Level and pseudo-Gorenstein path polyominoes (Q6635842)

Let \(a=(i,j)\), \(b=(k,\ell)\in \mathbb{N}^2\), with \(i\leq k\) and \(j\leq \ell\). The set \N\[\N[a,b]=\{(r,s)\in \mathbb{N}^2\;;\; i\leq r\leq k \;\text{and}\; j\leq s\leq \ell\},\N\]\Nis called an interval of \(\mathbb{N}^2\). If \(i<k\) and \(j<\ell\), \([a,b]\) is called a proper interval, and the elements \(a, b, c, d\) are called corners of \([a, b]\), where \(c = (i,\ell)\) and \(d = (k, j )\). In particular, \(a, b\) are called diagonal corners and \(c, d\) antidiagonal corners of \([a, b]\). The corner \(a\) (resp. \(c\)) is also called the left lower (resp. upper) corner of \([a, b]\), and d (resp. \(b\)) is the right lower (resp. upper) corner of \([a, b]\). A proper interval of the form \(C=[a,a+(1,1)]\) is called a cell. Its vertices \(V(C)\) are \(a, a + (1, 0), a + (0, 1), a + (1, 1)\). The sets \(\{a, a + (1, 0)\}, \{a, a + (0, 1)\}, \{a + (1, 0), a + (1, 1)\}\) and \(\{a + (0, 1), a + (1, 1)\}\) are called the edges of \(C\). Let \(\mathcal{P}\) be a finite collection of cells of \(\mathbb{N}^2\), and let \(C\) and \(D\) be two cells of \(\mathcal{P}\). Then \(C\) and \(D\) are said to be connected if there is a sequence of cells \(C = C_1,\ldots,C_m=D\) of \(\mathcal{P}\) such that \(C_i\cap C_{i+1}\) is an edge of \(C_i\) for \(i = 1,\ldots,m -1\). A collection of cells \(\mathcal{P}\) is called a polyomino if any two cells of \(\mathcal{P}\) are connected. We denote by \(V(\mathcal{P}) =\cup_{C\in \mathcal{P}} V(C)\) the vertex set of \(\mathcal{P}\).\N\NA polyomino \(\mathcal{Q}\) is called a subpolyomino of a polyomino \(\mathcal{P}\) if each cell belonging to \(\mathcal{Q}\) also belongs to \(\mathcal{P}\), and we write \(\mathcal{Q}\subseteq \mathcal{P}\). A proper interval \([a, b]\) is called an inner interval of \(\mathcal{P}\) if all cells of \([a, b]\) belong to \(\mathcal{P}\). We say that a polyomino \(\mathcal{P}\) is simple, if for any two cells \(C\) and \(D\) of \(\mathbb{N}^2\) not belonging to \(\mathcal{P}\), there exists a sequence of distinct cells \(C = C_1,\ldots,C_m=D\) such that \(C_i\cap C_{i+1}\) is an edge of \(C_i\) for \(i = 1,\ldots,m -1\), and \(C_i\notin\mathcal{P}\) for any \(i = 1,\ldots,m\).\N\N{Definition:} A simple polyomino \(\mathcal{P}\) is called a path if \(\mathcal{P}=\{C_1,\ldots,C_\ell\}\) such that\N\begin{itemize}\N\item[1.] \(C_i\cap C_{i+1}\) is a common edge for all \(i=1,\ldots,\ell\);\N\item[2.] \(C_i\neq C_j\) for all \(i\neq j\);\N\item[3.] For all \(i\in\{3,\ldots,\ell-2\}\) and \(j\notin\{i-2,i-1,i,i+1,i+2\}\), one has \(C_i\cap C_j=\emptyset\).\N\end{itemize}\N\NLet \(\mathcal{S}\) be a path polyomino and \(\mathcal{C}=\{I_1,I_2,\ldots,I_\lambda\}\) be the set of the maximal intervals of \(\mathcal{S}\), then we denote by \(l_k\) the length of \(I_k\), say \(l_k=|I_k|\) for \(k\in\{1,\ldots,s\}\). A path polyomino \(\mathcal{S}\) is called a stair, if \(\lambda\geq 3\) and \(l_i=2\) for all \(1<i<\lambda\). The length of the stair is \(\lambda\) and \(\mathcal{S}\) has odd (resp. even) length if \(\lambda\) is odd (resp. even).\N\NA path polyomino \(\mathcal{P}\) has a stair \(\mathcal{S}\), if \(\mathcal{S}\) is a stair subpolyomino of \(\mathcal{P}\), and \(\mathcal{S}\) is not a subpolyomino of any stair of greater length contained in \(\mathcal{P}\). Given a path polyomino \(\mathcal{P}\), we say that \(\mathcal{P}\) has no odd stairs if \(\mathcal{P}\) has no stairs of odd length.\N\N{Definition:} A stair \(\mathcal{P}\) of length \(\lambda\), with \(\lambda=4,6\) or \(\lambda\geq 8\) is called a bad stair.\N\NLet \(\mathcal{P}\) be a polyomino. Let \(\mathbb{K}\) be an arbitrary field and \(S = \mathbb{K}[x_v\;|\; v\in V(\mathcal{P})]\). The binomial \(x_ax_b-x_cx_d\in S\) is called an inner 2-minor of \(\mathcal{P}\) if \([a, b]\) is an inner interval of \(\mathcal{P}\), where \(c, d\) are the antidiagonal corners of \([a, b]\). We denote by \(\mathcal{M}\) the set of all inner 2-minors of \(\mathcal{P}\). The ideal generated by \(\mathcal{M}\) in \(S\) is said to be the polyomino ideal of \(\mathcal{M}\) and it is denoted by \(I_\mathcal{P}\). The properties of \(I_\mathcal{P}\) and \(S/I_\mathcal{P}\) arise from combinatorial properties of \(\mathcal{P}\).\N\NThe main results in the paper under review are the classification of level and pseudo-Gorenstein simple polyominoes that are paths. Indeed, the authors prove the following two results:\N\N{Theorem:} Let \(\mathcal{P}\) be a path polyomino. The followings are equivalent:\N\begin{itemize}\N\item[1.] \(S/I_\mathcal{P}\) is level;\N\item[2.] \(\mathcal{P}\) does not contain bad stairs.\N\end{itemize}\N\N{Theorem:} Let \(\mathcal{P}\) be a path polyomino with \(\mathcal{C}=\{I_1,I_2,\ldots,I_s\}\). Then \(S/I_\mathcal{P}\) is pseudo-Gorenstein if and only if either \(\mathcal{P}\) is a cell or the following conditions hold:\N\begin{itemize}\N\item[1.] \(l_1=l_s=2\) and \(l_k\leq 3\) for all \(2\leq k\leq s-1\);\N\item[2.] \(\mathcal{P}\) does not have odd stairs.\N\end{itemize}