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arxiv: 2604.04326 · v1 · submitted 2026-04-06 · 🧮 math.CO

Stingray Patterns of Dominant Weights

Tao Qin This is my paper

Pith reviewed 2026-05-10 20:29 UTC · model grok-4.3

classification 🧮 math.CO
keywords dominant weightse-corespartitionssimplicesstingray patternsalcove geometryaffine Weyl groupIwahori-Hecke algebras
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The pith

Dominant weights from partitions of fixed e-weight decompose into simplices with multiplicities from quotients.

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

The paper examines the sets of dominant weights for sl_r that come from partitions with a fixed e-weight w. It establishes that for w equal to zero these sets split into a disjoint union of simplices, each corresponding to a composition of the rank r. For positive w the sets consist of multiple copies of the same simplices, where the multiplicity of each simplex is read off from quotient data attached to the partitions. This decomposition immediately produces an explicit formula for the total number of weights in the set and reveals the stingray patterns in the geometry of the weight space. The same structure supplies a labeling of the relevant alcoves by weak compositions of w together with a partial action of the affine Weyl group realized by crossing walls, and it furnishes a geometric proof of the empty-runner removal theorem.

Core claim

We show that the set W_{r,e,0} of dominant weights arising from e-core partitions decomposes as a disjoint union of simplices indexed by compositions of r. For general w we prove that W_{r,e,w} is a disjoint union of copies of these simplices whose multiplicities are determined by the corresponding quotient data, yielding a closed counting formula for the cardinality of W_{r,e,w}. The geometry produces the stingray patterns in the title. More generally it yields a natural labeling of the dominant e-alcoves meeting W_{r,e,w} by weak compositions of w together with a compatible partial action of the affine Weyl group via wall crossing. Finally we give an explicit alcove-geometric proof of the空

What carries the argument

The sets W_{r,e,w} of dominant weights from fixed-e-weight partitions, decomposed via alcove geometry into simplices indexed by compositions and quotients, with wall-crossing partial action of the affine Weyl group.

If this is right

  • The cardinality of each W_{r,e,w} equals the sum, over the simplices, of their sizes multiplied by the multiplicities coming from the quotient partitions.
  • The dominant e-alcoves that intersect W_{r,e,w} admit a natural labeling by weak compositions of w.
  • The affine Weyl group acts partially on these labeled alcoves by the operation of crossing walls.
  • The empty-runner removal theorem for Iwahori-Hecke algebras follows directly from the alcove geometry.

Where Pith is reading between the lines

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

  • The simplicial decomposition supplies a concrete way to enumerate and visualize these weights inside the weight lattice.
  • The wall-crossing partial action may be compared with other combinatorial actions already known on partitions and cores.
  • The same geometric organization could be examined in other root systems to see whether analogous stingray patterns appear.

Load-bearing premise

The alcove geometry interacts with the affine Weyl group via wall crossing exactly as needed to produce the stated labeling and partial action on the dominant weights defined by the partitions.

What would settle it

An explicit enumeration of the dominant weights for concrete small values of r, e and w that fails to match the number obtained by summing the simplex cardinalities with the predicted multiplicities.

Figures

Figures reproduced from arXiv: 2604.04326 by Tao Qin.

Figure 1
Figure 1. Figure 1: r = 3, e = 10, w = 8 3.1. Simplicial structure of e-cores. We start with the case of e-cores (i.e., w = 0). To simplify notation, we define: Pr,e := Pr,e,0 and Wr,e := Wr,e,0. Thus Pr,e is the set of e-core partitions with at most r parts, and Wr,e = Ω(Pr,e) is the corresponding set of dominant weights. We fix a generic triple (r, e, 0) throughout this section. For each composition µ = (µ1, . . . , µj ) of… view at source ↗
Figure 2
Figure 2. Figure 2: r = 3, e = 2, w = 0 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: r = 3, e = 8, w = 8 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: r = 3, e = 12, w = 10 Corollary 3.13. The number of lattice points in a (j − 1)-dimensional simplex of dilation factor e − j is given by the binomial coefficient e−1 j−1  . Proof. By Theorem 3.6, the set of weights Wr,e,0(µ) corresponds bijectively to the set of gap variable tuples d = (d1, . . . , dj ) satisfying di ∈ Z≥0 and Pj i=1 di = e − j. This counts the number of non-negative integer solutions to … view at source ↗
Figure 6
Figure 6. Figure 6: Two stingray patterns and one regular pattern for (r, e, w) = (3, 8, 5). Our next goal is to show that a stingray tail is isolated in an appropriate sense, which explains why we regard it as a “tail”: it should be separated from the body, as in [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: r = 3, e = 12, w = 10 We now examine these labels in Example 3.52 more closely. First, in [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Level 3 [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: sl3 dominant weight pictures for e = 8 and w = 0, 1, . . . , 9 [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
read the original abstract

We study the set $W_{r,e,w}\ $ of dominant weights of $\mathfrak{sl}_r$ arising from partitions of fixed $e$-weight $w$. For $e$-cores, we show that $W_{r,e,0}\ $ decomposes as a disjoint union of simplices indexed by compositions of $r$. For general $w$, we prove that $W_{r,e,w}\ $ is a disjoint union of copies of these simplices, with multiplicities determined by the corresponding quotient data, yielding in particular a closed counting formula for $|W_{r,e,w}\ |\ $. The geometry gives rise to the stingray patterns appearing in the title. More generally, it yields a natural labeling of the dominant $e$-alcoves meeting $W_{r,e,w}\ $ by weak compositions of $w$, together with a compatible partial action of the affine Weyl group via wall crossing. Finally, we give an explicit alcove-geometric proof of the empty runner removal theorem for Iwahori-Hecke algebras.

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 studies the set W_{r,e,w} of dominant weights of sl_r arising from partitions of fixed e-weight w. For e-cores (w=0), it shows that W_{r,e,0} decomposes as a disjoint union of simplices indexed by compositions of r. For general w, it proves that W_{r,e,w} is a disjoint union of copies of these simplices with multiplicities from the corresponding quotient data, yielding a closed counting formula for |W_{r,e,w}|. The geometry produces stingray patterns; the work also gives a natural labeling of dominant e-alcoves meeting W_{r,e,w} by weak compositions of w, a compatible partial action of the affine Weyl group via wall crossing, and an explicit alcove-geometric proof of the empty runner removal theorem for Iwahori-Hecke algebras.

Significance. If the results hold, the paper supplies a precise combinatorial-geometric description of these weight sets together with explicit decompositions and an enumerative formula. The self-contained alcove-geometric proof of the runner-removal theorem strengthens the toolkit for affine Weyl group actions and Hecke algebra combinatorics. The stingray patterns and partial actions may have further applications in alcove geometry of type A.

minor comments (2)
  1. [Abstract] The abstract introduces the term 'stingray patterns' and the labeling by weak compositions without a brief definition or pointer to the relevant figure or section; adding one sentence in the introduction would improve immediate readability.
  2. [§2] The sets W_{r,e,w} are defined via dominant weights from partitions of e-weight w, but the precise interaction with the fundamental alcove and wall-crossing is assumed from standard references; a short recap of the relevant alcove geometry in §2 would make the decomposition statements fully self-contained for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, the assessment of its significance, and the recommendation of minor revision. No specific major comments or criticisms were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results on the decomposition of W_{r,e,0} into simplices indexed by compositions of r, and the extension to general w via quotient multiplicities, are derived directly from the standard combinatorial definition of e-weight partitions and the alcove geometry of the affine Weyl group. The explicit alcove-geometric proof of the empty runner removal theorem is self-contained within the manuscript and does not reduce any claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation. All steps rely on internal runner-removal combinatorics and wall-crossing actions without importing unverified external premises.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract relies on standard background from Lie theory and partition combinatorics; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (2)
  • standard math Standard action of the affine Weyl group on the weight lattice of sl_r and the geometry of e-alcoves
    Invoked for the wall-crossing partial action and alcove labeling.
  • domain assumption Definition of e-weight and e-cores for partitions
    Used to define the sets W_{r,e,w}.

pith-pipeline@v0.9.0 · 5465 in / 1396 out tokens · 41100 ms · 2026-05-10T20:29:16.687896+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Graded decomposition numbers for cyclotomic Hecke algebras

    [BK09] Jonathan Brundan and Alexander Kleshchev. Graded decomposition numbers for cyclotomic Hecke algebras. Advances in Mathematics, 222(6):1883–1942,

    work page 1942

  2. [2]

    Graded Specht modules.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2011(655),

    [BKW11] Jonathan Brundan, Alexander Kleshchev, and Weiqiang Wang. Graded Specht modules.Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2011(655),

    work page 2011