Incidence Geometry in a Weyl Chamber I: $GL_n$

TL;DR

<3-5 sentence high-level summary>This work studies the central hyperplane arrangement $I(\mathfrak{gl}_n)$ formed by the kernels of the weights of the vector and antisymmetric representations, restricted to the dual fundamental Weyl chamber. It derives generating functions $G(s,t)$ and $H(s,t)$ that count $k$-faces and $k$-flats, respectively, and proves precise recurrences for these counts via an extreme-ray combinatorial framework using the poset $\mathbb{E}_n^*$. The chambers are in bijection with sign choices $S\subseteq [n]$, and every $k$-face corresponds to a $k$-chain in $\mathbb{E}_n^*$, while flats correspond to $k$-ensembles, yielding a uniform combinatorial description. The results connect enumerative geometry of hyperplane arrangements with Coulomb-branch physics in five-dimensional gauge theories and with geometric engineering via elliptic fibrations, offering explicit data and recursion relations that illuminate the adjacency of resolutions and phase structure.

Abstract

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of $\mathfrak{gl}_n$ restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U(n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.

Figures (21)

• Figure 1: Sign tableau notation. The box labeled by $(i,j)$ corresponds to the sign of weight $x_i+x_j$. This convenient parameterization of chambers of $I(\mathfrak{gl}_n,V \oplus \bigwedge^2)$ was introduced in the physics literature Hayashi:2014kca.
• Figure 2: The $10$-ensemble $S$ in $\mathbb{E}^\ast_{12}$.
• Figure 3: A maximal chain of the ensemble $S$.
• Figure 5: The geometry $I(\mathfrak{gl}_{2},V \oplus \bigwedge^2)$. The geometry contains 4 chambers, 5 half-lines and the origin. There is one 2-flat (the half-plane below the line $x_1-x_2=0$), three 1-flats (the interior walls $C_1$, $C_2$, $C_3$), and one 0-flat (the origin).
• Figure 6: The interval $\lbrack P, Q\rbrack$ is the rectangle suspended between the points $P$ and $Q$.

Theorems & Definitions (49)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Example 1.7
• Corollary 1.8
• Example 1.9
• Theorem 1.10