Hilbert Series Of Polyomıno Ideals And Cohen-Macaulay Posets

Official URL: https://risc01.sabanciuniv.edu/record=b3400655

In this thesis, we study the algebraic and homological properties of polyomino ideals and characterize Cohen-Macaulay posets of dimension two. In 2012, Qureshi introduced a class of binomial ideals called polyomino ideals, related to combinatorial objects called polyominoes. The polyomino ideals are defined by associating each polyomino P with the ideal generated by the inner 2-minors of P in the polynomial ring SP = K[xv : v is a vertex of P]. A primary aim of this research is to investigate the algebraic and homological properties of K[P] = SP/IP depending on the shape of P. We introduce a new class of non-simple polyominoes called frame polyominoes, and demonstrate that its Hilbert series can be described in terms of certain rook arrangements in polyominoes. We also define a new collection of cells called zig-zag collection and its Hilbert series is similarly characterized by certain rook arrangements. For a zig-zag collection of cells P we provide a necessary condition for the coordinate ring K[P] to be Gorenstein. A key practical outcome of this research is the development of the PolyominoIdeals package for Macaulay2. This computational tool is tailored to assist in the study of polyomino ideals, enabling more effective exploration and analysis of these algebraic structures. Additionally, we provide a characterization of Cohen-Macaulay posets of dimension two. We demonstrate that these posets are shellable and strongly connected.