Stingray Patterns of Dominant Weights

TL;DR

This paper explores the geometric and combinatorial structure of dominant weights in $ ext{sl}_r$ related to partitions, revealing patterns, counting formulas, and connections to affine Weyl group actions.

Contribution

It introduces a geometric framework for understanding dominant weights, decomposes their structure into simplices, and provides a new alcove-geometric proof of the empty runner removal theorem.

Findings

Decomposition of $W_{r,e,0}$ into simplices indexed by compositions of $r$

Counting formula for the size of $W_{r,e,w}$

Labeling of dominant $e$-alcoves with weak compositions and affine Weyl group action

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.