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Incidence Geometry in a Weyl Chamber II: $SL_n$

Mboyo Esole, Steven Glenn Jackson, Ravi Jagadeesan, Alfred G. Noël

TL;DR

This work analyzes the incidence geometry $I(\mathfrak{sl}_n, V \oplus \bigwedge^2)$ inside the dual fundamental Weyl chamber, focusing on the polyhedral arrangement generated by hyperplanes orthogonal to the weights of the first two fundamental representations. It derives explicit generating functions that enumerate flats and faces by dimension, and characterizes the extreme rays and simplicial chambers, revealing a rich algebraic structure in contrast to the GL case. A key methodological hallmark is the reduction to the GL incidence geometry $I(\mathfrak{gl}_n, V \oplus \bigwedge^2)$ via bijections for faces and flats, enabling combinatorial descriptions in terms of null and bi-signed ensembles in the poset $\mathbb{E}^*_n$. The results include a non-rational generating function phenomenon for SL and a complete classification of simplicial chambers, with direct connections to networks of mixed Coulomb-Higgs branches and to crepant partial resolutions of elliptic fibrations, underscoring the physical relevance of these combinatorial structures. Overall, the paper advances the understanding of SL-type incidence geometries, provides exact counts for chambers, faces, and flats, and links abstract combinatorics to geometric transitions in string-theoretic settings.

Abstract

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of $sl_n$ inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces. From the perspective of supersymmetric gauge theories with 8 supercharges in five dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces is conjectured to be isomorphic to the network of crepant partial resolutions of an elliptic fibration with gauge algebra $sl_n$ and "matter representation" given by the sum of the first two fundamental representations.

Incidence Geometry in a Weyl Chamber II: $SL_n$

TL;DR

This work analyzes the incidence geometry inside the dual fundamental Weyl chamber, focusing on the polyhedral arrangement generated by hyperplanes orthogonal to the weights of the first two fundamental representations. It derives explicit generating functions that enumerate flats and faces by dimension, and characterizes the extreme rays and simplicial chambers, revealing a rich algebraic structure in contrast to the GL case. A key methodological hallmark is the reduction to the GL incidence geometry via bijections for faces and flats, enabling combinatorial descriptions in terms of null and bi-signed ensembles in the poset . The results include a non-rational generating function phenomenon for SL and a complete classification of simplicial chambers, with direct connections to networks of mixed Coulomb-Higgs branches and to crepant partial resolutions of elliptic fibrations, underscoring the physical relevance of these combinatorial structures. Overall, the paper advances the understanding of SL-type incidence geometries, provides exact counts for chambers, faces, and flats, and links abstract combinatorics to geometric transitions in string-theoretic settings.

Abstract

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces. From the perspective of supersymmetric gauge theories with 8 supercharges in five dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces is conjectured to be isomorphic to the network of crepant partial resolutions of an elliptic fibration with gauge algebra and "matter representation" given by the sum of the first two fundamental representations.

Paper Structure

This paper contains 38 sections, 49 theorems, 156 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Physical motivation
  3. Connection to the literature
  4. Results of this paper
  5. Outline of the paper
  6. Acknowledgments
  7. Preliminaries
  8. Introduction to hyperplane arrangements: chambers, flats, and faces
  9. Essential and central arrangements
  10. Flats, chambers, and faces
  11. Sign vectors
  12. Weyl chambers, roots and weights for gl and sl
  13. Structure of chambers, faces, flats, and extreme rays of I(gl(n))
  14. Notation
  15. Formal statements of results
  16. ...and 23 more sections

Key Result

Proposition 2.1

The signs in a tableau corresponding to a chamber satisfy the following conditions:

Figures (17)

  • Figure 1: The 2 chambers of$\mathrm{I}(\mathfrak{sl}_{3},V \oplus {\bigwedge}^2)$.
  • Figure 2: The 4 chambers of$\mathrm{I}(\mathfrak{sl}_{4},V \oplus {\bigwedge}^2)$.
  • Figure 3: The 12 chambers of$\mathrm{I}(\mathfrak{sl}_{5},V \oplus {\bigwedge}^2)$. See box.
  • Figure 5: $\mathrm{I}(\mathfrak{sl}_{2},V \oplus {\bigwedge}^2)$ consists of a half-line. See ESY1.
  • Figure 6: $\mathrm{I}(\mathfrak{sl}_{3},V \oplus {\bigwedge}^2)$ consists of two chambers separated by a half-line. See ESY1.
  • ...and 12 more figures

Theorems & Definitions (129)

  • Definition 1.1: gln-def:incGeoGeneral
  • Remark 1.2
  • Remark 1.3
  • Example 1.4
  • Example 1.5
  • Proposition 2.1: box
  • Remark 2.2
  • Definition 2.3: gln-def:subsetToChamber
  • Theorem 2.4: gln-thm:chamberStructure
  • Definition 2.5: gln-def:quarterPlanePoset
  • ...and 119 more