location of irrational values) that is needed beyond the Ŝn theorem. The dependence on the published Ŝn classification and on the classical restriction rules of Hoffman–Humphreys and James–Kerber is explicit and appropriate. Once the arithmetic constant is corrected, the paper is a solid, self-contained contribution of the expected length and depth for the subject.
major comments (1)
- [§3, proof of Theorem 1.1 (and the statement of Theorem 1.1 itself)] In the final paragraph of the proof of Theorem 1.1 (§3), the Ŝn result supplies ⟨α⟩ = 2^k χ_λ with k = ⌊max{r,s}/2⌋, hence ⟨α⟩↓Ân = 2^k φ when r ≠ s. When max is even (so α even) one has ⟨α⟩↓ = [α] + [α]^c and the two summands agree on all 2'-classes, so each equals (1/2)⟨α⟩↓ = 2^{k-1} φ. The text nevertheless asserts “[α] = [α]^c = 2^{⌊max/2⌋+1} φ as required,” and Theorem 1.1 encodes the same erroneous exponent via the formula 2^{⌊max{r,s}/2⌋+1-ε(α)}. The factor-of-four discrepancy is an internal arithmetic slip (the 1/2 was inverted to a factor of 2). The “if and only if” classification of partitions survives unchanged, but the explicit constant stated in the theorem is false for every even α arising in the r ≠ s case and must be corrected (e.g., to 2^{⌊max{r,s}/2⌋-1+ε(α)}).
minor comments (3)
- [Title page and abstract] The extracted manuscript text contains numerous spurious spaces and hyphenations inside words (e.g., “AL TERNA TING”, “PROPOR TIONAL”, “CHARAC TERISTIC2”). These are almost certainly PDF-extraction artefacts, but the production version should be checked for clean typesetting of the title, abstract and section headings.
- [Theorem 1.1] In the statement of Theorem 1.1 the two cases are written with nested “either riangleright riangleright” bullets; a slightly more conventional “Case 1 / Case 2” layout would improve readability without changing content.
- [Lemma 3.2] Lemma 3.2 is used only in one direction; a parenthetical remark that the converse fails (already illustrated by Remark 3.3) could be moved into the lemma statement itself for clarity.
Circularity Check
Minor self-citation of independent Sn classification; An result requires genuine extra case analysis and is not definitional.
-
self citation load bearing
[Proof of Theorem 1.1, first paragraph of §3]
"Then Lemma 3.2 implies that χλ is proportional to ⟨α⟩, and so by [FMcD25, Theorem 1.1] we have λ=[κa;(κr,κs)] and α=κ̄a⊔2(κr+κs) for some nonnegative integers a,r and s."
The only source of the explicit form of the partitions is the authors' own prior classification for Sn. While that prior result is independent and published, the present paper's 'if and only if' statement inherits its entire combinatorial content from the self-citation; the remaining argument merely transfers the proportionality via restriction.
full rationale
The paper's central claim (Theorem 1.1) is obtained by applying the classical Clifford/restriction rules for characters of Ân (Hoffman–Humphreys, James–Kerber) to the already-published Sn classification of Fayers–McDowell [FMcD25, Theorem 1.1]. Lemmas 3.1–3.2 and the subsequent case analysis on self-conjugacy of λ and parity of α are independent of that citation; they do not reduce the An statement to a tautology of the Sn statement. The cited Sn theorem is an external, peer-reviewed result (Annals of Representation Theory 2025) whose hypotheses do not include the present An claim, so the self-citation is ordinary scholarly dependence rather than circularity. No fitted parameters, self-definitional loops, or uniqueness theorems imported solely from the authors appear. Score 2 reflects only the presence of a non-load-bearing self-citation of a co-authored companion paper.
Assumptions & free parameters
assumptions (3)
- domain assumption Fayers–McDowell classification of proportional spin/non-spin 2-modular characters of the double cover of Sn (Theorem 1.1 of the 2025 Annals of Representation Theory paper).
- standard math Classical restriction and conjugacy rules for non-spin and spin characters of Ân (James–Kerber, Hoffman–Humphreys): self-conjugate partitions and even strict partitions split into conjugate pairs, and the pairs differ only on specific cycle types.
- standard math Elements of ÂSn outside ÂAn have even order, so 2-modular Brauer characters are determined by restriction to ÂAn (Lemma 3.1).
Cite this review
Pith. "Pith review of An isoperimetric inequality for mean shadow." pith.science (2026). https://pith.science/paper/MTEQHN73
@misc{pith2026260610608,
author = {Pith},
title = {Pith review of: An isoperimetric inequality for mean shadow},
year = {2026},
howpublished = {\url{https://pith.science/paper/MTEQHN73}},
note = {Machine review of arXiv:2606.10608}
}
read the original abstract
We prove an isoperimetric inequality in terms of the mean shadow area.
Reference graph
Works this paper leans on
-
[1]
Bogachev
V. Bogachev. Measure theory, Volume I . Springer, Berlin Heidelberg, 2007
2007
-
[2]
I. Laba. Recent progress on Favard length estimates for planar Cantor sets , in: Operator-Related Function Theory and Time-Frequency Analysis, Proceedings of the 2012 Abel Symposium, K. Grochenig, Y. Lyubarskii, K. Seip, eds., Springer 2015, pp. 117-145
2012
-
[3]
D. D a browski. Favard length and quantitative rectifiability. arXiv:2408.03919 [math.CA] (2024)
arXiv 2024
-
[4]
P. M. Gruber. Convex and Discrete Geometry , volume 336 of Grundlehren der mathematischen Wissenschaften . Springer Verlag, 2007
2007
-
[5]
I. Newton. Philosophiae naturalis principia mathematica . (London: Streater) 1687
-
[6]
Plakhov and V
A. Plakhov and V. Roshchina. Invisibility in billiards . Nonlinearity 24 , 847-854 (2011)
2011
-
[7]
A. Plakhov. Exterior billiards. Systems with impacts outside bounded domains . Springer, New York, 2012. xiv+284 pp. ISBN: 978-1-4614-4480-0
2012
-
[8]
A. Plakhov. The problem of camouflaging via mirror reflections . Proc. R. Soc. A 473 : 20170147. Published electronically (2017)
2017
Show all 12 references
-
[9]
Plakhov and V
A. Plakhov and V. Roshchina. The problem of optimal camouflaging . SIAM J. Math. Anal. 57 , 95-117 (2025)
2025
-
[10]
A. Plakhov. Plane sets invisible in finitely many directions . Nonlinearity 31 , 3914-3938 (2018)
2018
-
[11]
Fractal bodies invisible in 2 and 3 directions
A Plakhov and V Roshchina. Fractal bodies invisible in 2 and 3 directions . Discr. Contin. Dynam. Syst.-A 33 , 1615-1631 (2013)
2013
-
[12]
Bodies with mirror surface invisible from two points
A Plakhov and V Roshchina. Bodies with mirror surface invisible from two points. Nonlinearity 27 , 1193-1203 (2014)
2014
Reviewed July 12, 2026 · model on record in the stance chip above.
Discussion (0). Sign in to comment.