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no. 3
Bizonotopal graphical algebras
Kirillov, Anatol1; Nenashev, Gleb2; Shapiro, Boris3; Vaintrob, Arkady4
1Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China
2St. Petersburg State University, Department of Mathematics and Computer Science, St. Petersburg, 199178, Russia
3Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden, and Department of Mathematics with Computer Science, Guangdong Technion-Israel Institute of Technology, Shantou, China
4University of Oregon, Department of Mathematics, Eugene, OR, 97403, USA
Algebraic Combinatorics, Volume 9 (2026) no. 3, pp. 831-863

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 encode a wealth of combinatorial information about $G$. In this paper, we associate to $G$ a new family of algebras, which we call bizonotopal, since their definition involves doubling the set of edges of $G$. These algebras are monomial and exhibit intricate properties related, among other things, to the combinatorics of graphical parking functions and their associated polytopes.

Unlike classical zonotopal algebras, the Hilbert series of bizonotopal algebras are not specializations of the Tutte polynomial of $G$. Nevertheless, 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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.488
Classification: 05E16, 05C25, 05C60
Keywords: zonotopal algebras, parking functions, graph invariants, deletion-contraction relation

Kirillov, Anatol  1 ; Nenashev, Gleb  2 ; Shapiro, Boris  3 ; Vaintrob, Arkady  4

1 Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Huairou District, Beijing, China
2 St. Petersburg State University, Department of Mathematics and Computer Science, St. Petersburg, 199178, Russia
3 Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden, and Department of Mathematics with Computer Science, Guangdong Technion-Israel Institute of Technology, Shantou, China
4 University of Oregon, Department of Mathematics, Eugene, OR, 97403, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Kirillov, Anatol; Nenashev, Gleb; Shapiro, Boris; Vaintrob, Arkady. Bizonotopal graphical algebras. Algebraic Combinatorics, Volume 9 (2026) no. 3, pp. 831-863. doi: 10.5802/alco.488
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