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Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal Arrangements B(n,k)

Pragnya Das

Abstract

We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making it a partial cube and a median graph, with distances given by the Hamming distance and geodesics described by symmetric differences. We also prove a Poisson limit and a sharp threshold for overlaps of random circuit families, revealing an underlying hypercube geometry.

Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal Arrangements B(n,k)

Abstract

We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making it a partial cube and a median graph, with distances given by the Hamming distance and geodesics described by symmetric differences. We also prove a Poisson limit and a sharp threshold for overlaps of random circuit families, revealing an underlying hypercube geometry.
Paper Structure (21 sections, 14 theorems, 45 equations, 1 figure)

This paper contains 21 sections, 14 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. The discriminantal arrangement
  5. Intersection lattice
  6. Minimal defining families and Circuit support
  7. Rank structure
  8. The Johnson graph on circuits
  9. Definition of the circuit graph
  10. Basic properties
  11. Geometric interpretation
  12. Connection to the metric structure of the lattice
  13. Metric Structure via Circuit Supports
  14. Canonical support representation and Local moves
  15. Metric Structure and Isometric Embedding
  16. ...and 6 more sections

Key Result

Proposition 4.2

The circuit graph defined above is isomorphic to the Johnson graph $J(n,k+1)$.

Figures (1)

  • Figure 1: An interval $[X,Y]$ in $L(B(n,k))$ represented as a 3-dimensional cube. Each vertex corresponds to a feasible support of the form $F(X)\cup T$, where $T \subseteq \{a,b,c\} = F(Y)\setminus F(X)$. Edges correspond to adding a single circuit, and distances agree with Hamming distance.

Theorems & Definitions (33)

  • Definition 3.1
  • Definition 3.2: Feasible support
  • Remark 3.3
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • Proposition 5.1
  • proof
  • Lemma 5.2
  • ...and 23 more