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arxiv: 2510.17239 · v2 · submitted 2025-10-20 · 🧮 math.CO · math.AG

Bounded core partitions and Borel-Weil-Bott

Fern Gossow , Andrew Huchala This is my paper

Pith reviewed 2026-05-18 06:38 UTC · model grok-4.3

classification 🧮 math.CO math.AG
keywords bounded core partitionsBorel-Weil-Bott theoremHodge numbersGrassmannianhook-product statisticsemistandard tableauxNakano vanishingq-analogue
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The pith

Bounded core partitions yield explicit formulas for Hodge numbers on Grassmannians through the Borel-Weil-Bott decomposition.

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

The paper applies the Borel-Weil-Bott theorem to decompose the cohomology of twisted sheaves of holomorphic forms on the complex Grassmannian into irreducible representations of the general linear group. From this decomposition the authors extract two practical counting formulas for the associated Hodge numbers, one given by a new integer hook-product statistic on bounded partitions and the other by a count of semistandard tableaux. They also restate the condition for positivity of these numbers as the existence of partitions with specified properties, supply a combinatorial proof of the Nakano vanishing theorem by mapping core partitions to plane partitions, and produce a q-deformed version of the formulas.

Core claim

Analyzing the Borel-Weil-Bott decomposition of the cohomology provides two effective formulae for the Hodge numbers: a novel integer-valued hook-product statistic on bounded partitions and a count based on semistandard tableaux. This approach reformulates the positivity of the Hodge numbers as the existence of a partition satisfying certain properties, yields a combinatorial proof of the Nakano vanishing theorem for the Grassmannian via a map from core partitions to plane partitions, and extends the computation to a q-analogue.

What carries the argument

A novel integer-valued hook-product statistic on bounded partitions that directly computes the Hodge numbers from the representation-theoretic data supplied by Borel-Weil-Bott.

If this is right

  • Hodge numbers of twisted holomorphic forms on the Grassmannian equal the hook-product statistic evaluated on bounded partitions.
  • The numbers are positive precisely when a partition with the required properties exists.
  • Nakano vanishing for the Grassmannian follows from the existence of a map sending core partitions to plane partitions.
  • The same combinatorial counts admit a q-deformation that refines the ordinary Hodge numbers.

Where Pith is reading between the lines

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

  • The same partition-to-plane-partition map might be used to prove vanishing results on other homogeneous spaces.
  • The q-analogue could be compared with known deformations arising from quantum cohomology of the Grassmannian.
  • Improved bounds on the positivity range may help decide non-vanishing questions for related invariants such as Chern classes.

Load-bearing premise

The Borel-Weil-Bott theorem decomposes the cohomology in a way that isolates the Hodge numbers through counts on bounded core partitions and tableaux.

What would settle it

Direct computation of the Hodge numbers for the Grassmannian Gr(2,5) with a small twist by other means, followed by comparison to the value produced by the hook-product statistic on the corresponding bounded partitions.

Figures

Figures reproduced from arXiv: 2510.17239 by Andrew Huchala, Fern Gossow.

Figure 1
Figure 1. Figure 1: A pair of partitions related by the bijection between 3-core partitions to 2-bounded partitions. The 3-interior of λ is shaded, and the hook lengths labeled. Let ℓ be the number of unique entries in λ, and let r1, . . . , rℓ be the multiplicities of the entries in the order they appear. Similarly, let s1, . . . , sℓ be the multiplicities for λ T. On the partition diagram, these values are the heights and w… view at source ↗
Figure 2
Figure 2. Figure 2: If λ = (81 , 5 3 , 3 3 ) as above, then (r1, r2, r3) = (1, 3, 3) and (s1, s2, s3) = (3, 2, 3). For 1 ≤ p, q ≤ ℓ, we let Bp,q denote the block in row p and column q of this coarsening, which has size |Bp,q| = rpsq. A block is called internal if p + q ≤ ℓ, diagonal if p + q = ℓ + 1 and external if p + q ≥ ℓ + 2. We observe that the hook lengths of boxes within a block form a contiguous subset of Z. 2.2. Gras… view at source ↗
Figure 3
Figure 3. Figure 3: If λ is the above 9-core partition where ελ(9) has been shaded, then (α1, α2, α3) = (1, 2, 4) and (β1, β2, β3) = (1, 3, 4). For example, s ′ 2 = s2 + s3 = 2 + 1 = 3 is the number of columns in the second block of ελ(9). When i + j = N, we prove that any knijt-partition is a staircase of uniform rectangular blocks. Moreover, t will be the semiperimeter of each block when i > 0 (see [PITH_FULL_IMAGE:figures… view at source ↗
Figure 4
Figure 4. Figure 4: All knijt-partitions with k = 8, n = 12 and i + j = N. The t-interior of each is shaded with the possible values of t given. Proof. We use the injection τ between blocks of λ ′ := ελ(t) and λ C from the proof of Theorem 3.1. Lifting this to an injection between the boxes of these partitions, i + j = N − 1 gives that a single box is not in the image. If the bounding box is not sharp around λ, then k > λ1 or… view at source ↗
Figure 5
Figure 5. Figure 5: A knijt-partition λ satisfying i + j = N − 1 for n = 11 and i > 0. The only other such partition is λ T. Both require that i = 4 and t = 6. 4. The t-boundary We now state results related to the value j−i, which is the size of the t-boundary of λ. These improve upon the known bounds (S2)–(S5). Proposition 4.1. Every knijt-partition with i > 0 satisfies j − i ≥ t. Proof. If t = 0 then i = j, so suppose t > 0… view at source ↗
Figure 6
Figure 6. Figure 6: The region bounded by i + j ≤ N and the inequality i + j ≤ (j − i) 2 from Proposition 4.8. The bounds in Lemma 4.5, Proposition 4.6, Corollary 4.7, Proposition 4.8 and Corollary 4.9 are all sharp, with equality when λ = (k, k − 1, . . . , 1) is a staircase partition, with (k, n, i, j, t) = (k, 2k, k(k − 1)/2, k(k + 1)/2, 2). It would be an interesting problem to strengthen the inequalities in this section … view at source ↗
read the original abstract

The Borel-Weil-Bott theorem can be used to decompose the cohomology of twisted sheaves of holomorphic forms on the complex Grassmannian into irreducible representations of the general linear group. By analyzing this decomposition, we provide two effective formulae for computing the associated Hodge numbers, and give examples in special cases. One of these involves a novel integer-valued hook-product statistic on bounded partitions, and the other is based on semistandard tableaux. We reformulate Snow's observation that the positivity of the Hodge numbers is equivalent to the existence of a partition satisfying certain properties, and improve known bounds on when this occurs. This involves a combinatorial proof of the Nakano vanishing theorem for the Grassmannian utilizing a map from core partitions to plane partitions. Finally, we extend our computation of the Hodge numbers of twisted holomorphic forms on the Grassmannian to a q-analogue.

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 / 4 minor

Summary. The paper applies the Borel-Weil-Bott theorem to decompose the cohomology of twisted sheaves of holomorphic forms on the complex Grassmannian into irreducible representations of the general linear group. From this decomposition the authors derive two explicit combinatorial formulae for the associated Hodge numbers, one via a novel integer-valued hook-product statistic on bounded core partitions and the other via semistandard tableaux. They reformulate Snow's positivity criterion, improve known bounds on its occurrence, supply a combinatorial proof of the Nakano vanishing theorem by means of a map from core partitions to plane partitions, and extend the Hodge-number formulae to a q-analogue.

Significance. If the derivations hold, the work supplies effective, explicitly combinatorial methods for computing Hodge numbers of twisted holomorphic forms on the Grassmannian and a new combinatorial route to Nakano vanishing. The hook-product statistic and the q-analogue constitute concrete additions to the toolkit of representation-theoretic combinatorics.

minor comments (4)
  1. The definition of the hook-product statistic (around the statement of the first main formula) would benefit from an explicit small example that verifies both integrality and agreement with the representation-theoretic count.
  2. The map from core partitions to plane partitions used in the Nakano-vanishing argument is described combinatorially but lacks a short table or diagram illustrating the correspondence for a rank-3 or rank-4 Grassmannian.
  3. Notation for the q-analogue (final section) re-uses several symbols already employed for the ordinary case; a brief notational table or remark would prevent confusion.
  4. The abstract promises 'examples in special cases'; the manuscript contains two such examples, but they are presented without the intermediate weight-space calculations that would make the link to Borel-Weil-Bott fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary, positive assessment of significance, and recommendation of minor revision. We are pleased that the combinatorial formulae, hook-product statistic, q-analogue, and combinatorial proof of Nakano vanishing were viewed as concrete contributions.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies the standard Borel-Weil-Bott theorem to decompose cohomology on the Grassmannian and extracts Hodge numbers via direct combinatorial counting on bounded core partitions (hook-product statistic) and semistandard tableaux. These formulae follow from the representation-theoretic decomposition without any fitted parameters renamed as predictions or self-referential definitions. The reformulation of Snow's positivity criterion and the combinatorial Nakano vanishing proof via core-to-plane-partition maps are independent consequences of the same external setup, with no load-bearing self-citations or ansatz smuggling. The q-analogue extension is likewise a direct combinatorial lift. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on the applicability of the Borel-Weil-Bott theorem to the Grassmannian and on the equivalence between Hodge-number positivity and existence of certain partitions; the new statistic is introduced without independent evidence outside the paper.

axioms (1)
  • domain assumption Borel-Weil-Bott theorem decomposes the cohomology of twisted sheaves on the complex Grassmannian into irreducible representations of GL
    Invoked in the first sentence of the abstract as the starting point for the decomposition and subsequent formulas.
invented entities (1)
  • integer-valued hook-product statistic on bounded partitions no independent evidence
    purpose: To compute Hodge numbers via one of the two effective formulae
    Presented as novel in the abstract; no independent evidence or falsifiable prediction outside the paper is mentioned.

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