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 \(K[\Pc]\) with the classical rook theory. Some further related properties, as radicality, primary decomposition, and levelness are discussed, and a \texttt{Macaulay2 package, namely \texttt{PolyominoIdeals}, is also presented.