2.1. ASM varieties and ideals

We now outline the process for associating a variety and an ideal to an ASM.

Fix $A = (A_{a,b}) \in \tt ASM(n)$ . Define a rank function ${rk}_A$ on $[n] \times [n]$ by

\begin{equation*} {rk}_A(i,j)=\sum _{a=1}^i\sum _{b=1}^j A_{a,b}. \end{equation*}

The rank matrix of $A$ is the $n \times n$ matrix whose $(i,j)$ entry is ${rk}_A(i,j)$ . (Note that ${rk}_A(i,j)$ is typically not the rank of the submatrix of $A$ consisting of the first $i$ rows and first $j$ columns, unless $A \in S_n$ .)

Throughout this paper, we will let $Z = (z_{i,j})$ denote an $n \times n$ generic matrix, that is, a matrix of distinct variables, and we will let $R$ denote the polynomial ring over $\kappa$ in the entries of $Z$ . For $i,j \in [n]$ , let $Z_{[i],[j]}$ be the submatrix of $Z$ consisting of the first $i$ rows and $j$ columns. For $k \in \mathbb{Z}_+$ , let $I_k(Z_{[i],[j]})$ denote the ideal of $R$ generated by the $k \times k$ minors of $Z_{[i],[j]}$ .

Definition 2.2. For $A \in \tt ASM(n)$ , let

\begin{equation*} I_A = \sum _{(i,j) \in [n] \times [n]} I_{{rk}_A(i,j)+1}(Z_{[i],[j]}). \end{equation*}

We call $I_A$ the ASM ideal of $A$ . Let $X_A$ denote the variety of $I_A$ , which we call the ASM variety of $A$ .

Every ASM ideal $I_A$ is radical ([Reference Weigandt35, Proposition 5.4]). See also [Reference Fulton11, Proposition 3.3] for the case $A \in S_n$ . When $A \in S_n$ , $I_A$ is called a Schubert determinantal ideal and $X_A$ a matrix Schubert variety. An ASM variety is irreducible if and only if it is a matrix Schubert variety [Reference Fulton11, Proposition 3.3], [Reference Weigandt35, Proposition 5.4]. The codimension of $X_w$ is equal to the Coxeter length of $w$ , denoted $\ell (w)$ ([Reference Fulton11, Proposition 3.3]), also known as the inversion number of $w$ . See [Reference Fulton11] for further information on matrix Schubert varieties, including their close connection to Schubert varieties in the complete flag variety.

We call the generators described in Definition 2.2 the natural generators of $I_A$ . It is sometimes more convenient to work with a smaller set of generators called the Fulton generators, which we describe below, after giving some required auxiliary constructions. Fulton generators of ASM ideals were introduced by Weigandt [Reference Weigandt35], generalizing the description given by Fulton [Reference Fulton11] for Schubert determinantal ideals.

For $A \in \tt ASM(n)$ , let

\begin{equation*} D(A) = \left \{(i,j) \mid \sum _{k = 1}^i A_{k,j} = 0 = \sum _{\ell = 1}^j A_{i,\ell } \right \}. \end{equation*}

We call $D(A)$ the Rothe diagram of $A$ . (Some authors refer to $D(A)$ as the inversion set of $A$ .) Note that the definition of ASM forces $\sum _{k = 1}^i A_{k,j}, \sum _{\ell = 1}^j A_{i,\ell } \in \{0,1\}$ for all $i,j \in [n]$ .

Example 2.3. We give an example of a method for visualizing the Rothe diagram of an ASM. If $A=\begin{pmatrix} 0 &\quad 0 &\quad 1 &\quad 0\\ 1 &\quad 0 &\quad -1 &\quad 1 \\ 0 &\quad 1 &\quad 0 &\quad 0\\ 0 &\quad 0 &\quad 1 &\quad 0 \end{pmatrix}$ , we may visualize $D(A)$ by drawing

In an $n \times n$ grid, we have placed a solid dot for every $1$ in $A$ and an open dot for every $-1$ . Lines emanate down and to the right out of each solid dot, stopping before entering the cell of an open dot. Then $D(A) = \{(1,1), (1,2), (2,3)\}$ consists of the indices of cells without a line intersecting their interior.

Let $\texttt {Ess}(A) = \{(i,j) \in D(A) \mid (i,j+1), (i+1,j) \notin D(A)\}$ . We call $\texttt {Ess}(A)$ the essential set of $A$ and elements of $\texttt {Ess}(A)$ the essential cells of $A$ .

Returning to Example 2.3, we have $\texttt {Ess}(A) = \{(1,2),(2,3)\}$ . We may visualize $\texttt {Ess}(A)$ as the maximally southeast elements of connected components of $D(A)$ (where two cells are considered adjacent if they share an edge).

It is a theorem due to Fulton [Reference Fulton11, Lemma 3.10] for $A \in S_n$ and generalized for all $A \in \tt ASM(n)$ by Weigandt [Reference Weigandt35, Lemma 5.9] that the ASM ideal $I_A$ may be generated solely by considering rank conditions at essential cells:

\begin{equation*} I_A = \sum _{(i,j) \in \texttt {Ess}(A)} I_{{rk}_A(i,j)+1}(Z_{[i],[j]}). \end{equation*}

We call this set of generators of $I_A$ the Fulton generators.

Continuing with Example 2.3, the rank matrix of $A$ is ${rk}_A=\begin{pmatrix} 0 & \fbox {0} & 1 & 1\\ 1 & 1 & \fbox {1} & 2 \\ 1 & 2 & 2 & 3\\ 1 & 2 & 3 & 4 \end{pmatrix}$ , where the ranks in essential cells appear in boxes. The Fulton generators of $I_A$ are

\begin{align*} \left \{z_{1,1} , z_{1,2}, \begin{vmatrix} z_{1,1} &\quad z_{1,2} \\ z_{2,1} &\quad z_{2,2} \end{vmatrix}, \begin{vmatrix} z_{1,1} &\quad z_{1,3} \\ z_{2,1} &\quad z_{2,3} \end{vmatrix}, \begin{vmatrix} z_{1,2} &\quad z_{1,3} \\ z_{2,2} &\quad z_{2,3} \end{vmatrix}\right \}. \\[-10pt] \nonumber \end{align*}

The degree $1$ generators are determined by the essential cell $(1,2)$ , and the degree $2$ generators are determined by the essential cell $(2,3)$ .

For $A \in \tt ASM(n)$ , let $\texttt {Dom}(A) = \{(i,j) \in [n] \times [n] \mid {rk}_A(i,j) = 0\}$ , and call $\texttt { Dom}(A)$ the dominant part of $A$ . Note that $\texttt {Dom}(A) \subseteq D(A)$ and that $\texttt {Dom}(A)$ consists exactly of the indices of the degree $1$ Fulton generators of $I_A$ , that is, the generators which are single variables. In Example 2.3, $\texttt {Dom}(A) = \{(1,1),(1,2)\}$ .

3.1. Equidimensionality

If $A,B$ are $n \times n$ ASMs, define $A \ge B$ if ${rk}_A(i,j) \le {rk}_B(i,j)$ for all $1 \le i,j \le n$ . Restricted to permutation matrices, this is the (strong) Bruhat order on $S_n$ . Let

\begin{equation*} \textrm{Perm}(A) = \{w \in S_n : \text{$w \ge A$ and if $w \ge v \ge A$ for some $v \in S_n$, then $v=w$}\}. \end{equation*}

Proposition 3.1. [Reference Weigandt35, Proposition 5.4]. $I_A$ has the irredundant prime decomposition

\begin{equation*} I_A = \bigcap _{w \in \textrm{Perm}(A)} I_w. \end{equation*}

The codimension of $X_A$ is $\min \{\ell (w) \mid w \in \textrm{Perm}(A)\}$ .

Proof. If $A \in S_n$ , then the claim is trivial, and so we assume $A \notin S_n$ .

Suppose $w \in \textrm{Perm}(A)$ . We will first show that $1 \oplus w \in \textrm{Perm}(1 \oplus A)$ . From the definition of $1 \oplus A$ , we compute

\begin{equation*}{rk}_{1 \oplus A}(i,j) = \begin{cases} 1 & i=1 \mbox{ or } j=1\\ {rk}_A(i-1,j-1)+1 & \mbox{otherwise}. \end{cases} \end{equation*}

Thus,

(1) \begin{equation} \text{$w \gt A$ if and only if $1 \oplus w \gt {1 \oplus A}$.} \end{equation}

Suppose there exists a $v$ such that $1 \oplus w \ge v \gt 1 \oplus A$ . From the equations

\begin{equation*} {rk}_{1 \oplus w}(1,j) \le {rk}_{v}(1,j) \le {rk}_{1 \oplus A}(1,j) \end{equation*}

and

\begin{equation*} {rk}_{1 \oplus w}(1,j) = 1 = {rk}_{1 \oplus A}(1,j) \end{equation*}

for all $j \in [n+1]$ , we see that the first row of $v$ is the same as the first row of $1 \oplus A$ , which is the same as the first row of $1 \oplus w$ —specifically, the $1^{st}$ unit row vector. Similarly, the first column of all three matrices $1 \oplus w$ , $v$ , and $1 \oplus A$ must be the $1^{st}$ unit column vector.

Thus $v = 1 \oplus v'$ for some $v' \in S_n$ . By (1), $w \geq v'\gt A$ . By definition of $\textrm{Perm}(A)$ , the inequality $w \geq v' \gt A$ implies $w=v'$ , and so $1 \oplus w = 1 \oplus v'=v$ . Hence, $1 \oplus w \in \textrm{Perm}(1 \oplus A)$ .

In the other direction, fix some $u \in \textrm{Perm}(1 \oplus A)$ . We claim that $u = 1 \oplus u'$ for some $u' \in S_n$ . It suffices to show that $u(1) = 1$ .

Suppose for contradiction that $u(1) = j$ for some $j\gt 1$ and that $i$ is the least index so that $u(i)\lt j$ . Let $\nu = ut_{1,i}$ , where $t_{1,i}$ is the transposition $(1i)$ . That is, in matrix form, $\nu$ is the matrix obtained from $u$ by exchanging rows $1$ and $i$ .

We claim that $1 \oplus A \leq \nu \lt u$ , which will contradict the assumption $u \in \textrm{Perm}(1 \oplus A)$ . Note that ${rk}_\nu (\alpha ,\beta ) = {rk}_u(\alpha ,\beta )$ unless $\alpha \in [i-1]$ and $\beta \in [u(i), j-1]$ , in which case ${rk}_u(\alpha ,\beta ) = {rk}_\nu (\alpha ,\beta )-1$ (see Example 3.4 for a visualization). Thus, $\nu \lt u$ . Because $i$ was chosen minimally, ${rk}_u(\alpha ,\beta ) = 0$ for all $\alpha \in [i-1]$ and $\beta \in [u(i), j-1]$ , and so ${rk}_\nu (\alpha ,\beta ) = 1$ for all such $(\alpha , \beta )$ . But ${rk}_{1 \oplus A}(\alpha , \beta ) \ge 1$ for all $\alpha ,\beta$ . It follows from ${rk}_\nu (\alpha ,\beta ) = {rk}_u(\alpha ,\beta ) \leq {rk}_{1 \oplus A}(\alpha , \beta )$ unless $\alpha \in [i-1]$ and $\beta \in [u(i), j-1]$ together with ${rk}_\nu (\alpha ,\beta ) = 1 \leq {rk}_{1 \oplus A}(\alpha ,\beta )$ for $\alpha \in [i-1]$ and $\beta \in [u(i), j-1]$ that $1 \oplus A \leq \nu$ , completing the claim.

Hence, our arbitrary $u \in \textrm{Perm}(1 \oplus A)$ has the form $u = 1 \oplus u'$ for some $u' \in S_n$ . It remains to show that $u' \in \textrm{Perm}(A)$ . By (1), $u' \gt A$ . Hence, if $u' \notin \textrm{Perm}(A)$ , then there exists some $\tilde {u} \in S_n$ satisfying $A \lt \tilde {u}\lt u'$ . But then, by (1), $1 \oplus A \lt 1 \oplus \tilde {u}\lt u$ , contradicting the assumption $u \in \textrm{Perm}(1 \oplus A)$ .

Example 3.4. In order to help internalize the argument of Proposition 3.3, we will give a visualization of the region in which ${rk}_u$ and ${rk}_\nu$ differ for some $\nu = ut_{1,i}$ . Consider $u = 45213$ , in which case $j =4$ , $i = 3$ , $u(i) = 2$ , and $\nu = 25413$ . The region $[3-1] \times [2,4-1]$ , in which ${rk}_u$ and ${rk}_\nu$ differ, is shaded in yellow.

3.2. Vertex decomposition

We now review some basic definitions from simplicial complex theory. A simplicial complex $\Delta$ on $[n]$ is a set of subsets of $[n]$ such that, if $\sigma \in \Delta$ and $\tau \subseteq \sigma$ , then $\tau \in \Delta$ . We call the elements of $\Delta$ faces and the maximal faces (by inclusion) facets. The dimension of a face $\sigma \in \Delta$ is $|\sigma |-1$ , and the dimension of $\Delta$ is the maximum among dimensions of its faces. Faces of dimension $0$ are called $vertices$ . We call $\Delta$ pure if all of its facets have the same dimension.

Now, we discuss how these ideas from simplicial complex theory relate to the objects we want to study. The Stanley–Reisner correspondence is a bijection between simplicial complexes on $[n]$ and squarefree monomial ideals. For a subset $\sigma$ of $[n]$ , let $\mathbf{x}_\sigma = \prod _{i \in \sigma } x_i$ . The Stanley–Reisner ideal of the simplicial complex $\Delta$ is

\begin{equation*} I_{\Delta } = (\mathbf{x}_{\sigma } \mid \sigma \notin \Delta ). \end{equation*}

The complements of the facets of $\Delta$ correspond to the associated primes of $I_\Delta$ . Hence, $\Delta$ is pure if and only if the variety defined by $I_\Delta$ is equidimensional. We call $\Delta$ Cohen–Macaulay whenever $I_\Delta$ defines a Cohen–Macaulay quotient ring.

Given a simplicial complex $\Delta$ , we define the link of $\Delta$ at a face $\sigma$ of $\Delta$ by

\begin{equation*} \texttt {lk}_{\sigma }(\Delta ) = \{\tau \in \Delta \mid \tau \cap \sigma = \emptyset , \tau \cup \sigma \in \Delta \} \end{equation*}

and the deletion of $\Delta$ at a face $\sigma$ of $\Delta$ by

\begin{equation*} \texttt {del}_\sigma (\Delta ) = \{\tau \in \Delta \mid \tau \cap \sigma = \emptyset \}. \end{equation*}

Notice that $\texttt {lk}_\sigma (\Delta )$ is a subcomplex of $\texttt {del}_{\sigma }(\Delta )$ . When $\sigma = \{v\}$ is a vertex of $\Delta$ , we will often write $\texttt {lk}_{v}(\Delta )$ and $\texttt {del}_{v}(\Delta )$ for $\texttt { lk}_{\sigma }(\Delta )$ and $\texttt {del}_{\sigma }(\Delta )$ , respectively.

A simplicial complex $\Delta$ is vertex decomposable if $\Delta$ is pure and if either

1. (1) $\Delta = \{\emptyset \}$ , or if

2. (2) for some vertex $v$ in $\Delta$ , both $\texttt {del}_{v}(\Delta )$ and $\texttt {lk}_{v}(\Delta )$ are vertex decomposable.

Vertex decomposable simplicial complexes were introduced by Provan and Billera [Reference Provan and Billera29], who showed that vertex decomposable simplicial complexes are shellable, known more classically to be a virtue of a simplicial complex. Reisner [Reference Reisner31] gave a topological criterion characterizing Cohen–Macaulayness of $\Delta$ from which it follows that every shellable simplicial complex is Cohen–Macaulay, a desirable algebraic property we discuss further in Section 3.3.

We will use Stanley–Reisner theory to study initial ideals of ASM ideals. We refer the reader to [Reference Miller and Sturmfels27, Chapter 1] for more information on Stanley–Reisner theory and to [Reference Eisenbud10, Chapter 15] for general background on term orders and Gröbner bases.

Recall that $Z$ is a fixed $n \times n$ generic matrix. We call a term order antidiagonal if the lead term of the determinant of any submatrix $Y$ of $Z$ is the product of the entries along the antidiagonal of $Y$ . For each $A \in \tt ASM(n)$ , the Fulton generators of $I_A$ form a Gröbner basis with respect to any antidiagonal term order ([Reference Knutson and Miller17, Reference Knutson19, Reference Weigandt35]). (For $w \in S_n$ , Gao and Yong [Reference Gao and Yong12] recently determined a minimal generating set of $I_w$ which is already a Gröbner basis under any antidiagonal term order.) In particular, for any $A \in \tt ASM(n)$ , there is only one initial ideal arising from an antidiagonal term order. We call that initial ideal the antidiagonal initial ideal of $I_A$ , which we denote $\texttt {in} I_A$ .

Knutson and Miller [Reference Knutson and Miller17] showed that, for $w \in S_n$ , $\texttt {in} I_w$ is not only Cohen–Macaulay but, moreover, also the Stanley–Reisner ideal of a vertex decomposable simplicial complex. In fact, for a fixed $n$ , there is an ordering of the vertices (labeling the vertices by their associated variable)

\begin{equation*} z_{1,n} \gt z_{1,n-1} \gt \cdots \gt z_{1,1} \gt z_{2,n} \gt z_{2,n-1} \gt \cdots \gt z_{n,1}. \end{equation*}

by which the Stanley–Reisner complex of each $\texttt {in} I_w$ is vertex decomposable, independent of $w$ . We may hope that the same term order would show that every Stanley–Reisner ideal of every Cohen–Macaulay $\texttt {in} I_A$ is vertex decomposable. It turns out that that is false.

Example 3.6. For example, let

\begin{equation*} A = \begin{pmatrix} 0&\quad 1&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad 1\\ 1&\quad -1&\quad 1 & \quad0\\ 0&\quad 1&\quad 0& \quad 0 \end{pmatrix}, \;\; \texttt {in} I_A = (z_{11},z_{21},z_{12}z_{31},z_{31}z_{22},z_{22}z_{13}). \end{equation*}

The ideal of the deletion at that vertex corresponding to $z_{13}$ is

\begin{equation*} (z_{11},z_{21},z_{12}z_{31},z_{31}z_{22}) = (z_{11}, z_{21}, z_{12}, z_{22}) \cap (z_{11}, z_{21}, z_{31}), \end{equation*}

which has associated primes of different heights, which shows that the deletion of the Stanley–Reisner complex of $\texttt { in} I_A$ at the vertex corresponding to $z_{13}$ is not pure. Hence, the Stanley–Reisner complex of $\texttt{in} I_A$ is not vertex decomposable under the order used by Knutson and Miller.

If the Stanley–Reisner complex of $\texttt {in} I_A$ is vertex decomposable by the order dictated by [Reference Knutson and Miller17], we will call $A$ Knutson–Miller vertex decomposable, a definition which is local to this paper. We use the abbreviation CM for Cohen–Macaulay. The middle column of Figure 1 records the number of elements of $\tt ASM(n)$ that are obtained as $1 \oplus A$ from some $A \in \tt ASM(n-1)$ (which is equivalent to satisfying $A_{1,1} = 1$ ) and that are not Knutson–Miller vertex decomposable. Note, for example, that there are two such elements of $\tt ASM(5)$ but only one element of $\tt ASM(4)$ that is not Knutson–Miller vertex decomposable. Hence, there exists at least one element of $\tt ASM(5)$ of the form $1 \oplus A$ that is not Knutson–Miller vertex decomposable though $A$ is. Indeed, there is exactly one. The example is

\begin{equation*} A = \begin{pmatrix} 0 &\quad 0 &\quad 1 &\quad 0 \\ 0 &\quad 1 &\quad-1 &\quad 1 \\ 1 &\quad 0 &\quad 0 &\quad 0\\ 0 &\quad 0 &\quad 1 &\quad 0 \end{pmatrix}. \end{equation*}

This example shows that the operation $1 \oplus -$ does not preserve Knutson–Miller vertex decomposability. This failure also appears to become more frequent as $n$ grows. It is in part due to the frequency of this failure for small $n$ that we are meaningfully encouraged by the preservation of Cohen–Macaulayness under $1 \oplus -$ applied to all $A \in \tt ASM(n)$ for $n \leq 6$ , which we discuss further in Conjecture 3.15.

Figure 1.

Count of Cohen–Macaulay elements of $\tt ASM(n)$ that are and are not vertex decomposable by the Knutson–Miller ordering of the variables.

3.3. Cohen–Macaulayness

Cohen–Macaulayness of rings and varieties is a central consideration in commutative algebra and algebraic geometry, especially in the context of intersection theory. As Hochster said, “Life is really worth living in a Noetherian ring $R$ when all the local rings have the property that every [system of parameters] is an $R$ -sequence. Such a ring is called Cohen-Macaulay” [Reference Hochster13, p. 887]. For background on Cohen–Macaulay rings, we direct the reader to [Reference Bruns and Herzog5] and, especially for their import in combinatorial settings, to [Reference Hochster14].

In light of recent work of Conca and Varbaro [Reference Conca and Varbaro8], we will be able to study the Cohen–Macaulayness of each $R/I_A$ via the Cohen–Macaulayness of a suitable Gröbner degeneration.

Proposition 3.8. [Reference Weigandt35, Reference Knutson19, Reference Klein and Weigandt22]. If $w_1, \ldots , w_r, u_1, \ldots , u_k \in S_n$ and $A \in \tt ASM(n)$ are such that

\begin{equation*} I_{u_1}+\ldots +I_{u_k} = I_A = I_{w_1} \cap \cdots \cap I_{w_r}, \end{equation*}

then

\begin{equation*} \texttt {in} I_{u_1}+\ldots +\texttt {in} I_{u_k} = \texttt {in} I_A = \texttt {in} I_{w_1} \cap \cdots \cap \texttt {in} I_{w_r}. \end{equation*}

Our primary interest in what follows will be to infer Cohen–Macaulayness of $A$ from Cohen–Macaulayness of $1 \oplus A$ . Because the argument is essentially the same, we will prove something slightly stronger using the construction below. In the notation that follows, $1 \oplus A = \widetilde {A}(1,1)$ .

Fix $A \in \tt ASM(n)$ and $i, j \in [n]$ . We will build an element of $\tt ASM(n+1)$ , which we denote $\widetilde {A} = \widetilde {A}(i,j)$ , by the following rules:

\begin{equation*} \widetilde {A}_{a,b}=\begin{cases} A_{a,b}, & \text{if $a\lt i$ and $b\lt j$}\\ A_{a,b-1}, & \text{if $a\lt i$ and $b\gt j$}\\ A_{a-1,b}, & \text{if $a\gt i$ and $b\lt j$}\\ A_{a-1,b-1}, & \text{if $a\gt i$ and $b\gt j$}\\ 1, & \text{if $(a,b)=(i,j)$}\\ 0, & \text{if $a=i$ xor $b=j$}. \end{cases} \end{equation*}

The reader may verify that the Rothe diagram of $A$ embeds into the Rothe diagram of $\widetilde {A}$ . This close relationship may cause one to hope that the geometry of $X_A$ and $X_{\widetilde {A}}$ would be very closely related. However, as we discussed in Example 3.5, $X_A$ may be Cohen–Macaulay while $X_{\widetilde {A}}$ is not even unmixed. We now investigate special cases when $X_A$ and $X_{\widetilde {A}}$ are closely related.

As before, for $\tau \subseteq [n]$ , let $\mathbf{x}_\tau = \prod _{i \in \tau } x_i$ . Recall that, for an ideal $I$ and element $f$ of $R$ , we define the colon ideal $(I:f) = (r \in R \mid rf \in I)$ .

Lemma 3.12. Let $\Delta$ be a simplicial complex on $[n]$ . If $\sigma$ is a face of $\Delta$ , then

\begin{equation*} I_{\texttt {lk}_\sigma (\Delta )} = (I_{\Delta }: \mathbf{x}_\sigma )+(x_i \mid i \in \sigma ). \end{equation*}

Proof. Fix a squarefree monomial $\mathbf{x}_\tau$ corresponding to the subset $\tau$ of $[n]$ . If $\tau \cap \sigma \neq \emptyset$ , then $\mathbf{x}_\tau \in (x_i \mid i \in \sigma )$ . Also, by the definition of link, $\tau \notin \texttt{lk}_\sigma (\Delta )$ , and so $\mathbf{x}_\tau \in I_{\texttt {lk}_\sigma (\Delta )}$ .

Now assume $\tau \cap \sigma = \emptyset$ . Then

\begin{equation*} \mathbf{x}_\tau \in I_{\texttt {lk}_\sigma (\Delta )} \iff \tau \notin \texttt {lk}_\sigma (\Delta ) \iff \tau \cup \sigma \notin \Delta \iff \mathbf{x}_\tau \mathbf{x}_\sigma = \mathbf{x}_{\tau \cup \sigma } \in I_\Delta \iff \mathbf{x}_\tau \in I_\Delta :\mathbf{x}_\sigma . \end{equation*}

For $A \in \tt ASM(n)$ , we write $\Delta (A)$ for the Stanley–Reisner complex of $\texttt {in} I_A$ .

Proof. Recall that the antidiagonal initial ideals of ASM ideals are radical. Recall also that if a homogeneous ideal $I$ possesses a radical initial ideal $J$ then $I$ is a Cohen–Macaulay ideal if and only if $J$ is a Cohen–Macaulay ideal [Reference Conca and Varbaro8]. Hence, it suffices to show that, if $\texttt {in} I_{\widetilde {A}}$ is a Cohen–Macaulay ideal, then $\texttt {in} I_A$ is a Cohen–Macaulay ideal. We will show that, after a relabeling of vertices and deconing, $\Delta (A)$ is obtained as a link of $\Delta (\widetilde {A})$ , from which the result will follow because links of Cohen–Macaulay simplicial complexes are Cohen–Macaulay [Reference Reisner31].

For convenience, we will choose a non-standard indexing on the generic matrix from which equations for $I_{\widetilde {A}}$ are defined. Let $Z' = (z_{a,b})$ be an $(n+1) \times (n+1)$ generic matrix with $0 \leq a \leq n$ and $1 \leq b \leq n+1$ . Correspondingly, index the rows of $\widetilde {A}$ with the set $\{0,1,\ldots , n\}$ and the columns with $\{1,\ldots ,n+1\}$ , and consider $\Delta (\widetilde {A})$ as a complex on $\{0,\ldots ,n\} \times [n+1]$ with vertex $(a,b)$ corresponding to variable $z_{a,b}$ in the usual way. We retain the usual indexing for $A$ and use $Z = (z_{a,b})$ with $a,b \in [n]$ to define $I_A$ .

Example 3.14. If $A = \begin{pmatrix} \fbox {0} &\quad 1 &\quad 0\\ 1 &\quad \fbox {$-$1} &\quad 1\\ 0 &\quad 1 &\quad 0 \end{pmatrix}$ and $j=2$ , then $\widetilde {A} = \begin{pmatrix} {\color {blue}0} &\quad {\color {blue}1} &\quad {\color {blue}0} &\quad {\color {blue}0}\\ \fbox {0} &\quad {\color {blue}0} &\quad 1 &\quad 0\\ 1 &\quad {\color {blue}0} &\quad \fbox {$-$1} &\quad 1\\ 0 &\quad {\color {blue}0} &\quad 1 &\quad 0 \end{pmatrix}$ . With the prescribed indexing, $\texttt {Ess}(A) = \{(1,1),(2,2)\}$ and $\texttt {Ess}(\widetilde {A}) = \{(1,1),(2,3)\}$ . Note also that ${rk}_{\widetilde {A}}(1,1) ={rk}_A(1,1)$ (because column $1$ is to the left of column $j=2$ ) and that ${rk}_{\widetilde {A}}(2,3) = {rk}_A(2,2)+1$ (because column $3$ is to the right of column $j=2$ ). Then

\begin{equation*} I_A = \left (z_{1,1}, \begin{vmatrix} z_{1,1} &\quad z_{1,2}\\ z_{2,1} &\quad z_{2,2} \end{vmatrix}\right ) \mbox{, and } I_{\widetilde {A}} = \left (z_{0,1},z_{1,1}, \begin{vmatrix} z_{0,1} &\quad z_{0,2} &\quad z_{0,3}\\ z_{1,1} &\quad z_{1,2} &\quad z_{1,3}\\ z_{2,1} &\quad z_{2,2} &\quad z_{2,3} \end{vmatrix}\right ), \end{equation*}

where the matrix entries corresponding to essential cells are boxed.

We claim that, with this choice of indexing,

(2) \begin{align} \texttt {in} I_A+(z_{0,1}, \ldots ,z_{0,j-1}, z_{0,j+1}, \ldots , z_{0,n+1}) &= (\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1})+(z_{0,j+1}, \ldots , z_{0,n+1}) \notag \\ &= I_{\texttt {lk}_{\{(0,j+1), \ldots , (0,n+1)\}}\Delta (\widetilde {A})}. \end{align}

Because $I_A$ has a generating set that does not involve the variables $z_{0,1}, \ldots , z_{0,n+1}$ , $\texttt {in} I_A$ is a Cohen–Macaulay ideal if and only if $\texttt {in} I_A+(z_{0,1}, \ldots z_{0,j-1}, z_{0,j+1}, \ldots , z_{0,n+1})$ is. Similarly, $\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1}$ has a generating set that does not involve the variables $z_{0,j+1}, \ldots , z_{0,n+1}$ , and so $(\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1})+(z_{0,j+1}, \ldots , z_{0,n+1})$ is a Cohen–Macaulay ideal because $\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1}$ is. Hence, the desired result will follow from establishing Equation (2).

The latter equality of Equation (2) follows from Lemma 3.12. In order to establish the first equality of Equation (2), we will first show

\begin{equation*} \texttt {in} I_A+(z_{0,1}, \ldots , z_{0,j-1}, z_{0,j+1}, \ldots , z_{0,n+1}) \subseteq (\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1})+(z_{0,j+1}, \ldots , z_{0,n+1}). \end{equation*}

We note that the containments $(z_{0,1}, \ldots , z_{0,j-1}) \subseteq \texttt {in} I_{\widetilde {A}} \subseteq \texttt { in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1}$ are immediate from the definitions of rank function, ASM ideal, and colon ideal. Fix $\mu \in \texttt {in} I_A$ . We may assume that $\mu$ is the leading term of some Fulton generator $f$ of $I_A$ determined by essential cell $(a,b)$ of $A$ .

If $b\lt j$ , then $(a,b)$ is also an essential cell of $\widetilde {A}$ and ${rk}_A(a,b) = {rk}_{\widetilde {A}}(a,b)$ . Hence, $f$ is a Fulton generator of $I_{\widetilde {A}}$ , and so $\mu \in \texttt {in} I_{\widetilde {A}} \subseteq \texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1}$ .

If $b \geq j$ , then $(a,b+1)$ is an essential cell of $\widetilde {A}$ and ${rk}_A(a,b) = {rk}_{\widetilde {A}}(a,b+1)-1$ . Let $M$ be the $({rk}_A(a,b)+1) \times ({rk}_A(a,b)+1)$ submatrix of $Z$ so that $f = \det (M)$ . Form a submatrix $M'$ of $Z'$ by augmenting the set of rows of $M$ by $0$ and the set of columns of $M$ by $b+1$ . Then, $M'$ is a $({rk}_{\widetilde {A}}(a,b+1)+1) \times ({rk}_{\widetilde {A}}(a,b+1)+1)$ submatrix of $Z'$ weakly northwest of $(a,b+1)$ . Hence, $\det (M')$ is a Fulton generator of $I_{\widetilde {A}}$ . But $z_{0,b+1}\mu = \texttt {in} \det (M')$ , and so $\mu \in \texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1}$ , as desired.

It remains to show

\begin{equation*} (\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1})+(z_{0,j+1}, \ldots , z_{0,n+1}) \subseteq \texttt {in} I_A+(z_{0,1}, \ldots ,z_{0,j-1}, z_{0,j+1}, \ldots , z_{0,n+1}), \end{equation*}

for which it suffices to show $\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1} \subseteq \texttt {in} I_A+(z_{0,1}, \ldots ,z_{0,j-1}, z_{0,j+1}, \ldots , z_{0,n+1})$ . Fix a monomial $\nu \in \texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1}$ . First suppose $\nu \in \texttt {in} I_{\widetilde {A}}$ . We may assume that $\nu$ is the leading term of some Fulton generator $g$ of $I_{\widetilde {A}}$ . Then, there exist $(c,d) \in \texttt {Ess}(\widetilde {A})$ and $N'$ an $({rk}_{\widetilde {A}}(c,d)+1) \times ({rk}_{\widetilde {A}}(c,d)+1)$ submatrix of $Z'$ weakly northwest of $(c,d)$ so that $g = \det (N')$ .

If $d\lt j$ , then, either $\nu \in (z_{0,1}, \ldots , z_{0,j-1})$ , which occurs if $N'$ involves row $0$ , or $g$ is a Fulton generator of $I_A$ belonging to essential cell $(c,d)$ , which occurs if $N'$ does not involve row $0$ . If $d \gt j$ , then $\nu$ is equal to the product of the northeast entry of $N'$ with the lead term of Fulton generators of $A$ belonging to essential cell $(c,d-1)$ of $A$ . Note that $d \neq j$ because $\widetilde {A}$ has no essential cells in column $j$ .

Finally, suppose $\nu \in (\texttt {in} I_{\widetilde {A}}:z_{0,j+1}\cdots z_{0,n+1})\setminus \texttt {in} I_{\widetilde {A}}$ . Because the variables $z_{0,k}$ , $k \in [j+1,n+1]$ , all belong to the same row, each minimal generator of $\texttt {in} I_{\widetilde {A}}$ is divisible by at most one such $z_{0,k}$ . We may assume there exists $k\in [j+1,n+1]$ so that $z_{0,k}\nu$ is the product of antidiagonal entries of a submatrix of $Z'$ pertaining to an essential cell $(c,d)$ of $\widetilde {A}$ with $d\gt j$ . Then, $\nu$ is the product of antidiagonal entries of a submatrix of $Z$ pertaining to essential cell $(c,d-1)$ , which is to say that $\nu \in \texttt {in} I_A$ , completing the proof.

We have now seen that $A$ is equidimensional if and only if $1 \oplus A$ is equidimensional and that $1 \oplus A$ Cohen–Macaulay implies $A$ Cohen–Macaulay. A Macaulay2 computation confirms that, over the field of rational numbers, if $A \in \tt ASM(n)$ with $n \leq 6$ and $A$ is Cohen–Macaulay, then $1 \oplus A$ is Cohen–Macaulay. Based on this evidence, we make the following conjecture:

The truth of Conjecture 3.15 would have implications for any ASM obtained as a diagonal block sum of others. Our next goal is to state and prove that implication, for which we require a couple of lemmas.

The following is a routine exercise, which can be found, for example, in [Reference Villarreal34, Chapter 3].

Recall that the support of a monomial ideal $J$ , denoted $\texttt {Supp}(J)$ , is the set of variables dividing some minimal monomial generator of $J$ . If $J = \texttt {in} I_w$ for some $w \in S_n$ , then $\texttt {Supp}(J)$ is sometimes called the core of $w$ .

The following lemma is known to experts. We record it below for completeness.

Example 3.18. Consider $w = 31542$ , whose Rothe diagram is below. The cells $(i,j)$ with $z_{i,j} \in \texttt {Supp}(\texttt {in} I_w)$ are those with an orange line running across their antidiagonal.

Proof. Consider $R = \mathbb{C}[z_{i,j} \mid i, j \in [m+n]]$ as the ambient polynomial ring of $I_A$ . Consider the subrings $S_1 = \mathbb{C}[z_{i,j} \mid i+j \leq m]$ and $S_2 = \mathbb{C}[z_{i,j} \mid i+j \gt m]$ of $R$ . Then $R = S_1 \otimes _{\mathbb{C}} S_2$ .

For $k \geq 1$ , let $\mathbb{I}_k$ denote the $k \times k$ identity matrix, and observe that

\begin{equation*} I_A = I_{A_1 \oplus \mathbb{I}_n}+I_{\mathbb{I}_m \oplus A_2}. \end{equation*}

By Proposition 3.8,

\begin{equation*} \texttt {in} I_A = \texttt {in} I_{A_1 \oplus \mathbb{I}_n}+ \texttt {in} I_{\mathbb{I}_m \oplus A_2}. \end{equation*}

By Lemma 3.17, $\texttt {in} I_{A_1+\mathbb{I}_n}$ is supported only on variables $z_{i,j}$ with $i+j\leq m$ . Note that $\texttt {in} I_{\mathbb{I}_m \oplus A_2}$ is supported only on $z_{i,j}$ with $i+j\gt m+1$ . Specifically, $\texttt {in} I_{A_1 \oplus \mathbb{I}_n}$ and $\texttt {in} I_{\mathbb{I}_m \oplus A_2}$ are supported on disjoint sets of variables; the former on a subset of the variables of $S_1$ and the latter on a subset of the variables of  $S_2$ .

(a) By Lemma 3.16, the associated primes $P$ of $\texttt {in} I_A$ are exactly those ideals of the form $Q_1+Q_2$ where $Q_1$ is an associated prime of $\texttt {in} I_{A_1 \oplus \mathbb{I}_n}$ and $Q_2$ is an associated prime of $\texttt {in} I_{\mathbb{I}_m \oplus A_2}$ .

Combining Proposition 3.8 and Proposition 3.9, the associated primes of $\texttt {in} I_{A_1 \oplus \mathbb{I}_m}$ are exactly those ideals of the form $\texttt {in} I_{u'}$ for some $u' \in \textrm{Perm}(A_1 \oplus \mathbb{I}_n)$ , and the associated primes of $\texttt {in} I_{\mathbb{I}_m \oplus A_2}$ are exactly those of the form $\texttt {in} I_{v'}$ for some $v' \in \textrm{Perm}(\mathbb{I}_m \oplus A_2)$ . Because $I_{A_1 \oplus \mathbb{I}_n} = I_{A_1}R$ , those permutations $u'$ are exactly those of the form $u \oplus \mathbb{I}_m$ for some $u \in \textrm{Perm}(A_1)$ . By Proposition 3.3, those permutations $v'$ are exactly those of the form $\mathbb{I}_m \oplus v$ for some $v \in \textrm{Perm}(A_2)$ . Then $(u,v) \in \textrm{Perm}(A_1) \times \textrm{Perm}(A_2)$ if and only if $I_{u \oplus v} = I_{u \oplus \mathbb{I}_n}+I_{\mathbb{I}_m \oplus v}$ is an associated prime of $I_A$ , which is equivalent to $u \oplus v \in \textrm{Perm}(A)$ . This completes the proof of the bijection between $\textrm{Perm}(A_1) \times \textrm{Perm}(A_2)$ and $\textrm{Perm}(A)$ .

Because, for any $u \in S_m$ and $v \in S_n$ , $\ell (u \oplus v) = \ell (u)+\ell (v)$ , the codimension and equidimensionality claims now follow from Proposition 3.1.

(b) We turn to the Cohen–Macaulayness claim. By Theorem 3.10, $I_A$ , $I_{A_1 \oplus \mathbb{I}_n}$ , and $I_{\mathbb{I}_m\oplus A_2}$ define Cohen–Macaulay quotient rings if and only if $\texttt {in} I_A$ , $\texttt {in} I_{A_1 \oplus \mathbb{I}_n}$ , and $\texttt {in} I_{\mathbb{I}_m\oplus A_2}$ do, respectively. By Proposition 3.16, $\texttt {in} I_{A_1 \oplus \mathbb{I}_n}+\texttt {in} I_{\mathbb{I}_m\oplus A_2}$ defines a Cohen–Macaulay quotient if and only if both $\texttt {in} I_{A_1 \oplus \mathbb{I}_n}$ and $\texttt {in} I_{\mathbb{I}_m\oplus A_2}$ do. Because $I_{A_1 \oplus \mathbb{I}_n} = I_{A_1}R$ , $I_{A_1 \oplus \mathbb{I}_n}$ and $I_{A_1}$ define Cohen–Macaulay quotients or not alike. By Theorem 3.13, if $I_{\mathbb{I}_m\oplus A_2}$ defines a Cohen–Macaulay quotient ring, then $I_{A_2}$ does. If Conjecture 3.15 is true, the converse is true as well.

In Proposition 3.19, one may alternatively establish the codimension and equidimensionality claims by appealing to Lemma 3.16 together with the fact that codimension does not change under Gröbner degeneration (see, e.g., [Reference Eisenbud10, Chapter 15]).

Proposition 3.20. If $A$ admits a block-matrix decomposition of the form

\begin{equation*}A = \left (\begin{array}{@{}c|c@{}} 0 & A_1\\\hline A_2 & 0\\ \end{array}\right ), \end{equation*}

where both $A_1 \in \tt ASM(m)$ and $A_2 \in \tt ASM(n)$ are ASMs, then $A$ is unmixed (respectively, Cohen–Macaulay) if and only if $A_1$ and $A_2$ are both unmixed (respectively, Cohen–Macaulay).

In the next section, we will see through Proposition 4.3 that the special case of a block sum decomposition is not quite so specialized as it may seem in the sense that ASMs do not easily contain one another.