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Signed permutohedra, delta-matroids, and beyond

Christopher Eur, Alex Fink, Matt Larson, Hunter Spink

Abstract

We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.

Signed permutohedra, delta-matroids, and beyond

Abstract

We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.
Paper Structure (29 sections, 55 theorems, 174 equations, 1 figure)

This paper contains 29 sections, 55 theorems, 174 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main combinatorial consequences
  3. Underlying geometry
  4. Polytope algebras of delta-matroids
  5. The fan $\Sigma_{B_n}$ and the signed permutation group $\mathfrak S^B_n$
  6. The polytope algebra
  7. Schubert delta-matroids
  8. Intersecting with unit cubes
  9. Bases from Schubert delta-matroids
  10. The exceptional Hirzebruch--Riemann--Roch-type theorem
  11. $K$-rings and Chow rings of $X_{B_n}$
  12. The exceptional isomorphism
  13. Stellahedral geometry
  14. The Chow cohomology ring of $X_{B_n}$
  15. A Schubert basis
  16. ...and 14 more sections

Key Result

Theorem A

Let $P$ be a lattice $B_n$ generalized permutohedron (i.e., $P$ has vertices in $\mathbb Z^n$).

Figures (1)

  • Figure 1: The fans $(\Sigma_{B_1})^2$ (left), $\Sigma_{St_2}$ (middle), and $\Sigma_{B_2}$ (right)

Theorems & Definitions (131)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • Definition 1.3
  • Definition 1.4
  • Theorem B
  • Conjecture 1.5
  • Theorem C
  • Theorem D
  • Definition 2.1
  • ...and 121 more