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Stingray Patterns of Dominant Weights

Tao Qin

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.

Stingray Patterns of Dominant Weights

Abstract

We study the set of dominant weights of arising from partitions of fixed -weight . For -cores, we show that decomposes as a disjoint union of simplices indexed by compositions of . For general , we prove that 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 . The geometry gives rise to the stingray patterns appearing in the title. More generally, it yields a natural labeling of the dominant -alcoves meeting by weak compositions of , 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.

Paper Structure

This paper contains 24 sections, 25 theorems, 117 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and compositions
  4. Multi-partitions and counting formulas
  5. Beta numbers and the $e$–abacus
  6. Core, weight, and quotient
  7. Coxeter systems and Hecke algebras
  8. Anti-spherical Kazhdan–Lusztig polynomials
  9. Type $A$ Cartan data
  10. The geometric map $\Omega$
  11. Alcove geometry
  12. Affine hyperplanes and alcoves
  13. The level-$e$ action
  14. Shi coefficients
  15. Reindexing the anti-spherical KL polynomials
  16. ...and 9 more sections

Key Result

Theorem 1

In characteristic $0$, suppose $\lambda$ and $\mu$ lie in the same block and $\mu$ is $e$–regular. Form $\lambda^+$ and $\mu^+$ as above, then $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: $r=3$, $e=10$, $w=8$
  • Figure 2: $r=3$, $e=2$, $w=0$
  • Figure 3: $r=3$, $e=3$, $w=0$
  • Figure 4: $r=3$, $e=8$, $w=8$
  • Figure 5: $r=3$, $e=12$, $w=10$
  • ...and 5 more figures

Theorems & Definitions (75)

  • Example 1
  • Definition 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Theorem 1: jamesmathas-empty-runner-removal, Theorem 3.2
  • Remark 1
  • Theorem 2: goodmanwenzl-crystal-bases
  • Definition 2
  • ...and 65 more