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GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

Part of: Polytopes and polyhedra Algebraic combinatorics Graph theory Discrete geometry

Published online by Cambridge University Press:  04 December 2019

SPENCER BACKMAN ,
MATTHEW BAKER  and
CHI HO YUEN
SPENCER BACKMAN
Affiliation:
Department of Mathematics and Statistics, University of Vermont, Innovation Hall 82 University Place, Burlington, VT05405, USA; [email protected]
MATTHEW BAKER
Affiliation:
School of Mathematics, Georgia Institute of Technology Atlanta, GA 30332-0160, USA; [email protected]
CHI HO YUEN
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI02912, USA; [email protected]

Abstract

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Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$. This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$, many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$, and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$. (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$.) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$.

MSC classification

Primary: 52C40: Oriented matroids
Secondary: 05C31: Graph polynomials 05E18: Group actions on combinatorial structures 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 52B40: Matroids (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e45
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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