Recent Advances in the Theory of Polyomino Ideals

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 $\mathit{\text{Macaulay2}}$ package, namely $\mathtt{\text{PolyominoIdeals}}$, is also presented.