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Bizonotopal Graphical Algebras

Anatol Kirillov, Gleb Nenashev, Boris Shapiro, Arkady Vaintrob

Abstract

Zonotopal algebras (external, central, and internal) of an undirected graph G introduced by Postnikov-Shapiro and Holtz-Ron, are finite-dimensional commutative graded algebras whose Hilbert series contain a wealth of combinatorial information about G. In this paper, we associate to G a new family of algebras, which we call bizonotopal, because their definition involves doubling the set of edges of G. These algebras are monomial and have intricate properties related, among other things, to the combinatorics of graphical parking functions and their polytopes. Unlike the case of usual zonotopal algebras, the Hilbert series of bizonotopal algebras are not specializations of the Tutte polynomial of G. Still, we show that in the external and central cases these Hilbert series satisfy a modified deletion-contraction relation. In addition, we prove that the external bizonotopal algebra is a complete graph invariant.

Bizonotopal Graphical Algebras

Abstract

Zonotopal algebras (external, central, and internal) of an undirected graph G introduced by Postnikov-Shapiro and Holtz-Ron, are finite-dimensional commutative graded algebras whose Hilbert series contain a wealth of combinatorial information about G. In this paper, we associate to G a new family of algebras, which we call bizonotopal, because their definition involves doubling the set of edges of G. These algebras are monomial and have intricate properties related, among other things, to the combinatorics of graphical parking functions and their polytopes. Unlike the case of usual zonotopal algebras, the Hilbert series of bizonotopal algebras are not specializations of the Tutte polynomial of G. Still, we show that in the external and central cases these Hilbert series satisfy a modified deletion-contraction relation. In addition, we prove that the external bizonotopal algebra is a complete graph invariant.
Paper Structure (22 sections, 24 theorems, 127 equations)

This paper contains 22 sections, 24 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. External bizonotopal algebra
  3. Definitions and basic properties
  4. Conventions and notation
  5. Definition of $\mathscr{B}_G^e$
  6. Basis of $\mathscr{B}_G^e$
  7. External bizonotopal algebras distinguish graphs
  8. Defining relations of $\mathscr{B}_G^e$
  9. Score vector polytope
  10. Weak parking functions of a graph
  11. $r$-bizonotopal algebras and loopy deletion-contraction
  12. Definition of $r$-bizonotopal algebras
  13. Loopy deletion-contraction
  14. "Categorification" of the deletion-contraction relation
  15. Additional properties of central and internal bizonotopal algebras
  16. ...and 7 more sections

Key Result

Proposition 2.2

For each subset $\Sigma\subset \widehat{E}$, consider the monomial (1) The image of $x_\Sigma$ in $\widehat{\mathscr{E}}_G$ is nonzero if and only if $\Sigma$ is a partial orientation. Moreover, the images of the monomials $x_\Sigma$ corresponding to distinct partial orientations of $G$ form a basis of the algebra $\widehat{\mathscr{E}}_G$. (2) As a graded algebra, (3) The dimension of $\widehat{

Theorems & Definitions (60)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3: The algebra $\mathscr{B}^e_G$
  • Definition 2.4: Partial score vectors
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • Example 2.7
  • Theorem 2.8
  • ...and 50 more