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arxiv: 2606.28305 · v1 · pith:7O2AMLHJnew · submitted 2026-06-26 · 🧮 math.CO

Divisible Arm Lengths, Crystal Reflections, and Enumeration of Newly Found Decomposition Columns

David J. Hemmer , Pavel Turek This is my paper

Pith reviewed 2026-06-29 02:57 UTC · model grok-4.3

classification 🧮 math.CO
keywords partitionscrystal reflectionsbalanced partitionsdecomposition matricessymmetric groupsRoCK blockshook lengths
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The pith

Crystal reflections preserve the d-balanced property of partitions, so the number in each e-weight w block is given by an explicit binomial independent of the core.

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

The paper shows that affine crystal reflections on partitions preserve the d-balanced condition, where every hook length divisible by e has arm length divisible by d. This preservation immediately implies that the count of such e-regular partitions depends only on the weight w and not on which e-core labels the block. The authors then evaluate the count by restricting to RoCK blocks, where the enumeration reduces to a binomial coefficient. For the special case of odd primes p with d=2 and e=p the formula specializes to inom{w + (p-3)/2}{w} partitions per block of weight w. The same techniques also yield a generating function for the related odd sequences of partitions inside a block.

Core claim

For integers d,e > 1 the d-balanced e-regular partitions of e-weight w occur in the same number in every block; this common number is given by an explicit binomial formula obtained by first proving that crystal reflections preserve the d-balanced property and then counting inside RoCK blocks. In the representation-theoretic case d=2, e=p odd prime, each p-block of weight w therefore contains exactly inom{w + (p-3)/2}{w} partitions in which every hook of length divisible by p has even arm length.

What carries the argument

The d-balanced property on e-regular partitions together with the action of crystal (affine) reflections that preserve it, allowing reduction of the enumeration to RoCK blocks.

If this is right

  • The number of d-balanced e-regular partitions of weight w is the same in every block.
  • This number equals an explicit binomial coefficient that can be read off from the RoCK case.
  • The same reflection argument applies to the odd sequences of partitions and yields their generating function inside each block.
  • The enumerated partitions label certain columns of the decomposition matrices of symmetric groups in characteristic p.

Where Pith is reading between the lines

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

  • The independence result may extend to other hook-length divisibility conditions beyond arm-length divisibility.
  • The binomial formulas could be used to compare growth rates of these special partitions against the total number of e-regular partitions.
  • The reflection preservation might supply a combinatorial bijection between the special partitions in different blocks.

Load-bearing premise

Crystal reflections preserve the d-balanced property for every pair of integers d and e greater than 1.

What would settle it

An explicit pair d,e>1, a weight w, a d-balanced e-regular partition, and a crystal reflection that produces a partition violating the d-balanced condition.

Figures

Figures reproduced from arXiv: 2606.28305 by David J. Hemmer, Pavel Turek.

Figure 1
Figure 1. Figure 1: Let λ = (8, 5, 4, 2, 1). In the first Young diagram of λ, the nodes with the × symbol form the hook at node (2, 2). In the second Young diagram, these nodes form the rim hook at the same node. The hook length, arm length and leg length of both, this hook and rim hook, are 6, 3 and 2, respectively. Removing the rim hook from the second diagram yields the Young diagram of (8, 3, 1, 1, 1). Since λ has no repe… view at source ↗
Figure 2
Figure 2. Figure 2: Let e = 3 and d = 2. Partition (7, 5, 1) is 2-balanced since its four 3-divisible hooks – hooks at nodes (1, 1), (1, 3), (2, 1) and (2, 3), denoted by × in the diagram – have arm lengths 6, 4, 4 and 2, respectively, all divisible by 2. where we take λi to be zero for i > t. We set βi = λi − i + r, so that β1 > β2 > . . . . The partition may be recovered from the β-set by λi = βi − (r − i). The e-abacus of … view at source ↗
Figure 3
Figure 3. Figure 3: Unreduced 1-signature (bottom to top) +−++−+−, reduces to ++−. The 2-signature + + ++ and 0-signature − − −− are already reduced. Removable nodes corresponding to a “−” in the reduced i-signature are called i-normal, and addable nodes corresponding to a “+” are called i-conormal. In Kleshchev’s language [8], if it exists, the i-normal node corresponding to the leftmost − is called i-good and the i-conormal… view at source ↗
Figure 4
Figure 4. Figure 4: Abacus diagrams illustrating σi for εi(λ) < φi(λ) (left) and εi(λ) > φi(λ) (right). In either case, using the correspondence from Lemma 4.1, ‘new’ e-divisible hooks are created in two different ways. Beads that moved may have empty spaces at lower-numbered positions on their new runner. They also leave vacant positions that may be paired with beads at higher numbered positions on their original runner. The… view at source ↗
Figure 5
Figure 5. Figure 5: The RoCK block 5-core for w = 3, with runners 0, 1, 2, 3, 4 having 3, 6, 9, 12, 15 beads respectively. To prove this theorem we focus on the RoCK block of e-weight w. We represent its e-core on an abacus where runner i ∈ {0, 1, . . . , e − 1} has w(i + 1) beads, all slid up to the top. For example, [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of Lemma 5.2 with a = 4 and t = 2. The arm length is ai + t. Lemma 5.3. Suppose λ is a d-balanced partition in a RoCK block and i ∈ {0, 1, . . . , e − 1}. If d ∤ i then the beads on runner i are all flush to the top of the abacus. Assuming further that λ is also e-regular, the only runners, on which beads may slide down are d, 2d, . . . ,  e−1 d  d. Proof. If any bead on runner i can slide u… view at source ↗
read the original abstract

In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the present paper we enumerate these partitions and prove that each block of $p$-weight $w$ contains precisely \[ \binom{w+\frac{p-3}{2}}{w} \] such partitions. More generally, for any integers $d,e>1$, we study and enumerate $d$-balanced $e$-regular partitions -- partitions for which every hook of length divisible by $e$ has arm length divisible by $d$. Our first main result is that the crystal (affine) reflections preserve the $d$-balanced property for all $d,e > 1$. It follows that, for fixed $d$, $e$, and $w$, the number of $d$-balanced $e$-regular partitions in a block of $e$-weight $w$ is independent of the $e$-core. We then compute this number by working in RoCK blocks, obtaining an explicit binomial formula valid for every block. We also investigate closely related odd sequences of partitions. Among others, we find the generating function of the number of odd sequences occurring in a block. Alongside their representation-theoretic relevance, we expect these results to be of independent combinatorial interest.

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 proves that affine crystal reflections preserve the d-balanced property (every hook of length divisible by e has arm length divisible by d) for all d,e>1. This yields that the number of d-balanced e-regular partitions of e-weight w is independent of the e-core. The count is then computed explicitly via reduction to RoCK blocks, producing a binomial formula; a special case recovers the count inom{w+(p-3)/2}{w} of partitions with even arm lengths on p-divisible hooks. The paper also derives the generating function for the number of odd sequences of partitions occurring in a block and discusses representation-theoretic applications to decomposition matrices.

Significance. If the preservation theorem holds, the work supplies an explicit, core-independent binomial enumeration for a combinatorially defined class of partitions that label columns of symmetric-group decomposition matrices. The reduction to RoCK blocks and the explicit formulas constitute a concrete combinatorial advance; the generating-function result for odd sequences adds further value. These features make the results useful both for modular representation theory and for independent combinatorial study of partitions and crystals.

minor comments (2)
  1. The general binomial formula obtained in RoCK blocks is described as 'explicit' but is not written out in the abstract or introduction; stating the closed form (in terms of d,e,w) would improve immediate readability.
  2. Notation for the d-balanced condition and for odd sequences is introduced without a dedicated preliminary subsection; a short table or list of equivalent formulations would aid readers unfamiliar with the crystal operators.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit preservation proof and reduction to RoCK blocks

full rationale

The paper's central enumeration follows from two independent steps: (1) a direct proof that crystal reflections preserve the d-balanced property (first main result, stated without reference to the target count), yielding core-independence, and (2) explicit binomial computation inside RoCK blocks. No parameter is fitted to data and then relabeled a prediction, no definition is circular (d-balanced is defined directly via hook arm lengths), and the self-citation to prior decomposition-matrix work is used only for context, not as a load-bearing uniqueness theorem or ansatz. The binomial formula is obtained by direct counting after the reduction, not by construction from the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard facts about partitions, hook lengths, e-regularity, and affine crystal actions; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • domain assumption Affine crystal reflections act on the set of e-regular partitions and preserve the d-balanced condition for any d,e > 1.
    Invoked as the first main result to deduce independence from the e-core.
  • domain assumption RoCK blocks exist and allow explicit enumeration of d-balanced partitions for any fixed weight w.
    Used to obtain the binomial formula after reducing to these special blocks.

pith-pipeline@v0.9.1-grok · 5782 in / 1360 out tokens · 72752 ms · 2026-06-29T02:57:40.665255+00:00 · methodology

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