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PolyominoIdeals -- a package to work with binomial ideals associated with collections of cells


PolyominoIdeals is a package for making several computations with the inner 2-minor ideals attached to collections of cells.

In [AAQ2012] Ayesha Asloob Qureshi establishes a connection between Combinatorial Commutiative Algebra and collection of cells, assigning to every collection of cells the binomial ideal of its inner $2$-minors.

Consider the natural partial order on $\NN^2$ and let $a,b \in \N^2$ with $a\leq b$. The set $[a, b] = \{c \in \NN^2 : a \leq c \leq b\}$ is called an interval of $\NN^2$; moreover, if $b=a+(1,1)$, then $[a,b]$ is called a cell of $\NN^2$. An interval $C=[a, b]$, where $a = (i, j)$ and $b = (k, l)$, is said to be a proper interval if $i < k$ and $j < l$. The elements $a, b$ are said the diagonal corners of $C$ and $c = (k, j)$ and $d = (i, l)$ the anti-diagonal ones. If $C$ is a cell, then $V(C)=\{a,a+(1,1),a+(0,1),a+(1,0)\}$ is the set of the corners of $C$.
To each collection of cells $\mathcal{P}$, we attach an ideal $I_{\mathcal{P}}$ as following. Let $K$ be a field and $S=K[x_a: a \in V (\mathcal{P})$, where $V (\mathcal{P})$ is the union of the vertices sets of all cells of $\mathcal{P}$. A proper interval $[a, b]$ is called an inner interval of $\mathcal{P}$ if all cells of $[a, b]$ belong to $\mathcal{P}$. The binomial $f= x_ax_b − x_c x_d$ , where $c$ and $d$ are the anti-diagonal corners of $[a, b]$, is called an inner 2-minor of $\mathcal{P}$, if $[a, b]$ is an inner interval of $\mathcal{P}$. We denote by $I_{\mathcal{P}}$ the ideal generated in $S$ by the inner 2-minors of $\mathcal{P}$ and by $K [\mathcal{P}]$ the quotient ring $S/I_{\mathcal{P}}$, called the coordinate ring of $\mathcal{P}$.
The class of ideals attached to a collection of cells includes, for example, the ideals of 2-minors of two-sided ladders, but it is much more general. Interesting classes of collections of cells are the so-called polyominoes that are well studied in various combinatorial contexts. A collection of cells $\mathcal{P}$ is called a polyomino if for any two cells $A, B \in \mathcal{P}$ there exists a sequence of cells $A=C_1,\dots, C_m=B$ of $\mathcal{P}$ such that $C_i$ and $C_{i+1}$ have an edge in common. In such a case, $I_{\mathcal{P}}$ is called polyomino ideal of $\mathcal{P}$.

The aim of this package is to provide several tools to help mathematicians in the study of polyomino ideals. Every contribution is very welcome.




This documentation describes version 1.1 of PolyominoIdeals.

Source code

The source code from which this documentation is derived is in the file PolyominoIdeals.m2.


  • Functions and commands
    • polyoIdeal -- Ideal of inner 2-minors of a collection of cells
    • polyoMatrix -- Matrix attached to a collection of cells
    • polyoToric -- Toric ideal of a polyomino
  • Methods
    • polyoIdeal(List) -- see polyoIdeal -- Ideal of inner 2-minors of a collection of cells
    • polyoMatrix(List) -- see polyoMatrix -- Matrix attached to a collection of cells
    • polyoToric(List,List) -- see polyoToric -- Toric ideal of a polyomino
  • Symbols
    • Field -- optional argument for polyoIdeal
    • RingChoice -- optional argument for polyoIdeal
    • TermOrder -- optional argument for polyoIdeal

For the programmer

The object PolyominoIdeals is a package.