Stingray Patterns of Dominant Weights

Abstract: We study the set $W_{r,e,w}\$ of dominant weights of $\mathfrak{sl}<em>r$ arising from partitions of fixed $e$-weight $w$. For $e$-cores, we show that $W</em>{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.

• The paper establishes a geometric framework that embeds partitions into dominant weight lattices, revealing stingray and hexagonal patterns via alcove decompositions.
• It utilizes abacus combinatorics to classify blocks by e-core and e-weight, providing explicit simplicial decompositions and counting formulas.
• The study delivers a geometric proof of the empty runner removal theorem, offering fresh insights for modular representation theory and categorification.

Stingray Patterns and the Geometry of Dominant Weights for Affine Type A

Introduction

This work, "Stingray Patterns of Dominant Weights" (2604.04326), presents a comprehensive geometric and combinatorial study of the distribution of dominant weights associated with partitions of prescribed $e$-weight and at most $r$ parts, with a primary focus on affine type $A$ (i.e., the representation theory of symmetric and Iwahori–Hecke algebras). The analysis is grounded in the classification of blocks via $e$-core and $e$-weight, exploiting the interplay between weight lattices, abacus combinatorics, and the alcove geometry of the affine Weyl group. The author systematically characterizes the resulting patterns—termed "stingray patterns"—and situates them in the context of Kazhdan–Lusztig theory, culminating in a geometric proof of the empty runner removal theorem. This essay details the main results, methods, and implications of the work, with particular attention paid to the explicit geometric and combinatorial structures arising in this context.

Weight Lattice Embedding and Alcove Geometry

The paper's central construction is the embedding of the set of partitions with at most $r$ parts and fixed $e$-weight $w$ into the dominant weight lattice $P^+$ of $\mathfrak{sl}_r$, via the map

$r$0

where $r$1 are the fundamental weights and $r$2 are the parts of the partition extended by zeros as needed.

This geometric representation enables the translation of block-theoretic invariants—core, quotient, and $r$3-weight—into the language of alcove arrangements for the affine Weyl group acting on the real vector space $r$4.

The main combinatorial tool is the abacus model for partitions, whereby beads (beta-numbers) are placed on runners corresponding to residues mod $r$5. The configuration records the $r$6-core and $r$7-quotient data, which is then reflected in the geometry of the dominant weights.

Simplicial Decomposition and Counting

One of the principal results is the explicit simplicial decomposition of the set $r$8 of dominant weights. For $r$9-core partitions ($A$0), $A$1 is shown to be a disjoint union of simplices indexed by compositions $A$2 of $A$3: $A$4 where each $A$5 is the set of weights arising from configurations where the beads are placed on a set of runners specified by $A$6. Each such component forms the lattice points of a simplex whose dimension is one less than the length of $A$7 and whose dilation is $A$8.

For general $A$9, $e$0 is shown to be a union of multiple copies of these simplices, with multiplicities determined by the corresponding number of multipartitions of $e$1 of constrained lengths. Explicitly, the cardinalities are given by: $e$2 where $e$3 counts multipartitions of $e$4 of type $e$5 and $e$6 is the number of placements of runners. The combinatorics is managed via the abacus model and stars-and-bars arguments.

An immediate consequence is a closed formula for the number of core weights, which recovers known block counting results in Hecke algebra theory.

Visualization: Stingray and Hexagonal Patterns

A remarkable output of the geometric embedding is the emergence of stingray-like and hexagonal patterns in the set of dominant weights, particularly visible for low ranks. In type $e$7 ($e$8), as $e$9 increases, $e$0 grows in a manner that the new weights form elongated "tails" protruding from the body of the weight set (the "stingray" metaphor). In the interior, the pattern organizes into hexagonal configurations, reflecting the underlying Weyl group symmetries.

Figure 1: Example of a $e$1 stingray pattern in the dominant chamber of $e$2 showing colored simplices arising from the simplicial decomposition; the green simplices fill the interiors of $e$3-alcoves.

These patterns are rigorously described via the cell decomposition of the dominant chamber into alcoves, with stingray tails corresponding to affine vertices and the combinatorial structure of multipartitions controlling the interior cells.

Figure 2: Two stingray patterns and one regular pattern in $e$4, with the distinct "tails" corresponding to configurations where all beads lie on a single runner.

Indexing Alcoves and Weak Compositions

The interior points of $e$5 are in bijection with dominant $e$6-alcoves in the chamber, and, more strongly, the set of alcoves can be indexed by weak compositions of $e$7 of length $e$8. Each green simplex in the visualizations corresponds to a different composition, and moving between alcoves encodes the action of simple reflections.

This is reflected combinatorially in transformations of the multipartition data—swapping runner indices or shifting “bead” movements—corresponding to crossing the walls of alcoves under affine Weyl group action.

Figure 3: Example with $e$9, $r$0, $r$1, showing alcove decomposition and indexing of green interior simplices by weak compositions.

The formal connection between wall-crossing and Coxeter group action is established by computing the "Shi coefficients" associated to positions in the weight lattice and proving their invariance under the corresponding combinatorial moves.

Affine Weyl Group Action and Wall-Crossing

A further aspect detailed is the partially defined action of the affine Weyl group on these multipartition indexings. Simple reflections act as permutations (for finite Weyl group generators) or certain cyclic shifts with increment/decrement (for affine simple reflection $r$2). This action is compatible with wall-crossing between adjacent alcoves, as can be checked by explicit calculation of Shi coefficients.

The partiality arises because, in the truncated (length $r$3) abacus model, some runner swapping operations do not preserve the length constraint or weight region, so the action is only defined when the resulting multipartition remains viable.

Empty Runner Removal: Geometric and Combinatorial Proof

A highlight of the work is a direct geometric proof of the empty runner removal theorem—a result central in the block theory of Iwahori–Hecke algebras, previously established via more algebraic or canonical base methods [jamesmathas-empty-runner-removal]. The proof is formulated by identifying the effect of adding an empty runner to the abacus with a corresponding translation in the affine Weyl group action on alcoves, shown to yield a canonical bijection of dominant weights with the property that the anti-spherical Kazhdan–Lusztig polynomials are preserved under this operation.

This establishes the equality of graded decomposition numbers under runner insertion/removal, providing a conceptually clean proof and indicating the geometric (alcove) underpinnings of the block stability phenomena.

Practical and Theoretical Implications

The results have both combinatorial and categorical significance. The explicit formulas for the distribution and counting of dominant weights with prescribed parameters streamline the enumeration of blocks and facilitate detailed analysis of decomposition matrices. The alcove-geometric approach clarifies the conceptual basis of runner removal, essential for understanding structures in $r$4-Schur algebras, canonical bases, and higher-level generalizations, with potential for further categorification as noted by the author.

Practically, the visual patterns and indexing methods enable algorithmic enumeration and visualization of representation-theoretic data, which is valuable in the computational study of modular representation theory and the associated KLR and canonical basis modules.

Potential Extensions and Future Directions

The geometric model and the connection to alcove arrangements suggest several directions for further development:

• Extension to other types and higher-level analogues: The decomposition and visualization techniques may be extended to more general Lie types, higher-level Fock spaces, and Ariki–Koike or quiver Hecke settings, as indicated in related work [alice-full-runner-removal, alice-empty-runner-removal, qin-subdivision-runner-removal].
• Categorification and Soergel-theoretic models: The combinatorial stability under runner removal informs the search for explicit categorical equivalences, potentially simplifying equivalences between different KLRW (Khovanov–Lauda–Rouquier–Webster) algebras and realization of Soergel bimodules.
• Fine structure and Bruhat order: The identification of which pairs of affine vertices yield "good" or "bad" pairs (i.e., when the regular pattern completely fills an alcove versus when stingray tails are formed) is closely related to Bruhat order and cell decomposition, suggesting opportunities for finer granularity in the local structure of the weight patterns.

Conclusion

The paper provides an explicit and conceptually unified framework for describing the geometry of dominant weights arising from $r$5-core/weight data in type $r$6, with direct combinatorial and geometric tools for understanding block theory, alcove structures, and transformations under affine Weyl group actions. The visual "stingray" patterns, the simplicial decompositions, and the wall-crossing formalism bring new transparency to the structure of modular representation theory for symmetric and Hecke algebras, and the methods developed set the stage for further generalizations and applications within higher-level categorical representation theory.

Figure 4: The pattern of dominant weights for $r$7—the stingray pattern in the $r$8 dominant chamber, with each black ball a dominant weight in $r$9.

Figure 5: Visualization for $e$0, further amplifying the formation of stingray and hexagonal arrangements as $e$1 increases.

References: See (2604.04326) for detailed proofs, combinatorial arguments, and further context.