Recent Advances in the Theory of Polyomino Ideals

TL;DR

The paper surveys foundational and recent advances in polyomino ideals, focusing on primality, radicality, and Hilbert–Poincaré aspects. It unifies toric representations for simple polyominoes and extends to non-simple cases via toric parametrizations, zig--zag obstructions, and Gröbner-basis criteria. A central theme is the surprising link between Hilbert-Poincaré series and rook theory, including the switching rook polynomial, with broad results across L-convex, grid, closed-path, and frame polyominoes, plus canonical-module and Gorenstein-type properties. The survey culminates in practical computational tools through a Macaulay2 package that implements structural, rook-theoretic, and algebraic operations for polyominoes, enabling explicit computation and exploration of these combinatorial–algebraic structures.

Abstract

Polyomino ideals, defined as the ideals generated by the inner $2$-minors of a polyomino, are a class of binomial ideals whose algebraic properties are closely related to the combinatorial structure of the underlying polyomino. We provide a unified account of recent advances on two central themes: the characterization of prime polyomino ideals and the emerging connection between the Hilbert-Poincaré series and Gorensteinness of $K[\mathcal{P}]$ with the classical rook theory. Some further related properties, as radicality, primary decomposition, and levelness are discussed, and a \textit{Macaulay2} package, namely \texttt{PolyominoIdeals}, is also presented.

Figures (21)

• Figure 1: A polyomino, a weakly connected collection of cells with two connected components, and a collection of cells with three weakly connected components.
• Figure 2: A Ferrer diagram, a stack, a parallelogram and a directed convex.
• Figure 3: A polyomino $\mathcal{P}$.
• Figure 4: Polyominoes $\mathcal{P}^{c}$.
• Figure 5: An example of a zig-zag walk.

Theorems & Definitions (48)

• Remark 1.1
• Theorem 2.1
• Conjecture 3.1: HibiQureshi2015
• Theorem 3.2: HibiQureshi2015, Theorem 4.1
• Theorem 3.3: HibiQureshi2015, Theorem 3.1
• proof : Sketch of proof
• Theorem 3.4
• proof
• Remark 3.5
• Theorem 3.6