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The signed Varchenko Determinant for Complexes of Oriented Matroids

Winfried Hochstättler, Sophia Keip, Kolja Knauer

TL;DR

This paper extends the Varchenko determinant factorization from hyperplane arrangements to complexes of oriented matroids (COMs), providing a unified treatment of signed and unsigned versions. The authors define the signed Varchenko matrix ${\mathfrak V}$ for a COM and prove the factorization $\det({\mathfrak V})=\prod_{Y\in\mathcal{L}}(1-a(Y))^{b_Y}$ with $a(Y)=\prod_{e\in z(Y)} x_e^+ x_e^-$ and explicitly computable $b_Y$. The main method blends combinatorial topology (order complexes, Möbius functions) with a block factorization via matrices ${\mathcal M}^e$ and Quillen-type fiber arguments to control Möbius numbers. Applications to distributive lattices and linear extensions yield explicit determinant formulas, illustrating the reach to poset-derived COMs and supporting broader conjectures about supertopes of oriented matroids.

Abstract

We generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids.

The signed Varchenko Determinant for Complexes of Oriented Matroids

TL;DR

This paper extends the Varchenko determinant factorization from hyperplane arrangements to complexes of oriented matroids (COMs), providing a unified treatment of signed and unsigned versions. The authors define the signed Varchenko matrix for a COM and prove the factorization with and explicitly computable . The main method blends combinatorial topology (order complexes, Möbius functions) with a block factorization via matrices and Quillen-type fiber arguments to control Möbius numbers. Applications to distributive lattices and linear extensions yield explicit determinant formulas, illustrating the reach to poset-derived COMs and supporting broader conjectures about supertopes of oriented matroids.

Abstract

We generalize the (signed) Varchenko matrix of a hyperplane arrangement to complexes of oriented matroids and show that its determinant has a nice factorization. This extends previous results on hyperplane arrangements and oriented matroids.
Paper Structure (8 sections, 20 theorems, 71 equations, 1 figure)

This paper contains 8 sections, 20 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Varchenko Determinant and Complexes of Oriented Matroids
  3. Preparation
  4. Main Proof
  5. Applications
  6. Distributive Lattices
  7. Linear extensions
  8. Conclusion

Key Result

Theorem 1.1

Let $\mathcal{A}$ be a real hyperplane arrangement, $B_{\mathcal{A}}$ its Varchenko matrix, and $L(\mathcal{A})$ the geometric lattice formed by the intersections of hyperplanes in $\mathcal{A}$, then where $w_F=\prod_{F \subset H_e} w_e$ and $m_F$ are positive integers depending only on $L(\mathcal{A})$.

Figures (1)

  • Figure 1: A poset $\mathcal{Q}$, its lattice $L(\mathcal{Q})$ of ideals and its set $X(\mathcal{Q})$ of linear extensions. Edges in the graphs in the middle and on the right are drawn if endpoints correspond to topes with separator consisting of a single element. Edges corresponding to the same element are parallel.

Theorems & Definitions (46)

  • Theorem 1.1: Varchenko 1993 V
  • Definition 2.1
  • Definition 2.2: Complex of Oriented Matroids (COM)
  • Definition 2.3: Oriented Matroid (OM)
  • Remark 2.1
  • Definition 2.4: Graphic OM of a directed $n$-cycle
  • Example 2.1: Graphic OM of a directed triangle
  • Definition 2.5: Signed Varchenko Matrix of a COM
  • Example 2.2: continued
  • Theorem 2.1
  • ...and 36 more