\section*{Introduction}

Gr\"obner bases are the most widely applicable
computational tool available in the context of commutative
algebra and algebraic geometry.
However they also are an important theoretical tool, as they can be
used to establish properties such as, e.g., primality,
normality, Cohen-Macaulayness, and to give formulas for the height of
an ideal. Liaison theory, or linkage, on the other hand, is mostly
regarded as a classification tool. In fact, much effort has been
devoted in recent years to the study of liaison classes, in particular
to deciding which ideals belong to the G-liaison class of a complete
intersection. However, a clear understanding of the liaison pattern of
an ideal often allows us to recursively compute invariants such as its
Hilbert function and graded Betti numbers.

In this paper we introduce liaison-theoretic methods as a tool in the
theory of Gr\"obner bases. More precisely, we deduce that a certain set
${\mathcal G}$ of polynomials is a Gr\"obner basis for the ideal $I$ that it
generates by understanding the linkage pattern of the ideal $I$ and of
the monomial ideal generated by the initial terms of the elements of
${\mathcal G}$. Concretely, we apply this reasoning to ideals generated by minors
or pfaffians, whose liaison pattern we understand.

Ideals generated by minors or pfaffians have been studied extensively
by both commutative algebraists and algebraic geometers. The study of
determinantal rings and varieties is an active area of research per
se, but it has also been instrumental to the development of new
techniques, which have become part of the commonly used tools in
commutative algebra. Ideals generated by minors and pfaffians are
studied in invariant theory and combinatorics, and are relevant in
algebraic geometry. In fact, many classical varieties such as the
Veronese and the Segre varieties are cut out by minors or
pfaffians. Degeneracy loci of morphisms between direct sums
of line bundles over projective space have a determinantal
description, as do Schubert varieties, Fitting schemes, and some toric
varieties. Ideals generated by minors or pfaffians are
often investigated by commutative algebraists by
means of Gr\"obner basis techniques (see, e.g., \cite{br88},
\cite{st90}, \cite{he92}). Using such tools, many families of ideals
generated by minors or pfaffians have been shown to enjoy properties
such as primality, normality, and Cohen-Macaulayness.
A different approach to the study of ideals generated by minors is via
flags and degeneracy loci, and was initiated in~\cite{fu91}. Such an
approach allows one to establish that these ideals are normal,
Cohen-Macaulay and have rational singularities (see, e.g.,
\cite{kn05}, \cite{kn09}, \cite{kn10}).

In recent years, much progress has been made towards understanding
determinantal ideals and varieties also from the point of view of
liaison theory. A central open question in liaison theory asks whether
every arithmetically Cohen-Macaulay scheme is glicci, i.e., whether it
belongs to the G-liaison class of a complete intersection. In~\cite{ga53},
\cite{kl01}, \cite{go07a}, \cite{go07b}, \cite{go08}, \cite{de09},
and~\cite{go10}, several families of ideals generated by minors or
pfaffians are shown to be glicci. More precisely, it is shown that
they can be obtained from an ideal generated by linear forms via a
sequence of ascending elementary G-biliaisons. Moreover, each of the
elementary G-biliaisons takes place between two ideals which both
belong to the family in question. Since the linkage steps are
described very explicitly, in theory it is possible to use the linkage
results to recursively compute invariants or establish properties of
these ideals. This has been done, e.g., in~\cite{co09} using the
linkage results from~\cite{go07b}.

Rather than contributing to the theory of liaison (see for
instance~\cite{mi98}), in this paper we give a new method
of using liaison as a tool.
More precisely, we consider large families of ideals generated by
minors or pfaffians in a matrix or a ladder, namely pfaffian ideals of
ladders, mixed determinantal, and symmetric mixed determinantal
ideals. Combining the liaison results from~\cite{go07b}, \cite{de09},
and~\cite{go10} with a Hilbert function computation, we are able to
prove that the pfaffians or the minors are a reduced Gr\"obner basis
for the ideal that they generate, with respect to any anti-diagonal or
diagonal term-order.  Moreover, we show the simplicial complex
corresponding to the initial ideal of any ideal in the family that we
consider is vertex decomposable. Vertex decomposability is a strong
property, which in particular implies shellability of the complex and
Cohen-Macaulayness of the associated initial ideal.

In Section~\ref{mainlemma}, we prove a lemma which will be central to
the subsequent arguments (Lemma~\ref{inid}). The lemma gives a
sufficient criterion for a monomial ideal to be the initial ideal of a
given ideal $J$. Both the ideal $J$ and the ``candidate'' initial
ideal are constructed via Basic Double Linkage or elementary
biliaison. In Section~\ref{ex_sect} we use Lemma~\ref{inid} to prove
that the maximal minors of a matrix of indeterminates are a Gr\"obner
basis of the ideal that they generate with respect to any diagonal (or
anti-diagonal) term-order. Although the result is well-known, we wish
to illustrate our method by showing how it applies to this simple
example. In Sections~\ref{pfaff_sect}, \ref{symm_sect}, and
\ref{ladd_sect}, we apply our technique to ideals generated by:
pfaffians of mixed size in a ladder of a skew-symmetric matrix of
indeterminates, minors of mixed size in a ladder of a symmetric matrix
of indeterminates, and minors of mixed size in a ladder of a
matrix of indeterminates. We prove that the natural generators of
these ideals are a Gr\"obner basis with respect to any diagonal (in
the case of minors of a symmetric matrix) or anti-diagonal (in the
case of pfaffians or minors in a generic matrix) term-order. We also
prove that the corresponding initial ideals are Cohen-Macaulay, and
that the associated simplicial complexes are vertex decomposable.
While Sections~\ref{mainlemma} and~\ref{ex_sect} are meant to be read
first, Sections~\ref{pfaff_sect}, \ref{symm_sect}, and~\ref{ladd_sect}
can be read independently of each other, and in any order. In the
appendix, we indicate how our liaison-theoretic approach can be made
self-contained in order to derive also all the classical
Gr\"obner basis results about ladder determinantal ideals from
one-sided ladders.

\section{Linkage and Gr\"obner bases}\label{mainlemma}

Let $K$ be an arbitrary field and let $R$ be a standard graded
polynomial ring in finitely many indeterminates over $K$.
In this section, we give a sufficient condition for a set ${\mathcal G}$
of polynomials to be a Gr\"obner basis with respect to a given
term-order for the ideal $I$ that it generates. Our criterion depends
on the linkage pattern of the ideal $I$ and of the monomial ideal
generated by the initial terms of the elements of ${\mathcal G}$.

In order to use geometric language, we need to consider the
algebraic closure of the field $K$. Notice however that restricting
the field of coefficients does not affect the property of being a
Gr\"obner basis, as long as the polynomials are defined over the
smaller field. More precisely, if $I=(g_1,\ldots,g_s)\subset
K[x_0,\ldots,x_n]$ and $g_1,\ldots,g_s$ have coefficients in a
subfield $k$ of $K$, then: $g_1,\ldots,g_s$ are a Gr\"obner
basis of $I\subseteq K[x_0,\ldots,x_n]$ if and only if they are a
Gr\"obner basis of $I\cap k[x_0,\ldots,x_n]$.
In this sense, the property of being a Gr\"obner basis
does not depend on the field of definition. Therefore, while proving
that a set ${\mathcal G}$ of polynomials is a  Gr\"obner basis with respect to a
given term-order for the ideal $I$ that it generates, we may pass to
the algebraic closure without loss of generality. We shall therefore
assume without loss of generality that the field $K$ is algebraically
closed.

\begin{notat}
Fix a term-order $\sigma$. Let $I\subset R$ be an ideal and let
${\mathcal G}$ be a set of polynomials in $R$. We denote by $in(I)$ the initial
ideal of $I$ with respect to $\sigma$, and by $in({\mathcal G})$ the set of
initial terms of the elements of ${\mathcal G}$ with respect to $\sigma$.
\end{notat}

For the convenience of the reader, we recall the definition of
diagonal and anti-diagonal term-order.

\begin{defn}
Let $X$ be a matrix (resp. a skew-symmetric or a symmetric matrix) of
indeterminates. Let $\sigma$ be a term-order on the set of terms of
$K[X]$. The term-order $\sigma$ is {\bf diagonal} if the leading term
with respect to $\sigma$ of the determinant of a submatrix of $X$ is
the product of the indeterminates on its diagonal. It is {\bf
  anti-diagonal} if the leading term with respect to $\sigma$ of the
determinant of a submatrix of $X$ is the product of the indeterminates
on its anti-diagonal.
\end{defn}

\begin{notat}
Let $A$ be a finitely generated, graded $R$-module. We denote by
$H_A(d)$ the {\bf Hilbert function} of $A$ in degree $d$, i.e., the
dimension of $A_d$ as a $k$-vector space.
\end{notat}

In this paper we study large families of ideals generated by minors or
pfaffians in a matrix or a ladder, where the size of the minors or
pfaffians is allowed to vary in different regions of the matrix or the
ladder. We study their initial ideals with respect to a diagonal or
anti-diagonal term-order, and we prove that the associated simplicial
complexes are vertex decomposable. In particular, the initial ideals in
question are Cohen-Macaulay. For the convenience of the reader, we now
recall the main definitions.

\begin{defn}
A {\bf simplicial complex} $\Delta$ on $n+1$ vertices, is a collection
of subsets of $\{0,\ldots,n\}$ such that for any $F\in\Delta$, if
$G\subseteq F$, then $G\in\Delta$. An $F\in\Delta$ is called a {\bf
  face} of $\Delta$. The dimension of a face $F$ is $\dim F=|F|-1$,
and the {\bf dimension} of the complex is
$$\dim\Delta=\max\{\dim F\mid F\in\Delta\}.$$
The complex $\Delta=2^{\{0,\ldots,n\}}$ is called a {\bf simplex}.

The {\bf vertices} of $\Delta$ are the subsets of $\{0,\ldots,n\}$ of
cardinality one. The faces of $\Delta$ which are maximal with respect
to inclusion are called {\bf facets}. A complex is {\bf pure} if all
its facets have dimension equal to the dimension of the complex.
\end{defn}

\begin{notat}
To each face $F\in\Delta$ we associate the following two
simplicial subcomplexes of $\Delta$: the {\bf link} of $F$
$$\lk_F(\Delta)=\{G\in\Delta\mid F\cup G\in\Delta, F\cap
G=\emptyset\}$$ and the {\bf deletion}
$$\Delta-F=\{G\in\Delta\mid F\cap G=\emptyset\}.$$
If $F=\{k\}$ is a vertex, we denote the link of $F$ and the deletion by
$\lk_k(\Delta)$ and $\Delta-k$, respectively.
\end{notat}

\begin{defn}
A simplicial complex $\Delta$ is {\bf vertex decomposable} if it is a
simplex, or it is the empty set, or there exists a vertex $k$ such that
$\lk_k(\Delta)$ and $\Delta-k$ are both pure and vertex decomposable,
and $$\dim\Delta=\dim(\Delta-k)=\dim\lk_k(\Delta)+1.$$
\end{defn}

In this article, we show that the simplicial complexes associated to
the initial ideals of the family of ideals that we consider are vertex
decomposable.

\begin{defn}
The {\bf Stanley-Reisner ideal} associated to a complex $\Delta$ on
$n+1$ vertices is the squarefree monomial ideal
$$I_{\Delta}=(x_{i_1},\ldots,x_{i_s}\mid
\{i_1,\ldots,i_s\}\not\in\Delta)\subset K[x_0,\ldots,x_n].$$
Conversely, to every squarefree monomial ideal $I\subseteq
K[x_0,\ldots,x_n]$ one can associate the unique simplicial complex
$\Delta(I)$ on $n+1$ vertices, such that $I_{\Delta(I)}=I.$
\end{defn}

\begin{rmk}
\begin{enumerate}
\item A vertex $\{k\}\in\Delta$ is called a {\bf cone point} if for every
face $F\in\Delta$, $F\cup\{k\}\in\Delta$. If $\Delta$ has a cone point
$\{k\}$, then $$I_{\Delta}=I_{\Delta-k}K[x_0,\ldots,x_n].$$
Moreover, $\Delta$ is vertex decomposable if and only if $\Delta-k$
is. Therefore, {\em we will not distinguish between a complex and a
  cone over it}.
\item Notice that, if $\Delta$ is a complex on $n+1$ vertices, then
  both $\lk_k(\Delta)$ and $\Delta-k$ are complexes on $n$ (or fewer)
  vertices. However, since we do not distinguish between a complex and
  a cone over it, we will regard them as complexes on $n+1$ vertices.
\item On the side of the associated Stanley-Reisner ideals, let $I$
  and $J$ be squarefree monomial ideals such that the generators of $I$
  involve fewer variables than the generators of $J$. Then we may associate
  to $I$ and $J$ simplicial complexes $\Delta(I)$ and $\Delta(J)$ on
  the same number of variables. This amounts to regarding $I$ and $J$
  as ideals in the same polynomial ring.
\end{enumerate}
\end{rmk}

We now recall some definitions from liaison theory that will be
fundamental throughout the paper.

\begin{defn}\label{g0}
Let $J\subset R$ be a homogeneous, saturated ideal. We say that $J$
is {\bf Gorenstein in codimension $\leq$ c} if the localization
$(R/J)_P$ is a Gorenstein ring for any prime ideal $P$ of $R/J$ of
height smaller than or equal to $c$. We often say that $J$ is
$G_c$. We call {\bf generically Gorenstein}, or $G_0$, an ideal $J$
which is Gorenstein in codimension 0.
\end{defn}

\begin{defn}\label{bdl}
Let $A\subset B\subset R$ be homogeneous ideals such that $\hgt A=\hgt
B- 1$ and $R/A$ is Cohen-Macaulay. Let $f\in R_d$ be a homogeneous element
of degree $d$ such that $A:f=A$. The ideal $C:=A+fB$ is called a {\bf Basic
Double Link} of degree $d$ of $B$ on $A$. If moreover $A$ is $G_0$ and
$B$ is unmixed, then $C$ is a {\bf Basic Double G-Link} of $B$ on $A$.
\end{defn}

\begin{defn}\label{bilid}
Let $I,J\subset R$ be homogeneous, saturated, unmixed ideals, such
that $\hgt(I)=\hgt(J)=c$. We say that $J$ is obtained by an {\bf
elementary biliaison} of height $\ell$ from $I$ if there exists a
Cohen-Macaulay ideal $N$ in $R$ of height $c-1$
such that $N\subseteq I\cap J$ and $J/N\cong [I/N](-\ell)$ as
$R/N$-modules. If in addition the ideal $N$ is $G_0$, then $J$ is
obtained from $I$ via an {\bf elementary G-biliaison}. If $\ell>0$ we
have an {\bf ascending} elementary G-biliaison.
\end{defn}

We refer to~\cite{mi98}, \cite{kl01}, and~\cite{ha07} for the basic
properties of Basic Double Linkage and elementary biliaison. Notice in
particular that, if $C$ is a Basic Double Link of $B$ on $A$, then it is
not known in general whether $B$ and $C$ belong to the same G-liaison
class. On the other side, if $C$ is a Basic Double G-Link of $B$ on
$A$, then $B$ and $C$ can be G-linked in two steps.

We are now ready to state a sufficient condition for a set
of polynomials to be a Gr\"obner basis (with respect to a given
term-order) for the ideal that they generate.

\begin{lemma}\label{inid}
Let $I,J,N\subset R$ be homogeneous, saturated, unmixed ideals, such that
$N\subseteq I\cap J$ and $\hgt(I)=\hgt(J)=\hgt(N)+1$. Assume that $N$
is Cohen-Macaulay.
Let $A,B,C\subset R$ be monomial ideals
such that $C\subseteq in(J)$, $A=in(N)$ and $B=in(I)$ with respect to
some term-order $\sigma$. Assume that $A$ is Cohen-Macaulay and
that $\hgt(B)=\hgt(A)+1$. Suppose that $J$ is obtained from $I$ via an
elementary biliaison of height $\ell$ on $N$, and that $C$ is a Basic
Double Link of degree $\ell$ of $B$ on $A$. Then $C=in(J)$.
\end{lemma}

\begin{proof}
Since $C\subseteq in(J)$, it suffices to show that $H_C(d)=H_J(d)$ for
all $d\in{\mathbb Z}$. This is indeed the case, since
$$H_C(d)=H_B(d-\ell)+H_A(d)-H_A(d-\ell)=H_I(d-\ell)+H_N(d)-H_N(d-\ell)=H_J(d).$$
\end{proof}

\begin{rmks}
\begin{enumerate}
\item Notice that, if $J$ is obtained from $I$ via an elementary
  biliaison on $N$, we do not know in general whether they belong
  to the same G-liaison class. However, if in addition $N$ is
  generically Gorenstein, then $J$ is obtained from $I$ via an
  elementary G-biliaison on $N$. In particular, it can be obtained
  from $I$ via two Gorenstein links on $N$.
\item If in addition $A$ is generically Gorenstein, then $C$ is a
  Basic Double G-Link of $B$ on $A$. In particular, it can be obtained
  from $B$ via two Gorenstein links on $A$.
\item The concepts of Basic Double Linkage and biliaison are
  interchangeable in the statement of Lemma~\ref{inid}. More
  precisely, Basic Double Linkage is a special case of
  biliaison. Moreover, it can be shown that if $J$ is obtained from
  $I$ via an elementary biliaison of height $\ell$ on $N$, then there
  exist an ideal $H$ and a $d\in{\mathbb Z}$ s.t. $H$ is a Basic Double Link
  of degree $d+\ell$ of $I$ on $N$ and also a Basic Double Link of
  degree $d$ of $J$ on $N$. Then it is easy to verify that the lemma
  holds under the weaker assumption that $C$ is obtained from $B$ via
  an elementary biliaison of height $\ell$ on $A$.
\end{enumerate}
\end{rmks}

In the next section, we use the lemma to prove that the maximal minors
of a matrix of indeterminates are a Gr\"obner basis of the ideal that
they generate with respect to any diagonal term-order. Although the
result is well-known, we wish to illustrate our method by showing how it
applies to this simple example.
In Sections~\ref{pfaff_sect}, \ref{symm_sect}, and \ref{ladd_sect}, we
apply the lemma to ideals generated by:
pfaffians of mixed size in a ladder of a skew-symmetric matrix of
indeterminates, minors of mixed size in a ladder of a symmetric matrix
of indeterminates, and minors of mixed size in a one-sided ladder of a
matrix of indeterminates. We prove that the natural generators of
these ideals are a Gr\"obner basis with respect to any diagonal (in
the case of minors of a symmetric matrix) or anti-diagonal (in the
case of pfaffians or minors in a generic matrix) term-order. We also
prove that their initial ideals are squarefree and that they can be
obtained from an ideal generated by indeterminates via a sequence of
Basic Double G-links of degree 1, which only involve squarefree
monomial ideals. In particular, they are glicci (i.e., they can be
obtained from a complete intersections via a sequence of G-links),
hence they are Cohen-Macaulay. Moreover, we prove that the simplicial
complexes associated to their initial ideals are vertex decomposable.

Notice that, if we knew a priori that the simplicial complexes
associated to the initial ideals are vertex decomposable, then we
could deduce that the corresponding squarefree monomial ideals are
glicci by the following result of Nagel and R\"omer. However, we
cannot directly apply their result in our situation, since we need to
first produce the Basic Double G-links on the squarefree monomial
ideals, in order to deduce that the associated simplicial complexes
are vertex decomposable.

\begin{thm}[\cite{na08}, Theorem~3.3]
Let $\Delta$ be a simplicial complex on $n+1$ vertices and let
$I_{\Delta}\subset K[x_0,\ldots,x_n]$ be the Stanley-Reisner ideal of
$\Delta$. Assume that $\Delta$ is (weakly) vertex decomposable. Then
$I_{\Delta}$ can be obtained from an ideal generated by indeterminates
via a sequence of Basic Double G-links of degree 1, which only involve
squarefree monomial ideals.
\end{thm}

Notice that, although the statement above is slightly stronger than
Theorem~3.3 in~\cite{na08}, the result above follows from the proof
in~\cite{na08}.

\section{A simple example: ideals of maximal minors}\label{ex_sect}

This section is meant to illustrate the idea and the method of our
proof on a simple example. We prove that the maximal minors
of a matrix of indeterminates are a Gr\"obner basis of the ideal that
they generate with respect to any diagonal term-order. Notice that for
the case of minors in a matrix, diagonal term-orders are the same as
anti-diagonal ones, up to transposing the matrix.

\begin{thm}\label{maxmin}
Let $X=(x_{ij})$ be an $m\times n$ matrix whose entries are distinct
indeterminates, $m\leq n$. Let $K[X]=K[x_{ij} \;|\; 1\leq i,j\leq n ]$
be the polynomial ring associated to $X$. Let ${\mathcal G}_m(X)$ be the set of
maximal minors of $X$ and let $I_m(X)\subset K[X]$ be the ideal
generated by ${\mathcal G}_m(X)$. Let $\sigma$ be a diagonal
term-order and let $in(I_m(X))$ be the initial ideal of $I_m(X)$
with respect to $\sigma$. Then ${\mathcal G}_m(X)$ is a reduced Gr\"obner basis
of $I_m(X)$ with respect to $\sigma$ and $in(I_m(X))$ is a squarefree,
Cohen-Macaulay ideal. Moreover, the simplicial complex $\Delta_X$
associated to $in(I_m(X))$ is vertex decomposable.
\end{thm}

\begin{proof}
We proceed by induction on $mn=|X|$. If $|X|=1$, then the ideal
$I_1(X)$ is generated by one indeterminate. Hence
${\mathcal G}_1(X)$ is a reduced Gr\"obner basis of $I_1(X)$ with
respect to any term ordering and $I_1(X)=in(I_1(X))$ is generated by
indeterminates. The associated simplicial complex $\Delta_X$ is the
empty set, hence it is vertex decomposable.

By induction hypothesis, in order to prove the thesis for a matrix
with $m$ rows and $n$ columns, we may assume that it holds for any
matrix with fewer than $mn$ entries. If $m=1$, then ${\mathcal G}_1(X)$ consists
of indeterminates, hence it is a reduced Gr\"obner basis of $I_1(X)$ with
respect to any term ordering. Moreover, $I_1(X)=in(I_1(X))$ is
generated by indeterminates. The associated simplicial complex
$\Delta_X$ is the empty set, hence it is vertex decomposable.

If $m\geq 2$, let $C\subseteq in(I_m(X))$
be the ideal generated by the initial terms of ${\mathcal G}_m(X)$. We claim
that $C=in(I_m(X))$. In fact, let $Z$ be the $m\times(n-1)$ matrix
obtained from $X$ by deleting the last column, and let $Y$ be the
$(m-1)\times(n-1)$ matrix obtained from $Z$ by deleting the last
row. Let $A=in({\mathcal G}_m(Z))$ be the ideal generated by the initial terms
of the elements of ${\mathcal G}_m(Z)$, and let $B=in({\mathcal G}_{m-1}(Y))$ be the ideal
generated by the initial terms of the elements of
${\mathcal G}_{m-1}(Y)$. By the induction hypothesis, ${\mathcal G}_m(Z)$ is a Gr\"obner
basis for $I_m(Z)$ and ${\mathcal G}_{m-1}(Y)$ is a Gr\"obner basis for
$I_{m-1}(Y)$. In other words, $A=in(I_m(Z))$ and $B=in(I_{m-1}(Y))$. Notice that
$$in({\mathcal G}_m(X))=in({\mathcal G}_m(Z))\cup x_{mn}in({\mathcal G}_{m-1}(Y))$$ where $x_{mn}{\mathcal G}$
denotes the set of products $x_{mn}g$ for $g\in{\mathcal G}$. Since
$x_{mn}$ does not appear in $in({\mathcal G}_m(Z))$, we have $A:x_{mn}=A$. Therefore,
$$A+x_{mn}B=C\subseteq in(I_m(X))$$ and $C$ is a
Basic Double G-Link of degree 1 of $B$ on $A$. $A$ and $B$ are
squarefree and glicci by induction hypothesis, therefore $C$ is
squarefree and glicci.
It follows from~\cite[Theorem 3.6]{kl01} that $I_m(X)$ is obtained from
$I_{m-1}(Y)$ via an elementary G-biliaison of height 1 on
$I_m(Z)$. By Lemma~\ref{inid} the maximal minors of $X$ are a
Gr\"obner basis of $I_m(X)$ with respect to $\sigma$, and
$C=in(I_m(X))$.

Finally, let $\Delta_Z,\Delta_Y,\Delta_X$ be the
simplicial complexes associated to $A,B,C$, respectively. Since
$A+x_{mn}B=C$, $$\Delta_Z=\Delta_X-mn\;\;\;\mbox{and}\;\;\;
\Delta_Y=\lk_{mn}(\Delta_X).$$
Since $\Delta_Y$ and $\Delta_Z$ are vertex decomposable by induction
hypothesis, so is $\Delta_X$.
\end{proof}

\begin{rmk}
Theorem~\ref{maxmin} gives in particular a new proof of the fact that
the maximal minors of a generic matrix are a Gr\"obner basis for the
ideal that they generate, with respect to a diagonal or anti-diagonal
term-order. This is a classical result. While previous proofs have a
combinatorial flavor, our proof is completely algebraic, and
independent of all the previous Gr\"obner basis results.
\end{rmk}

\section{Pfaffian ideals of  ladders}\label{pfaff_sect}

In this section, we study Gr\"obner bases with respect to an
anti-diagonal term-order of ideals generated by pfaffians. We always
consider pfaffians in a skew-symmetric matrix whose entries are
distinct indeterminates.

Pfaffians of size $2t$ in a skew-symmetric matrix are known to be a
Gr\"obner basis for the ideal that they generate, as shown by Herzog
and Trung in~\cite{he92} and independently by Kurano in~\cite{ku91}.
In~\cite{de98}, De Negri generalized this
result to pfaffians of size $2t$ in a symmetric ladder. In this
section, we extend these results to pfaffians of mixed size in a
symmetric ladder. In other words, we consider ideals generated by
pfaffians, whose size is allowed to vary in different regions of the
ladder (see Definition~\ref{idealpf}). In Theorem~\ref{gbpf} we prove
that the pfaffians are a reduced Gr\"obner basis with respect to any
anti-diagonal term-order for the ideal that they generate, and that the
corresponding initial ideal is Cohen-Macaulay and
squarefree. Moreover, the associated simplicial complex is vertex
decomposable. The proof that we give is not a generalization of the
earlier ones. Instead, we use our liaison-theoretic approach and the
linkage results of~\cite{de09}.

In the recent paper~\cite{de09u}, De Negri and Sbarra consider a
different family of ideals generated by pfaffians of mixed size in a
skew-symmetric matrix, namely cogenerated ideals. They are able to
show that the pfaffians are almost never a Gr\"obner basis of the
ideal that they generate with respect to an anti-diagonal
term-order. The family of ideals that they study and the family that we
consider in this article have a small overlap, which consists of ideals
of pfaffians of size $2t$ in a symmetric ladder, and of ideals
generated by $2t$-pfaffians in the first $m$ rows and columns of the
matrix and $(2t+2)$-pfaffians in the whole matrix. For the ideals in the
overlap, the pfaffians are a Gr\"obner basis for the ideal that they
generate. This follows from Theorem~2.8 of~\cite{de09u}, as well as
from our Theorem~\ref{gbpf}. The results in~\cite{de09u} and those in
this article are obtained independently and by following a completely
different approach. Nevertheless, we feel that they complement each
other nicely, giving a more complete picture of the behavior of
Gr\"obner bases of pfaffian ideals and of their intrinsic complexity.

Pfaffian ideals of ladders were introduced and studied by De Negri and
the first author in~\cite{de09}. From the point of view of liaison
theory, this is a very natural family to consider.
In this section, we prove that pfaffians of
mixed size in a ladder of a skew-symmetric matrix are a Gr\"obner
basis with respect to any anti-diagonal term-order for the ideal that
they generate. We start by introducing the relevant definitions and
notation.

Let $X=(x_{ij})$ be an $n\times n$ skew-symmetric matrix of indeterminates.
In other words, the entries
$x_{ij}$ with $i<j$ are indeterminates, $x_{ij}=-x_{ji}$ for
$i>j$, and $x_{ii}=0$ for all $i=1,...,n$.
Let $K[X]=K[x_{ij} \;|\; 1\leq i<j\leq n ]$ be the polynomial ring associated
to $X$.

\begin{defn}\label{laddpf}
A {\bf symmetric ladder} $\mathcal L$ of $X$ is a subset of the set
${\mathcal X}=\{(i,j)\in{\mathbb N}^2 \;|\; 1\le i,j\le n\}$
with the following properties :
\begin{enumerate}
\item if $(i,j)\in {\mathcal L}$ then $(j,i)\in {\mathcal L}$,
\item if  $i<h,j>k$ and $(i,j),(h,k)\in\mathcal L$, then also
$(i,k),(i,h),(h,j),(j,k)\in\mathcal L$.
\end{enumerate}

We do not assume that the ladder ${\mathcal L}$ is connected, nor that
$X$ is the smallest skew-symmetric matrix having ${\mathcal L}$ as ladder.
It is easy to see that any symmetric ladder can be decomposed as a
union of square subladders
\begin{equation}\label{decomppf}
{\mathcal L}={\mathcal X}_1\cup\ldots\cup {\mathcal X}_s
\end{equation}
where
 $${\mathcal X}_k=\{(i,j)\;|\; a_k\le i,j \le b_k\},$$
for some  integers $1\leq a_1\leq\ldots\leq a_s\leq n$ and $1\leq
b_1\leq\ldots\leq b_s\leq n$ such that $a_k<b_k$ for all $k$.
We say that ${\mathcal L}$ is the ladder with {\bf upper
corners} $(a_1,b_1),\ldots,(a_s,b_s)$, and that ${\mathcal X}_k$ is the
square subladder of ${\mathcal L}$ with upper outside corner $(a_k,b_k)$. See
Figure~\ref{decomp_fig}.
\begin{figure}[h!]
\input{figs/pfaff1a.pstex_t}
\caption{An example of a symmetric ladder with its decomposition as a
  union of skew-symmetric matrices and the corresponding upper
  corners.}
\label{decomp_fig}
\end{figure}
We allow two upper corners to have the
same first or second coordinate, however we assume that no two
upper corners coincide. We assume moreover that all upper corners
belong to the border of the ladder, i.e.,
$(a_k-1,b_k+1)\not\in{\mathcal L}$. Notice that with these conventions a
ladder does not have a unique decomposition of the form
(\ref{decomppf}). In other words, a symmetric ladder does not
correspond uniquely to a set of upper corners
$(a_1,b_1),\ldots,(a_s,b_s)$. However, any symmetric ladder is
determined by its upper corners as in (\ref{decomppf}).
Moreover, the upper corners of $\mathcal L$ determine the
submatrices ${\mathcal X}_k$. We assume that every symmetric ladder comes with its
set of upper corners and the corresponding decomposition as a
union of square submatrices as in (\ref{decomppf}).
Notice that the set of upper corners as given in our definition
contains all the usual upper outside corners, and may contain some of
the usual upper inside corners, as well as other elements of the
ladder which are not corners of the ladder in the usual sense.

Given a ladder $\mathcal L$ we set $L=\{x_{ij}\in X\;|\; (i,j)\in
{\mathcal L},\; i<j\}$. If $p$ is a positive integer, we let
$I_{2p}(L)$ denote the ideal generated by the set of the
$2p$-pfaffians of $X$ which involve only indeterminates of
$L$. In particular $I_{2p}(X)$ is the ideal of $K[X]$ generated by
the $2p$-pfaffians of $X$.
\end{defn}

\begin{defn}\label{idealpf}
Let ${\mathcal L}={\mathcal X}_1\cup\ldots\cup {\mathcal X}_s$ be a symmetric ladder.
Let $X_k=\{x_{i,j}\;|\; (i,j)\in{\mathcal X}_k,\; i<j\}$ for $k=1,\dots,s$.
Fix a vector $t=(t_1,\ldots,t_s)$, $t\in{\mathbb Z}_+^s$.
The {\bf ladder pfaffian ideal} $I_{2t}(L)$ is by definition the sum of
pfaffian ideals $I_{2t_1}(X_1)+\ldots+I_{2t_s}(X_s)$. We also
refer to these ideals as {\bf pfaffian ideals of ladders}. For ease of
notation, we regard all ladder pfaffian ideals as ideals in $K[X]$.
\end{defn}

This family of ideals was introduced and studied in~\cite{de09}. From
the point of view of G-biliaison, this appears to be the right family
to consider. Notice that it does not coincide with the family
of cogenerated pfaffian ideals as defined, e.g., in~\cite{de95}.

\begin{notat}\label{genspf}
Denote by ${\mathcal G}_{2t_k}(X_k)$ the set of the $2t_k$-pfaffians of $X$
which involve only indeterminates of $X_k$ and
let $${\mathcal G}_{2t}(L)={\mathcal G}_{2t_1}(X_1)\cup\ldots\cup{\mathcal G}_{2t_s}(X_s).$$
The elements of ${\mathcal G}_{2t}(L)$ are a minimal system of generators of
$I_{2t}(L)$. We sometimes refer to them as ``natural generators''.
\end{notat}

\begin{notat}\label{ladderheight}
For a symmetric ladder ${\mathcal L}$ with upper corners
$(a_1,b_1),\ldots,(a_s,b_s)$ and $t=(t_1,\ldots,t_s)$, we denote by
$\tilde{{\mathcal L}}$ the symmetric ladder with upper corners
$(a_1+t_1-1,b_1-t_1+1),\ldots,(a_s+t_s-1,b_s-t_s+1)$. See
Figure~\ref{ladd_hgt_fig}.
\begin{figure}[h!]
\input{figs/pfaff2.pstex_t}
\caption{An example of a ladder ${\mathcal L}$ with five upper corners and
  $t=(2,3,4,2,3)$. The corresponding $\tilde{{\mathcal L}}$ is shaded.}
\label{ladd_hgt_fig}
\end{figure}
\end{notat}

The ladder $\tilde{{\mathcal L}}$ computes the height of the ideal $I_{2t}(L)$
as follows.

\begin{prop}[Proposition~1.10, \cite{de09}]
Let ${\mathcal L}$ be the symmetric ladder with  upper corners
$(a_1,b_1), \ldots,$ $(a_s,b_s)$ and $t=(t_1,\ldots,t_s)$. Let
$\tilde{{\mathcal L}}$ be as in Notation~\ref{ladderheight}. Then $\tilde{{\mathcal L}}$
is a symmetric ladder and the height of $I_{2t}(L)$ is equal to the
cardinality of $\{(i,j)\in\tilde{{\mathcal L}} \;|\; i<j\}$.
\end{prop}

The following is the main result of~\cite{de09}. Its proof consists of
an explicit description of the G-biliaison steps, which will be used in the
proof of Theorem~\ref{gbpf}.

\begin{thm}[Theorem~2.3, \cite{de09}]\label{biliaisonpf}
Any pfaffian ideal of ladders can be obtained from an ideal generated
by indeterminates by a finite sequence of ascending elementary
G-biliaisons.
\end{thm}

By combining Lemma~\ref{inid} and Theorem~\ref{biliaisonpf}, we
prove that the pfaffians are a Gr\"obner basis of the ideal that they
generate with respect to any anti-diagonal term-order.

\begin{thm}\label{gbpf}
Let $X=(x_{ij})$ be an $n\times n$ skew-symmetric matrix of
indeterminates. Let ${\mathcal L}={\mathcal X}_1\cup\ldots\cup{\mathcal X}_s$ and
$t=(t_1,\ldots,t_s)$. Let $I_{2t}(L)\subset K[X]$ be the
corresponding ladder pfaffian ideal and let ${\mathcal G}_{2t}(L)$ be the set of
pfaffians that generate it. Let $\sigma$ be any anti-diagonal
term-order. Then ${\mathcal G}_{2t}(L)$ is a reduced Gr\"obner
basis of $I_{2t}(L)$ with respect to $\sigma$. Moreover, the initial
ideal of $I_{2t}(L)$ with respect to $\sigma$ is squarefree, and the
associated simplicial complex is vertex decomposable. In particular,
the initial ideal of $I_{2t}(L)$ is Cohen-Macaulay.
\end{thm}

\begin{proof}
Let $$I_{2t}(L)=I_{2t_1}(X_1)+\cdots+I_{2t_s}(X_s)\subset K[X]$$ be
the pfaffian ideal of the ladder ${\mathcal L}$ with $t=(t_1,\ldots,t_s)$ and
upper corners $(a_1,b_1),\ldots,$ $(a_s,b_s)$. Let ${\mathcal G}_{2t}(L)$ be the
set of pfaffians that generate $I_{2t}(L)$. We proceed by induction on
$\ell=|{\mathcal L}|$.

If $\ell=|{\mathcal L}|=1$, then ${\mathcal L}=\tilde{{\mathcal L}}$ and $t=1$. ${\mathcal G}_{2t}(L)$
consists only of one indeterminate, in particular it is a reduced
Gr\"obner basis with respect to any term-order $\sigma$ of the ideal
that it generates. Since $in(I_2(L))=I_2(L)$ is generated by
indeterminates, it is squarefree and Cohen-Macaulay, and the
associated simplicial complex is the empty set.

We assume that the thesis holds for ideals associated to ladders ${\mathcal N}$
with $|{\mathcal N}|<\ell$ and we prove it for an ideal $I_{2t}(L)$ associated
to a ladder ${\mathcal L}$ with $|{\mathcal L}|=\ell$. If
$t_1=\ldots=t_s=1$, then ${\mathcal G}_{2t}(L)$ consists only of
indeterminates. In particular, it is a reduced Gr\"obner basis of the
ideal that it generates, with respect to any term-order
$\sigma$. Moreover, $in(I_2(L))=I_2(L)$ is generated by
indeterminates, hence it is squarefree and Cohen-Macaulay. The
associated simplicial complex is the empty set. Otherwise, let
$k\in\{1,\ldots,s\}$ such that
$t_k=\max\{t_1,\ldots,t_s\}\geq 2$. Let $\mathcal L'$ be the ladder
with upper corners $$(a_1,b_1),\ldots,(a_{k-1},b_{k-1}),(a_k+1,b_k-1),
(a_{k+1},b_{k+1}),\ldots,(a_s,b_s)$$
and let $t'=(t_1,\ldots,t_{k-1},t_k-1,t_{k+1},\ldots,t_s)$.
Let $I_{2t'}(L')\subset K[X]$ be the associated ladder pfaffian
ideal. Let ${\mathcal G}_{2t'}(L')$ be the set of pfaffians which minimally
generate $I_{2t'}(L')$. Since $|{\mathcal L}'|<\ell$, by
induction hypothesis ${\mathcal G}_{2t'}(L')$ is a reduced Gr\"obner
basis of $I_{2t'}(L')$ with respect to any anti-diagonal
term-order. Hence $$in(I_{2t'}(L'))=(in({\mathcal G}_{2t'}(L'))).$$
Let $\mathcal M$ be the ladder obtained from ${\mathcal L}$ by removing
$(a_k,b_k)$ and $(b_k,a_k)$. ${\mathcal M}$ has upper corners
$$(a_1,b_1),\ldots,(a_{k-1},b_{k-1}),(a_k,b_k-1),
(a_k+1,b_k),(a_{k+1},b_{k+1}),\ldots,(a_s,b_s)$$ and
$u=(t_1,\dots,t_{k-1},t_k,t_k,t_{k+1},\dots,t_s)$.
Let $I_{2u}(M)\subset K[X]$ be the associated ladder pfaffian
ideal. Let ${\mathcal G}_{2u}(M)$ be the set of pfaffians which minimally
generate $I_{2u}(M)$. Since $|{\mathcal M}|=\ell-1$, by induction hypothesis
${\mathcal G}_{2u}(M)$ is a reduced Gr\"obner basis of $I_{2u}(M)$ with respect
to any anti-diagonal term-order,
and $$in(I_{2u}(M))=(in({\mathcal G}_{2u}(M))).$$

It follows from~\cite{de09}, Theorem~2.3 that $I_{2t}(L)$ is
obtained from $I_{2t'}(L')$ via an ascending elementary G-biliaison of
height $1$ on $I_{2u}(M)$. The ideals $in(I_{2t'}(L'))$ and
$in(I_{2u}(M))$ are Cohen-Macaulay by induction hypothesis. Moreover
$$in({\mathcal G}_{2t}(L))=in({\mathcal G}_{2u}(M))\cup x_{a_k,b_k}in({\mathcal G}_{2t'}(L')),$$
where $x_{a,b}{\mathcal G}$ denotes the set of products $x_{a,b}g$ for $g\in{\mathcal G}$. Since
$x_{a_k,b_k}$ does not appear in $in({\mathcal G}_{2u}(M))$, it does not divide zero
modulo the ideal $in(I_{2u}(M))$. Therefore,
\begin{equation}\label{cpx_pfaff}
I:=(in({\mathcal G}_{2t}(L)))=in(I_{2u}(M))+x_{a_k,b_k}in(I_{2t'}(L'))
\subseteq in(I_{2t}(L))
\end{equation}
and $I$ is a Basic Double G-Link of degree 1 of $in(I_{2t'}(L'))$ on
$in(I_{2u}(M))$. Therefore $I$ is a squarefree Cohen-Macaulay
ideal. By Lemma~\ref{inid} $I=in(I_{2t}(L))$, hence ${\mathcal G}_{2t}(L)$ is a
Gr\"obner basis of $I_{2t}(L)$ with respect to any anti-diagonal
term-order.

Let $\Delta$ be the simplicial complex associated to
$in(I_{2t}(L))$. By (\ref{cpx_pfaff}) the simplicial complexes
associated to $in(I_{2t'}(L')$ and $in(I_{2u}(M))$ are
$\lk_{(a_k,b_k)}(\Delta)$ and $\Delta-(a_k,b_k)$,
respectively. $\Delta$ is vertex decomposable, since
$\lk_{(a_k,b_k)}(\Delta)$ and $\Delta-(a_k,b_k)$ are by induction
hypothesis.
\end{proof}

\begin{rmks}\label{blahblah}
\begin{enumerate}
\item From the proof of the theorem it also follows that $I_{2t}(L)$
  is obtained from an ideal generated by indeterminates via a sequence
  of degree 1 Basic Double G-links, which only involve squarefree
  monomial ideals. Hence in particular it is glicci.
  Since any vertex decomposable complex is
  shellable, it also follows that the associated simplicial
  complex is shellable (see Section~5 of~\cite{na08} for a summary
  of the implications among different properties of simplicial
  complexes, such as vertex decomposability, shellability,
  Cohen-Macaulayness, etc).
\item The proof of Theorem~\ref{gbpf} given above does not
constitute a new proof of the fact that the $2t$-pfaffians in a matrix
or in a symmetric ladder are a Gr\"obner basis with
respect to any anti-diagonal term-order for the ideal that they
generate. In fact, our proof is based on Theorem~2.3 in~\cite{de09},
which in turn relies on the fact that pfaffians all of the same size
in a ladder of a skew-symmetric matrix generate a prime ideal. Primality
of the ideal is classically deduced from the fact that the pfaffians
are a Gr\"obner basis. So we are extending (and not
reproving) the results in~\cite{he92}, \cite{ku91}, and~\cite{de98}.
\end{enumerate}
\end{rmks}

\section{Symmetric mixed ladder determinantal ideals}\label{symm_sect}

In this section, we study ideals generated by minors contained in a
ladder of a generic symmetric matrix. We show that the minors are
Gr\"obner bases for the ideals that they generate, with respect to a
diagonal term-order. We also show that the corresponding initial ideal
is glicci (hence Cohen-Macaulay) and squarefree, and that the
associated simplicial complex is vertex decomposable.

Cogenerated ideals of minors in a symmetric matrix of indeterminates
or a symmetric ladder thereof were studied by Conca in~\cite{co94c}
and~\cite{co94a}. We refer to~\cite{co94c} and~\cite{co94a} for the
definition of cogenerated determinantal ideals in a symmetric
matrix. In those articles Conca proved among other things that the
natural generators of cogenerated ideals of ladders of a symmetric
matrix are a Gr\"obner bases with respect to any diagonal term-order.
In this section, we study the family of symmetric mixed ladder
determinantal ideals. This family strictly contains the family of
cogenerated ideals. Symmetric mixed ladder determinantal ideals have been
introduced and studied by the first author in~\cite{go10}. This is a
very natural family to study, from the point of view of liaison theory.
In this paper we extend the result of Conca and prove that the natural
generators of symmetric mixed ladder determinantal ideals are a
Gr\"obner bases with respect to any diagonal term-order.

Let $X=(x_{ij})$ be an $n\times n$ symmetric
matrix of indeterminates. In other words, the entries $x_{ij}$ with
$i\leq j$ are distinct indeterminates, and $x_{ij}=x_{ji}$ for
$i>j$. Let $K[X]=K[x_{ij}\mid 1\leq i\leq j\leq n]$ be the
polynomial ring associated to the matrix $X$. In the sequel, we study
ideals generated by the minors contained in a ladder of a generic
symmetric matrix. Throughout the section, we let
$${\mathcal X}=\{(i,j)\mid 1\leq i,j\leq n\}.$$
We let ${\mathcal L}$ be a symmetric ladder (see Definition~\ref{laddpf}). We
can restrict ourselves to
symmetric ladders without loss of generality, since the ideal
generated by the minors in a ladder of a symmetric matrix coincides
with the ideal generated by the minors in the smallest symmetric
ladder containing it. We do not assume that ${\mathcal L}$ is connected, nor that
that $X$ is the smallest symmetric matrix having ${\mathcal L}$ as a ladder.
Let $${\mathcal X}^+=\{(i,j)\in{\mathcal X}\mid 1\leq i\leq j\leq n\}\;\;\;\mbox{and}\;\;\;
{\mathcal L}^+={\mathcal L}\cap{\mathcal X}^+.$$
Since ${\mathcal L}$ is symmetric, ${\mathcal L}^+$ determines ${\mathcal L}$ and vice versa. We
will abuse terminology and call ${\mathcal L}^+$ a ladder.
Observe that ${\mathcal L}^+$ can be written as
$${\mathcal L}^+=\{(i,j)\in{\mathcal X}^+\mid i\leq c_l \mbox{ or } j\leq d_l \mbox{ for }
l=1,\ldots,r\; \mbox{ and }$$ $$ i\geq a_l \mbox{ or } j\geq b_l
\mbox{ for } l=1,\ldots,u \}$$
for some  integers $1\leq a_1<\ldots<a_u\leq n$, $n\geq b_1>\ldots>b_u\geq 1$,
$1\leq c_1<\ldots<c_r\leq n$, and $n\geq d_1>\ldots>d_r\geq 1$, with
$a_l\leq b_l$ for $l=1,\ldots,u$ and $c_l\leq d_l$ for $l=1,\ldots,r$.

The points $(a_1,b_2),\ldots,(a_{u-1},b_u)$ are the {\bf upper outside
corners} of the ladder, $(a_1,b_1),\ldots,(a_u,b_u)$ are the
{\bf upper inside corners}, $(c_2,d_1),\ldots,(c_r,d_{r-1})$ the {\bf
lower outside corners}, and $(c_1,d_1),\ldots,(c_r,d_r)$ the {\bf lower inside
corners}. If $a_u\neq b_u$, then $(a_u,a_u)$ is an upper outside corner
and we set $b_{u+1}=a_u$. Similarly, if $c_r\neq d_r$ then $(d_r,d_r)$
is a lower outside corner, and we set $c_{r+1}=d_r$. A ladder has at least
one upper and one lower outside
corner. Moreover, $(a_1,b_1)=(c_1,d_1)$ is both an upper and a lower
inside corner. See Figure~\ref{corners}.
\begin{figure}[h!]
\begin{center}
\input{figs/symm1.pstex_t}
\caption{An example of ladder with tagged lower and upper corners.}
\label{corners}
\end{center}
\end{figure}
The {\bf lower border} of ${\mathcal L}^+$
consists of the elements $(c,d)$ of ${\mathcal L}^+$ such that either $c_l\leq
c\leq c_{l+1}$ and $d=d_l$, or $c=c_l$ and $d_l\leq d\leq d_{l-1}$ for
some $l$. See Figure~\ref{lowerborder}.
\begin{figure}[h!]
\begin{center}
\input{figs/symm2.pstex_t}
\caption{The lower border of the same ladder.}
\label{lowerborder}
\end{center}
\end{figure}
All the corners belong to ${\mathcal L}^+$. In fact, the ladder ${\mathcal L}^+$
corresponds to its set of lower and upper outside (or equivalently
lower and upper inside) corners. The lower corners
of a ladder belong to its lower border.

Given a ladder ${\mathcal L}$ we set $L=\{x_{ij}\in X\mid (i,j)\in{\mathcal L}^+\}$. For
$t$ a positive integer
we let $I_t(L)$ denote the ideal generated by the set of the
$t$-minors of $X$ which involve only indeterminates of
$L$. In particular $I_t(X)$ is the ideal of $K[X]$ generated by
the minors of $X$ of size $t\times t$.

\begin{notat}\label{decompsym}
Let ${\mathcal L}^+$ be a ladder. For $(v,w)\in{\mathcal L}^+$ let
$${\mathcal L}^+_{(v,w)}=\{(i,j)\in{\mathcal L}^+\mid i\leq v,\; j\leq w\},
\;\;\;\;\; L_{(v,w)}=\{x_{ij}\in X\mid (i,j)\in{\mathcal L}^+_{(v,w)}\}.$$
Notice that ${\mathcal L}^+_{(v,w)}$ is a ladder and
$${\mathcal L}^+=\bigcup_{(v,w)\in{\mathcal U}}{\mathcal L}^+_{(v,w)}$$
where ${\mathcal U}$ denotes the set of lower outside corners of ${\mathcal L}^+$.
\begin{figure}[h!]
\begin{center}
\input{figs/symm3.pstex_t}
\caption{The ladder ${\mathcal L}^+$ with a shaded subladder ${\mathcal L}^+_{(v,w)}$.}
\label{region}
\end{center}
\end{figure}
\end{notat}

\begin{defn}\label{ldi}
Let $\{(v_1,w_1),\ldots,(v_s,w_s)\}$ be a subset of the
lower border of ${\mathcal L}^+$ which contains all the lower outside
corners. We order them so that $1\leq v_1\leq\ldots\leq v_s\leq n$
and $n\geq w_1\geq\ldots\geq w_s\geq 1$. Let
$t=(t_1,\ldots,t_s)\in{\mathbb Z}_+^s$. Denote ${\mathcal L}^+_{(v_k,w_k)}$ by
${\mathcal L}^+_k$, and $L_{(v_k,w_k)}$ by $L_k$.
The ideal $$I_t(L)=I_{t_1}(L_1)+\ldots+I_{t_s}(L_s)\subset
K[X]$$ is a {\bf symmetric mixed ladder determinantal ideal}.
Denote $I_{(t,\ldots,t)}(L)$ by $I_t(L)$.
We call $(v_1,w_1),\ldots,(v_s,w_s)$ {\bf distinguished points} of
${\mathcal L}^+$.
\end{defn}
If $t=(t,\ldots,t)$, then $I_t(L)$ is the ideal generated by the
$t$-minors of $X$ that involve only indeterminates from $L$. These
ideals have been classically studied (see, e.g., \cite{co94c}, \cite{co94a},
\cite{co94b}). It is not hard to show
(see~\cite{go10}, Examples~1.5) that the family of symmetric mixed
ladder determinantal ideals contains the family of cogenerated ideals
in a ladder of a symmetric matrix, as defined in~\cite{co94a}.

\begin{notat}\label{genssym}
Denote by ${\mathcal G}_{t_k}(L_k)$ the set of the $t_k$-minors of $X$
which involve only indeterminates of $L_k$ and
let $${\mathcal G}_t(L)={\mathcal G}_{t_1}(L_1)\cup\ldots\cup{\mathcal G}_{t_s}(L_s).$$
The elements of ${\mathcal G}_t(L)$ are a minimal system of generators of
$I_t(L)$. We sometimes refer to them as ``natural generators''.
\end{notat}

\begin{notat}
Let ${\mathcal L}$ be a ladder with distinguished points
$(v_1,w_1),\ldots,(v_s,w_s)$. We denote
by $$\tilde{{\mathcal L}}^+=\{(i,j)\in{\mathcal L}^+\mid i\leq v_{k-1}-t_{k-1}+1 \mbox{
  or } j\leq w_k-t_k+1 \mbox{ for } k=2,\ldots,s,$$ $$j\leq
w_1-t_1+1,\; i\leq v_s-t_s+1\}$$ and by
$$\tilde{{\mathcal L}}=\tilde{{\mathcal L}}^+\cup\{(j,i)\mid (i,j)\in\tilde{{\mathcal L}}^+\}.$$
See Figure~\ref{lplus}.

\begin{figure}[h!]
\begin{center}
\input{figs/symm4.pstex_t}
\caption{An example of ${\mathcal L}^+$ with three distinguished points and
  $t=(3,6,4)$. The corresponding $\tilde{{\mathcal L}}^+$ is shaded.}
\label{lplus}
\end{center}
\end{figure}
\end{notat}

The ladder $\tilde{{\mathcal L}}$ computes the height of $I_t(L)$ as follows.

\begin{prop}[\cite{go10}, Proposition~1.8]
Let ${\mathcal L}$ be a ladder with distinguished points
$(v_1,w_1),\ldots,(v_s,w_s)$ and let $\tilde{{\mathcal L}}$ and $\tilde{{\mathcal L}}^+$
be as above. Then $\tilde{{\mathcal L}}$ is a symmetric ladder and
$$\hgt I_t(L)=|\tilde{{\mathcal L}}^+|.$$
\end{prop}

The result about Gr\"obner bases will follow by combining the next
theorem with Lemma~\ref{inid}.

\begin{thm}[\cite{go10}, Theorem~2.4]\label{biliaisonsym}
Any symmetric mixed ladder determinantal ideal can be obtained from an ideal
generated by indeterminates by a finite sequence of ascending
elementary G-biliaisons.
\end{thm}

The following is the main result of this section. We prove that the
natural generators of symmetric mixed ladder determinantal ideals are
a Gr\"obner basis with respect to any diagonal term-order, and that
the simplicial complexes associated to their initial ideals are vertex
decomposable. In particular, the initial ideals are Cohen-Macaulay.

\begin{thm}\label{gbsym}
Let $X=(x_{ij})$ be an $n\times n$ symmetric matrix of
indeterminates. Let ${\mathcal L}^+={\mathcal L}^+_1\cup\ldots\cup{\mathcal L}^+_s$ and
$t=(t_1,\ldots,t_s)$. Let $I_t(L)\subset K[X]$ be the
corresponding symmetric mixed ladder determinantal ideal and let
${\mathcal G}_t(L)$ be the set of minors that generate it. Let $\sigma$ be any
diagonal term-order. Then ${\mathcal G}_t(L)$ is a reduced Gr\"obner basis of
$I_t(L)$ with respect to $\sigma$. Moreover, the initial ideal of
$I_t(L)$ with respect to $\sigma$ is squarefree and Cohen-Macaulay,
and the associated simplicial complex is vertex decomposable.
\end{thm}

\begin{proof}
Let $$I_t(L)=I_{t_1}(L_1)+\cdots+I_{t_s}(L_s)\subset K[X]$$ be
the symmetric mixed ladder determinantal ideal with ladder ${\mathcal L}$,
$t=(t_1,\ldots,t_s)$ and distinguished points
$(v_1,w_1),\ldots,(v_s,w_s)$. Let ${\mathcal G}_t(L)$ be the set of natural
generators of $I_t(L)$. We proceed by induction on
$\ell=|{\mathcal L}^+|$.

If $\ell=1$, then ${\mathcal G}_t(L)$ consists of one indeterminate, in
particular it is a Gr\"obner basis of the ideal that it generates with
respect to any term-order $\sigma$. Moreover,
$in(I_1(L))=I_1(L)$ is generated by indeterminates, and the simplicial
complex associated to it is the empty set.

We now assume that the thesis holds for ladders ${\mathcal N}$ with
$|{\mathcal N}^+|<\ell$ and prove it for a ladder ${\mathcal L}$ with
$|{\mathcal L}^+|=\ell$. If $t_1=\ldots=t_s=1$, then ${\mathcal G}_t(L)$ consists only of
indeterminates. In particular it is a Gr\"obner basis of the ideal
that it generates with respect to any term-order $\sigma$. Moreover,
$in(I_1(L))=I_1(L)$ is generated by indeterminates, and the simplicial
complex associated to it is the empty set. Otherwise, let
$k\in\{1,\ldots,s\}$ such that $t_k=\max\{t_1,\ldots,t_s\}\geq 2$. Let
${\mathcal L}'$ be the ladder with distinguished points
$$(v_1,w_1),\ldots,(v_{k-1},w_{k-1}),(v_k+1,w_k-1),
(v_{k+1},w_{k+1}),\ldots,(v_s,w_s)$$
and let $t'=(t_1,\ldots,t_{k-1},t_k-1,t_{k+1},\ldots,t_s)$.
Let $I_{t'}(L')\subset K[X]$ be the associated symmetric mixed ladder
determinantal ideal. Let ${\mathcal G}_{t'}(L')$ be the set of minors which minimally
generate $I_{t'}(L')$. Since $|{\mathcal L}'^+|<\ell$, by
induction hypothesis ${\mathcal G}_{t'}(L')$ is a reduced Gr\"obner
basis of $I_{t'}(L')$ with respect to
$\sigma$. Hence $$in(I_{t'}(L'))=(in({\mathcal G}_{t'}(L'))).$$
Let $\mathcal M$ be the ladder obtained from ${\mathcal L}$ by removing
$(v_k,w_k)$ and $(w_k,v_k)$. Let
$$(v_1,w_1),\ldots,(v_{k-1},w_{k-1}),(v_k,w_k-1),(v_k+1,w_k),
(v_{k+1},w_{k+1}),\ldots,(v_s,w_s)$$  be the distinguished points of
$\mathcal M$ and let $u=(t_1,\dots,t_{k-1},t_k,t_k,t_{k+1},\dots,t_s)$.
Let $I_u(M)\subset K[X]$ be the associated symmetric mixed ladder
determinantal ideal. Let ${\mathcal G}_u(M)$ be the set of minors which minimally
generate $I_u(M)$. Since $|{\mathcal M}^+|=\ell-1<\ell$, by induction hypothesis
${\mathcal G}_u(M)$ is a reduced Gr\"obner basis of $I_u(M)$ with respect to any
diagonal term-order. Hence $$in(I_u(M))=(in({\mathcal G}_u(M))).$$

It follows from~\cite{go10}, Theorem~2.4 that $I_t(L)$ is
obtained from $I_{t'}(L')$ via an ascending elementary G-biliaison of
height $1$ on $I_u(M)$. The ideals $in(I_{t'}(L'))$ and $in(I_u(M))$
are squarefree and Cohen-Macaulay by induction hypothesis. Moreover
$$in({\mathcal G}_t(L))=in({\mathcal G}_u(M))\cup x_{v_k,w_k}in({\mathcal G}_{t'}(L')),$$
where $x_{u,v}{\mathcal G}$ denotes the set of products $x_{u,v}g$ for $g\in{\mathcal G}$. Since
$x_{v_k,w_k}$ does not appear in $in({\mathcal G}_u(M))$, it does not divide zero
modulo the ideal $in(I_u(M))$. Therefore,
\begin{equation}\label{cpx_symm}
I:=(in({\mathcal G}_t(L)))=in(I_u(M))+x_{v_k,w_k}in(I_{t'}(L'))\subseteq
in(I_t(L))
\end{equation}
and $I$ is a Basic Double G-Link of degree 1 of $in(I_{t'}(L'))$ on
$in(I_u(M))$. Hence $I$ is a squarefree Cohen-Macaulay ideal. By
Lemma~\ref{inid} $I=in(I_t(L))$, hence ${\mathcal G}_t(L)$ is a Gr\"obner
basis of $I_t(L)$ with respect to any diagonal term-order.

Let $\Delta$ be the simplicial complex associated to
$in(I_t(L))$. By (\ref{cpx_symm}) the simplicial complexes
associated to $in(I_{t'}(L')$ and $in(I_{u}(M))$ are
$\lk_{(v_k,w_k)}(\Delta)$ and $\Delta-(v_k,w_k)$,
respectively. $\Delta$ is vertex decomposable, since
$\lk_{(v_k,w_k)}(\Delta)$ and $\Delta-(v_k,w_k)$ are by induction
hypothesis.
\end{proof}

\begin{rmks}
\begin{enumerate}
\item From the proof of the previous theorem it also follows that
  $in_t(L)$ is obtained from an ideal generated by indeterminates via
  a sequence of degree 1 Basic Double G-links which only involve
  squarefree monomial ideals. In particular, it is
  glicci. Moreover, the associated simplicial complex is shellable.
\item The proof of Theorem~\ref{gbsym} given above does not constitute
  a new proof of the fact that the $t$-minors in a symmetric matrix or
  in a symmetric ladder are a Gr\"obner basis with respect to any
  diagonal term-order for the ideal that they generate. In fact, our
  proof is based on Theorem~2.4 in~\cite{go10}, which in turn relies
  on the fact that minors all of the same size in a ladder of a
  symmetric matrix generate a prime ideal. Primality of this ideal is
  classically deduced from the fact that the minors are a Gr\"obner
  basis. So we are extending (and not providing a new proof of) the
  results in~\cite{co94c}.
\item Our argument, however, gives a new proof of the fact that the
  minors generating a cogenerated ideal in a symmetric matrix or in a
  ladder thereof are a Gr\"obner basis with respect to a diagonal
  term-order, knowing that minors all of the same size in a ladder of
  a symmetric matrix are a Gr\"obner basis of the ideal that they
  generate.
\end{enumerate}
\end{rmks}

\section{Mixed ladder determinantal ideals}\label{ladd_sect}

In this section, we prove that minors of mixed size in one-sided
ladders are Gr\"obner bases for the ideals that they generate, with
respect to any anti-diagonal term order. Moreover, the associated
simplicial complex is vertex decomposable. These results are already
known, and were established in different levels of generality
in~\cite{na86}, \cite{co93}, \cite{co95}, \cite{go00}, \cite{kn05},
\cite{go07b}, \cite{kn09} and~\cite{kn09p}.
The papers~\cite{kn05}, \cite{kn09} and~\cite{kn09p} follow a
different approach than the others. The family that they treat
strictly contains that of one-sided mixed ladder determinantal
ideals. The paper~\cite{go07b} follows essentially the same approach
as the the first four papers, extending it to the family of two-sided
mixed ladder determinantal ideals.
The proof we give here is different and independent of all the
previous ones: we use the result that we established in
Section~\ref{mainlemma} and the liaison results which were established
in~\cite{go07b}.

In~\cite{go07b} the first author approached the
study of ladder determinantal ideals from the opposite point of view:
she first proved that the minors were Gr\"obner bases of the ideals
that they generated. From the Gr\"obner basis result, she deduced that
the ideals are prime and Cohen-Macaulay, and computed their height. Finally,
she proved the liaison result. Here we  wish to take the opposite
approach: namely, deduce the fact that the minors are a Gr\"obner
basis for the ideal that they generate from the liaison result. In
order to do that, we need to show how to obtain the liaison result
independently of the computation of a Gr\"obner basis. We do this in
Appendix~\ref{ladd_app} following the approach of~\cite{de09}
and~\cite{go10}. In this section, we deduce the result about Gr\"obner
bases from the G-biliaison result.

We start by introducing the relevant notation. Let $X=(x_{ij})$ be an
$m\times n$ matrix whose entries are distinct indeterminates, $m\leq n$.

\begin{defn}\label{ladd}
A {\bf one-sided ladder} ${\mathcal L}$ of $X$ is a subset of the set
${\mathcal X}=\{(i,j)\in{\mathbb N}^2 \mid 1\le i\le m,\ 1\le j\le n \}$
with the properties:
\begin{enumerate}
\item $(1,m)\in{\mathcal L}$,
\item if $i<h,j>k$ and $(i,j),(h,k)\in{\mathcal L}$, then
$(i,k),(i,h),(h,j),(j,k)\in{\mathcal L}$.
\end{enumerate}
\end{defn}

We do not make any connectedness assumption on the ladder ${\mathcal L}$.
For ease of notation, we also do not assume that $X$ is the
smallest matrix having ${\mathcal L}$ as a ladder.
Observe that ${\mathcal L}$ can be written as
$${\mathcal L}=\bigcup_{k=1}^u \{(i,j)\in{\mathcal X}\mid i\leq c_k \mbox{ and } j\geq
d_k\}$$ for some  integers $1\leq c_1<\ldots<c_u\leq m$, $1\leq
d_1<\ldots<d_u\leq n$.

We call $(c_1,d_1),\ldots,(c_u,d_u)$ {\bf lower outside
corners} and $(c_1,d_2),\ldots,(c_{u-1},d_u)$ {\bf lower inside
corners} of the ladder ${\mathcal L}$. A one-sided ladder has at least one
lower outside corner. A one-sided ladder which has exactly one lower
outside corner is a matrix. All the corners belong to ${\mathcal L}$, and the
ladder ${\mathcal L}$ corresponds to its set of lower outside (or equivalently,
lower inside) corners. The {\bf lower border} of ${\mathcal L}$ consists of the
elements $(c,d)$ of ${\mathcal L}$ such that $(c+1,d-1)\not\in{\mathcal L}$. See
Figure~\ref{lb}. Notice that the lower corners of a ladder belong to
its lower border.
\begin{figure}[h!]
\begin{center}
\input{figs/ladd1.pstex_t}
\caption{An example of a ladder with shaded lower border.}
\label{lb}
\end{center}
\end{figure}

Given a ladder ${\mathcal L}$ we set $L=\{x_{ij}\in X\mid
(i,j)\in{\mathcal L}\}$. We denote by $|{\mathcal L}|$ the cardinality of the ladder.
We let $I_t(L)$ denote the ideal generated by the set of the
$t$-minors of $X$ which involve only indeterminates of
$L$. In particular, $I_t(X)$ is the ideal of $K[X]$ generated by
the $t\times t$-minors of $X$.

\begin{defn}
Let $\{(a_1,b_1),\ldots,(a_s,b_s)\}$ be a subset of the
lower border of ${\mathcal L}$ which contains all the lower outside
corners. We order them so that $1\leq a_1\leq\ldots\leq a_s\leq m$
and $1\leq b_1\leq\ldots\leq b_s\leq n$.
Let $t=(t_1,\ldots,t_s)$ be a vector of positive integers. For
$k=1,\ldots,s$, denote by
$${\mathcal L}_k=\{ (i,j)\in{\mathcal X}\mid i\leq a_k \mbox{ and } j\geq
b_k\}\;\; \mbox{ and }\;\; L_k=\{x_{i,j}\mid (i,j)\in{\mathcal L}_k\}.$$ Notice
that ${\mathcal L}_k\subseteq{\mathcal L}$ and $L_k\subseteq
L$. Moreover, ${\mathcal L}=\cup_{k=1}^s{\mathcal L}_k.$ The ideal
$$I_t(L)=I_{t_1}(L_1)+\ldots+I_{t_s}(L_s)$$
is a {\bf mixed ladder determinantal ideal}.
We denote $I_{(t,\ldots,t)}(L)$ by $I_t(L)$.
We call $(a_1,b_1),\ldots,(a_s,b_s)$ {\bf distinguished points} of ${\mathcal L}$.
Notice that a ladder is uniquely determined by the set of its
distinguished points, but it does not determine them.
\end{defn}

\begin{notat}\label{gensladd}
Denote by ${\mathcal G}_{t_k}(L_k)$ the set of the $t_k$-minors of $X$
which involve only indeterminates of $L_k$ and
let $${\mathcal G}_t(L)={\mathcal G}_{t_1}(L_1)\cup\ldots\cup{\mathcal G}_{t_s}(L_s).$$
The elements of ${\mathcal G}_t(L)$ are a minimal system of generators of
$I_t(L)$. We sometimes refer to them as ``natural generators''.
\end{notat}

We will need the following result. See the appendix for a self
contained proof.

\begin{thm}[\cite{go07b}, Theorem~2.1]\label{biliaisonladd}
Any mixed ladder determinantal ideal can be obtained from an ideal
generated by indeterminates by a finite sequence of ascending
elementary G-biliaisons.
\end{thm}

We now prove that the natural generators of a mixed ladder
determinantal ideal are a Gr\"obner basis with respect to any anti-diagonal
term-order.

\begin{thm}\label{gbladd}
Let $X=(x_{ij})$ be an $m\times n$ matrix whose entries are distinct
indeterminates, $m\leq n$, and let ${\mathcal L}$ be a one-sided ladder of
$X$. Let $t\in{\mathbb N}^s$ and let $(a_1,b_1),\ldots,(a_s,b_s)$ be the
distinguished points of the ladder. Let $I_t(L)$ be the corresponding
ladder determinantal ideal. Denote by ${\mathcal G}_t(L)$ be the set of minors
that generate $I_t(L)$. Then ${\mathcal G}_t(L)$ is a reduced Gr\"obner basis of
$I_t(L)$ with respect to any anti-diagonal term ordering. Moreover,
the initial ideal $in(I_t(L))$ is squarefree Cohen-Macaulay, and the
associated simplicial complex is vertex decomposable.
\end{thm}

\begin{proof}
We proceed by induction on $\ell=|{\mathcal L}|$. If $\ell=1$, then ${\mathcal G}_1(L)$
consists of one indeterminate. Hence ${\mathcal G}_1(L)$ is a reduced Gr\"obner
basis for $I_1(L)$ with respect to any term-order. Moreover,
$I_1(L)=in(I_1(L))$ is generated by indeterminates, hence the
associated simplicial complex is the empty set.

We now assume by induction that the statement holds for ladders ${\mathcal H}$
with $|{\mathcal H}|<\ell$, and we prove the statement for a ladder
${\mathcal L}$ with $|{\mathcal L}|=\ell$. If $t=(1,\ldots,1)$, then ${\mathcal G}_1(L)$
consists of indeterminates. Hence ${\mathcal G}_1(L)$ is a reduced Gr\"obner
basis for $I_1(L)$
with respect to any term-order. Moreover, $I_1(L)=in(I_1(L))$ is
generated by indeterminates, hence the associated simplicial complex
is the empty set. Otherwise, let $C\subseteq in(I_t(L))$ be the ideal
generated by the initial terms of
${\mathcal G}_t(L)$. It suffices to show that $C=in(I_t(L))$ and that $C$ is
Cohen-Macaulay. Let $(a_1,b_1),\ldots,(a_s,b_s)$ be the distinguished
points of the ladder and choose $k\in\{1,\ldots,s\}$ so that
$t_k=\max\{t_1,\ldots,t_s\}\geq 2$. Let
${\mathcal M}={\mathcal L}\setminus\{(a_k,b_k)\}$, and let
$$(a_1,b_1),\ldots,(a_{k-1},b_{k-1}),(a_k-1,b_k),(a_k,b_k+1),
(a_{k+1},b_{k+1}),\ldots,(a_s,b_s)$$ be the distinguished points of
${\mathcal M}$. Let $p=(t_1,\ldots,t_{k-1},t_k,t_k,t_{k+1},\ldots,t_s)\in
{\mathbb N}^{s+1}$. Let ${\mathcal N}$ be the ladder with distinguished points
$$(a_1,b_1),\ldots,(a_{k-1},b_{k-1}),(a_k-1,b_k+1),(a_{k+1},b_{k+1}),
\ldots,(a_s,b_s)$$ and let $q=(t_1,\ldots,t_{k-1},t_k-1,
t_{k+1},\ldots,t_s)\in{\mathbb N}^s$.
As shown in the proof of Theorem~\ref{biliaisonladd}, $I_t(L)$ is obtained
from $I_q(N)$ via an elementary G-biliaison of height 1 on
$I_p(M)$.

Let $A=in({\mathcal G}_p(M))$ and let $B=in({\mathcal G}_q(N))$.
The induction hypothesis applies to both $I_p(M)$ and $I_q(N)$, hence
${\mathcal G}_p(M)$ is a Gr\"obner basis for $I_p(M)$, ${\mathcal G}_q(N)$ is a Gr\"obner
basis for $I_q(N)$ and $A=in(I_p(M)),B=in(I_q(N))$ are Cohen-Macaulay
ideals. Notice that
$$in({\mathcal G}_t(L))=in({\mathcal G}_p(M))\cup x_{a_k,b_k}in({\mathcal G}_q(N))$$ where $x_{a_k,b_k}{\mathcal G}$
denotes the set of products $x_{a_k,b_k}g$ for $g\in{\mathcal G}$. Since
$x_{a_k,b_k}$ does not appear in $in({\mathcal G}_p(M))$, it does not divide zero
modulo the ideal $A=in(I_p(M))$. Therefore,
\begin{equation}\label{cpx_ladd}
in(I_p(M))+x_{a_k,b_k}in(I_q(N))=C\subseteq in(I_t(L))
\end{equation}
and $C$ is a Basic Double G-Link of degree 1 of $in(I_q(N))$ on
$in(I_p(M))$. $C$ is Cohen-Macaulay and squarefree, since $A$ and $B$
are. By Lemma~\ref{inid} we
conclude that $C\subseteq in(I_t(L))$ and that ${\mathcal G}_t(L)$ is a
Gr\"obner basis of $I_t(L)$ with respect to any anti-diagonal
term-order.

Let $\Delta$ be the simplicial complex associated to
$in(I_{t}(L))$. By (\ref{cpx_ladd}) the simplicial complexes
associated to $in(I_{q}(N)$ and $in(I_{p}(M))$ are
$\lk_{(a_k,b_k)}(\Delta)$ and $\Delta-(a_k,b_k)$,
respectively. $\Delta$ is vertex decomposable, since
$\lk_{(a_k,b_k)}(\Delta)$ and $\Delta-(a_k,b_k)$ are by induction
hypothesis.
\end{proof}

\begin{rmks}
\begin{enumerate}
\item From the proof of the theorem it also follows that the ideal
  $in(I_t(L))$ is obtained from an ideal generated by indeterminates
  via a sequence of degree 1 Basic Double G-links, which only involve
  squarefree monomial ideals. Hence in particular it is glicci. Moreover, the
  associated simplicial complex is shellable.
\item Notice that, in contrast to Theorem~\ref{gbpf} and
Theorem~\ref{gbsym}, Theorem~\ref{gbladd} does constitute a new proof
of the fact that $t$-minors in a generic matrix or in a one-sided
ladder are a Gr\"obner basis with respect to any anti-diagonal term-order
for the ideal that they generate. In fact, in Theorem~\ref{laddall} we
give a proof of primality for mixed ladder determinantal ideals which
is independent of any previous Gr\"obner basis results.
\end{enumerate}
\end{rmks}

By following the same approach as in the previous sections and using the
result of Narasimhan from~\cite{na86}, we can prove that the natural
generators of mixed ladder determinantal ideals from two-sided ladders
are a Gr\"obner basis for the ideal that they generate with respect to
any anti-diagonal term-order (see~\cite{go07b} for the relevant
definitions). This is, to our knowledge, the largest
family of ideals generated by minors in a ladder for which the minors
are a Gr\"obner basis for the ideal that they generate. Notice, e.g.,
that cogenerated ladder determinantal ideals all belong to this
family. The result was already established by the first author
in~\cite{go07b}, but a different proof can be given using the
techniques discussed in this paper. Notice moreover that we also show
that the simplicial complex associated to the initial ideal is vertex
decomposable. In particular, it is shellable. Since the proof is
completely analogous to the previous ones, we omit it.

\begin{thm}
Let $X=(x_{ij})$ be an $m\times n$ matrix whose entries are distinct
indeterminates, $m\leq n$, and let ${\mathcal L}$ be a ladder of
$X$. Let $t\in{\mathbb N}^s$ and let $(a_1,b_1),\ldots,(a_s,b_s)$ be the
distinguished points of the ladder. Let $I_t(L)$ be the corresponding
ladder determinantal ideal. Denote by ${\mathcal G}_t(L)$ be the set of minors
that generate $I_t(L)$. Then ${\mathcal G}_t(L)$ is a reduced Gr\"obner basis of
$I_t(L)$ with respect to any anti-diagonal term ordering, and the
initial ideal $in(I_t(L))$ is squarefree Cohen-Macaulay. Moreover, the
associated simplicial complex is vertex decomposable.
\end{thm}


\section{INTRODUCTION}
Materials exhibiting very high values in dielectric permittivity ($\varepsilon' > 10^3$) are often coined as ``colossal dielectric constant" (CDC) materials\cite{Lunkenheimer2010, Lunkenheimer2002}. They bear enormous potential for enhancing the capacitance, e.g. in multilayer ceramic or low-temperature co-fired capacitors \cite{Kishi2003, C9TC02921D, Li2020}. Typically, proper ferroelectric materials\cite{Toledano}, such as BaTiO$_3$\cite{Merz1949, Devon1949} or Pb(Zr$_x$,Ti$_{1-x}$)O$_3$\cite{Shirane1952}, are used as their dielectric permittivity exceeds $10^3$ at ambient temperatures and the loss tangent -- indicating the dielectric loss -- is rather low (tan$\delta < 10^{-2}$)\cite{C7TA05392D, C9TC02921D,Elissalde2001}. The frequency stability of these parameters within a certain temperature range with respect to the applied voltages are key quantities for technical applicability.

Another approach towards CDCs is to use thin layers with reduced conductivity -- so called barrier-layer capacitances (BLC) -- in bulk ceramics and single crystals. These BLCs can be internal layers, like insulating grain boundaries in polycrystalline ceramics\cite{Adams2002,Zhao2004,Frey1998}, or surface layers formed, e.g., due to the depletion zone of Schottky diodes arising at metal-semiconductor contacts\cite{Krohns2007,Krohns2009}. Both mechanisms are sensitive to variations in preparation (i.e., size of grains and conductivity of grain boundaries\cite{Zhao2004}) or the contact area of the Schottky barrier. BLCs appear as a step-like decrease in $\varepsilon'$ accompanied by a peak in $\varepsilon''$ in a frequency-dependent representation mimicking a classical `Debye-like' relaxation process (see \textcite{Lunkenheimer2010} and references therein for more details). The electrical heterogeneity
is responsible for the first relaxation-like feature in the dielectric properties, called Maxwell-Wagner relaxation.

Recently, a reduced conductivity at the DWs and a related BLC effect was observed in h-YMnO$_3$\cite{Ruff2017ConductivityYMnO3} suggesting a high dielectric constant $\varepsilon' > 200$.
However, a systematic analysis that can confirm the connection between the high dielectric constant and DW driven BLCs remains illusive.

Here, we provide a dielectric analysis of an h-ErMnO$_3$ single crystal for which the polarisation is parallel to the applied contact electrodes (in-plane). Two distinct relaxations are observed in this sample: the first leads to a high dielectric constant in the order of 300, and the second to a CDC of more than $5\times 10^3$. So far, mainly the dielectric properties of out-of-plane polarized samples were investigated, for which often only one relaxation was reported leading to CDC\cite{Holstad2018, Ruff2018APL, Schaab2018ElectricalWalls}.
To disentangle the contribution of various BLCs arising in the sample and at the surface of the sample, we simulate the dielectric spectra by an equivalent circuit model, analogous to previous studies on the CDC prime-example, CaCu$_3$Ti$_4$O$_{12}$\cite{Krohns2008,Lunkenheimer2010,Lunkenheimer2002}. This approach, in combination with a distinct modification of the electrode contact area and the thickness of the sample, allows to distinguish BLCs arising from internal and surface effects.
Furthermore, we use local probe analysis by piezo-force response (PFM) and conductive atomic force microscopy (cAFM) to determine electronic DWs properties at the sample surface and estimate the volume fraction of insulating DWs\cite{Meier2012,Schoenherr2018}. Our systematic analysis provides new insight into the dielectric properties of hexagonal manganites, corroborating that insulating DWs act as BLCs, playing a key role for the high or even colossal dielectric constants observed in this class of materials.

\section{EXPERIMENT}

High-quality hexagonal h-ErMnO$_3$ single crystals were grown by the pressurized floating zone technique\cite{Yan2015}. The sample was cut into a disc (area = 2.38\,mm$^2$, thickness = 0.61\,mm) with the polar axis lying parallel to the surface, i.e. (110)-oriented. For dielectric spectroscopy we used a plate capacitor geometry, coating both top and bottom surfaces either with silver paint or sputtered gold. We performed the measurements using a Alpha Analyzer (Novocontrol, Montabaur, Germany), which covers the frequency range of 1\,Hz to 1\,MHz. This analysis was conducted in a closed-cycle refrigerator between 150\,K and 300\,K.

The microscopic data was recorded on the same sample at room-temperature using an NT-MDT (NTEGRA, Apeldoorn, Netherlands) atomic force microscope (AFM), using diamond tips (DDESP-10, Bruker, Billerica, MA, USA).
The sample was lapped with a 9\,\textmu m-grained Al$_2$O$_3$ water suspension and polished using silica slurry (Ultra-Sol® 2EX, Eminess Technologies, Scottsdale, AZ, USA) to produce a flat surface with a root mean square roughness (RMS) of about 1.65\,nm (determined over a $25\times25\,$\textmu m$^2$ area).
For domain imaging by PFM, an ac-excitation voltage of 10\,V was applied to the back electrode at a frequency of 61\,kHz while the tip was grounded. Local transport data was gained by cAFM with a dc-voltage of 2\,V applied to the back electrode.

\section{RESULTS \& DISCUSSION}

\subsection{Two relaxation-like dielectric features}

Figure \ref{fig1} shows the temperature dependent dielectric constant $\varepsilon'$ \textbf{a} and loss-tangent tan$\delta$ \textbf{b} for various frequencies from 1\,Hz to 1\,MHz. The electrodes where made with silver paint. For almost all frequencies $\varepsilon'$ exhibits a distinct two-step increase from about 30 to 200 -- 400 and further to $\varepsilon' > 5\times 10^3$. These steps in $\varepsilon'$ are accompanied by a peak in tan$\delta$, e.g., for the 4\,kHz curve at about 185\,K and 255\,K, respectively. This behaviour corresponds to a relaxation process in the temperature-dependent representation. Such prominent relaxation-like features are well known in oxide materials\cite{Lunkenheimer2010}, often originating from BLCs, e.g., Schottky-diodes forming a depletion-zone that acts as a thin insulating layer.
Further, the rather high values of the loss tangent (tan$\delta > 0.01$) corroborate the framework of BLC mechanisms responsible for the dielectric features.

\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{Figures/Fig1.pdf}
\caption{Temperature-dependence of $\varepsilon'$ \textbf{a} and tan$\delta$ \textbf{b} of h-ErMnO$_{3}$ with in-plane polarisation at selected frequencies from 1\,Hz to 1\,MHz. The inset in \textbf{a} compares $\Delta\varepsilon' = \varepsilon'/\varepsilon_{\infty}$ of the in-plane (open symbols) and out-of-plane (lines) polarisation of samples from the same batch for two different frequencies (colours refer to the respective frequencies in \textbf{a}). The data for the out-of-plane sample were taken from \textcite{Ruff2018APL}. Numbers (\textcircled{\footnotesize{1}} and \textcircled{\footnotesize{2}}) indicate the relaxation-like features (dashed lines give a guide to the eyes for the temperature-dependency of these features). The inset in \textbf{b} compares the temperature dependent conductivity $\sigma'$ for these samples for two frequencies (again, colours refer to the respective frequencies in \textbf{a}).
}
\label{fig1}
\end{figure}

Recently, it was reported that in h-YMnO$_3$ and h-ErMnO$_3$ the Schottky-barriers give rise to CDCs\cite{Schaab2018ElectricalWalls,Ruff2017ConductivityYMnO3,Holstad2018,Ruff2018APL}. Similar to this work (Fig.\,\ref{fig1}), two distinct relaxation-like features on h-YMnO$_3$ were measured\cite{Ruff2017ConductivityYMnO3}; one attributed to an internal BLC mechanism, possibly originating from insulating DWs. In contrast to the previously published data gained on samples with out-of-plane polarisation, the measurements presented in Fig.\,\ref{fig1} show well separated features, facilitating a more detailed analysis. We illustrate this difference with the inset in Fig.\,\ref{fig1}\,\textbf{a}, comparing the change in dielectric constant $\Delta\varepsilon'$ for the present in-plane oriented sample to an out-of-plane oriented one, published in \textcite{Ruff2018APL}. The CDC feature \textcircled{\footnotesize{2}} is the same, while the high dielectric constant feature \textcircled{\footnotesize{1}} appears as a distinct increase only for the in-plane sample. It is important to note, that for this comparison of $\varepsilon'$ we used different frequencies for the in-plane (259\,Hz and 4\,kHz) and out-of-plane (4\,kHz and 67\,kHz) orientation. For BLC-driven mechanisms the bulk dc-conductivity has a strong impact on the temperature and frequency range, where this feature dominates the dielectric properties\cite{Lunkenheimer2010}.
The inset of Fig.\,\ref{fig1}\,\textbf{b} shows $\sigma'$ for two representative frequencies, indicating a significant decrease of the conductivity of the out-of-plane oriented sample corroborating the above mentioned shift in the BLC driven feature. This is confirmed by the frequency-dependent dielectric analysis of h-(Er$_{0.99}$Ca$_{0.01}$)MnO$_3$ shown in Fig.\,S1.
This is a first hint of an anisotropic BLC feature, which is either based on the change in dc-conductivity or might be due to differences in DW density or conductance. For the latter, it is already well established via local probe measurements, that the conductance of the DWs strongly depend on the orientation of the polarisation\cite{Meier2012,Mosberg2019}.

\subsection{Quantifying the DW density}

We use PFM and cAFM to estimate the density of DWs, which provide a possible origin for the observed high dielectric constant feature.
To approximate the density of these
DWs in our h-ErMnO$_3$ crystal, we map the domain distribution at the sample surface using PFM (in-plane contrast) as presented in Fig.\,\ref{fig2}\,\textbf{a}. The PFM scan reveals the typical domain distribution, characteristic for hexagonal manganites\cite{Choi2010,Jungk2010,Safrankova1967}.
To determine the domain / DW density, we evaluate multiple test lines (one line is shown in Fig.\,\ref{fig2}\,\textbf{b} and further lines in Fig.\,S3) applying the procedure outlined in \textcite{Hubert1998}.
Measurements by cAFM (Fig.\,\ref{fig2}\,\textbf{c}) confirm the presence of DWs with enhanced (tail-to-tail) and reduced (head-to-head) conductance\cite{Meier2012}.
From this analysis we find $1 \pm 0.1$ DWs per \textmu m with enhanced or reduced conductance in comparison to the bulk. The same evaluation was also performed for h-(Er$_{0.99}$Ca$_{0.01}$)MnO$_3$ depicted in Fig.\,S2, providing a similar domain / DW fraction.

\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{Figures/Fig2.png}
\caption{\textbf{a} Calibrated in-plane PFM scan where the yellow (blue) represents ferroelectric domains pointing to the left (right). These form the characteristic domain pattern present in hexagonal manganite single crystals, where the ferroelectric $180^{\circ}$ domains come together at sixfold vertex points. \textbf{b} A representative line-profile from \textbf{a} where high values are from domains with \textit{$\leftarrow$P} and low values are from \textit{$\rightarrow$P} domains, used to get an approximate density of DWs per length. \textbf{c} A cAFM image from the area marked be the red box in \textbf{a}. Light colors indicated areas of enhanced conductance (tail-to-tail walls) while dark areas indicate areas of lower conductance (head-to-head walls).}
\label{fig2}
\end{figure}

\subsection{Dielectric features due to barrier layers}

To disentangle surface and internal contributions, a frequency-dependent analysis of the dielectric response is required, which is shown in Fig.\,\ref{fig3}. The frequency-dependent dielectric permittivity (Fig.\,\ref{fig3}\,\textbf{a}) exhibits two distinct relaxations for selected temperatures varying from 170\,K to 294\,K. The first one in the 210\,K-curve evolves at 1\,kHz indicated by a step-like increase in $\varepsilon'$ from $\sim20$ to $\sim300$. The lower $\varepsilon'$-value at high frequencies denotes the contribution of ionic and electronic polarizability to the so-called intrinsic $\varepsilon_{\infty}$, confirming literature values in the order of $20 - 40$\cite{Holstad2018,Ruff2018APL}.
The upper plateau of the second step for the 210\,K-curve begins at $\nu < 100$\,Hz and settles at an $\varepsilon'$ value in the order of $5 \times 10^3$. Both relaxations are accompanied by steps in $\sigma' (\nu)$. The plateaus in $\sigma'$ indicate roughly the dc-conductivity of the BLCs and the bulk. However, the dc-conductivity of step two -- most likely the Schottky barrier -- is almost shifted out of the measured frequency range. The curvature of $\sigma'$(294\,K) for $\nu < 10$\,Hz indicates the onset of this dc-plateau of approximately $\sigma_{dc} \approx 3\times 10^{-9}$\,$\Omega^{-1}$cm$^{-1}$. The $\sigma_{dc}$-plateau of the first BLC feature emerges, e.g., for the 210\,K-curve between 100 -- 1000\,Hz at $2\times 10^{-8} $\,$\Omega^{-1}$cm$^{-1}$. Finally, at higher frequencies, e.g., $\nu > 10$\,kHz for 210\,K-curve, $\sigma_{dc}$ of the bulk evolves, which is for low temperatures superimposed by a contribution of an universal dielectric response (UDR) feature\cite{Jonscher1977}, giving rise to a frequency-dependent increase in the overall conductivity.

\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{Figures/Fig3.pdf}
\caption{Frequency-dependence of $\varepsilon$' \textbf{a} and $\sigma$' \textbf{b} of h-ErMnO$_{3}$ with in-plane polarisation at various temperatures (again numbers \textcircled{\footnotesize{1}} and \textcircled{\footnotesize{2}} as well as the blacked dashed lines indicate the relaxation-like features). The coloured dashed lines represent fits for the measured data, obtained by considering the equivalent circuit model, sketched in the corner of \textbf{b}. The equivalent circuit model consists of three $RC$-circuits connected in series.}
\label{fig3}
\end{figure}

In oxide materials there exist different descriptions for these observed dielectric relaxation-like features: (i) hopping conductivity, (ii) displacements of DWs in excitation fields far below the coercive field, (iii) charged DWs acting as conductive inclusions\cite{Esin2017}, (iv) Maxwell-Wagner type electrical inhomogeneities or (v) Schottky-barriers at the electrode contacts. Hopping conductivity in disordered systems can be responsible for a continuous increase in $\varepsilon'$ at low frequencies (typical $\nu < 1$\,Hz)\cite{Lunkenheimer2010}. In the present study, we observe distinct relaxation-like features strongly shifting with temperature, excluding mechanism (i). Further, the DWs are strongly pinned by the vortex structure of the hexagonal manganites and the coercive-fields for polarisation reversal in h-ErMnO$_3$ are typically $> 3$\,kV/mm\cite{Choi2010,Ruff2018APL}, which is orders in magnitude higher than the used voltage of 1\,V/mm for the dielectric spectroscopy. Therefore, irreversible DW motion does not contribute to the dielectric permittivity\cite{Lewis1959,Hall1999}. In addition, the dielectric relaxation related to reversible domain wall motion typically occurs in the GHz range\cite{Kittel1951} and is temperature independent\cite{Arlt1994}. The strong temperature-dependence of the dielectric relaxation observed in our samples (Fig.\,\ref{fig1}) also allows to exclude reversible domain wall motion as an origin.

Due to these robust domain structure, mechanism (ii) seems to be very unlikely. Interestingly, explanation model (iii) requires charged conductive DWs, which are discussed to enhance the dielectric permittivity in (K,Na)NbO$_3$-based ferroelectrics\cite{Esin2017}. More specific, a higher density of DWs leads to an increase in dielectric permittivity. Charged DWs with enhanced conductivity are also present in our samples (c.f. Fig.\,\ref{fig2}\,\textbf{c}). However, in contrast to the above mentioned ferroelectric, we find the contrary behaviour of the relation of DW density to the dielectric permittivity. Here, $\varepsilon'$ decreases with increasing DW density\cite{Ruff2017ConductivityYMnO3}, excluding charged conductive DWs as origin for the observed BLC feature. Thus, we focus in the following only on the latter two mechanisms of internal (iv) and surface (v) BLCs. For the present study we investigated a single crystal, for which we can further exclude a BLC mechanism due to insulating grain boundaries, as observed in oxide ceramics\cite{Adams2002,Lunkenheimer2010,Moulson}.

The dashed lines in Fig.\,\ref{fig3} represent fits using the equivalent circuit model depicted in the inset of Fig.\,\ref{fig3}\,\textbf{b}. This model uses the approach of Maxwell and Wagner\cite{Wagner1914,Maxwell1873}, for which volume fractions of the overall sample with certain capacitances and conductivities can be described by discrete $RC$-circuits connected in series. In a nutshell, we deploy $RC$-circuits for step two (surface BLC) and step one (internal BLC) in series to the bulk properties. For the latter $RC$-circuit (bulk) we use an additional frequency-dependent resistor accounting for the power-law contribution of UDR to $\sigma'$\cite{Jonscher1977}. From these fits parameters of the dielectric constants and dc-conductivities ($\varepsilon_{\infty}$', $\varepsilon_{1}$', $\varepsilon_{2}$', and temperature-dependent values for $\sigma_{dc}, \sigma_1'$ and $\sigma_2'$) are gained as listed in Table S1. The fitting parameters confirm the temperature-dependent evolution of the dielectric properties of the BLC contributions and the semiconducting behaviour of the bulk.

While fitting with an equivalent circuit model allows for analyzing and disentangling single BLC contributions, it cannot provide information about the underlying mechanism. The formation of a Schottky barrier at the interface of the metal electrode and the semiconducting bulk leads to a depletion layer that acts as a thin capacitor at the sample surface. A proven approach\cite{Krohns2007,Lunkenheimer2010} to establish such Schottky BLCs is to measure the dielectric properties using electrodes of different wettings, e.g. painted \textit{vs}. sputtered electrodes: for Schottky BLCs an increased wetting will provide an enhanced CDC. Internal BLCs and the intrinsic dielectric properties, however, should not be affected by this change. Another way to proof a surface BLC is to reduce the sample thickness, as both $\varepsilon'$ and $\sigma'$ are geometry independent values, so the dielectric response should not be affected.

\begin{table*}[htb]
\caption{Calculated and measured contribution of the DWs to the dielectric response for h-\textit{R}MnO$_3$ (\textit{R} = Y, Er) with in-plane (IP) and out-of-plane (OOP) polarisation. The values are estimated from the dielectric spectroscopy (Figs. \ref{fig3} \& S1) and local-probe analyses (Figs. \ref{fig2}, S2 \& S3).}
\label{tab1}
\begin{tabular}{l|c|c|c|c|c|c}
sample & dir. of P & $\varepsilon_{\infty}$ & $n_{DW}$ [$1/$\textmu\text{m}] & $V_{DW}$ [\%] & estim. $\varepsilon_{1}$ & meas. $\varepsilon_{1}$ \\
\hline
h-ErMnO$_3$ (Figs. \ref{fig2}, \ref{fig3} \& S3) & IP & 32 $(\pm 3)$ & 0.5 $(\pm 0.05)$ & 7.5 $(\pm 0.75)$ & 400 $(\pm 60)$ & 250 $(\pm 50)$  \\
h-(Er$_{0.99},$Ca$_{0.01}$)MnO$_3$ (Figs. S1 \& S2) & IP & 18 & 0.46 & 7 & $260$ & $220$ \\
\hline
h-YMnO$_3$ sample 1 (data from \textcite{Ruff2017ConductivityYMnO3}) & OOP & 20 & 0.17 & 2.5 & $780$ & $670$ \\% $1\times 10^{-7}$ \\
h-YMnO$_3$ sample 2 (data from  \textcite{Ruff2017ConductivityYMnO3}) & OOP & 20 & 2.5 & 37.5 & 54 & 40 \\
\end{tabular}
\end{table*}

Figure\,\ref{fig4} shows frequency-dependent $\varepsilon'$ \textbf{a} and $\sigma'$ \textbf{b} for both strategies indicated by different symbols (open symbols $\rightarrow$ silverpaint \& $d_{\text{thick}}$, closed symbols $\rightarrow$ silverpaint \& $d_{\text{thin}}$, crosses $\rightarrow$ sputtered gold \& $d_{\text{thick}}$). Analogous to Fig.\,\ref{fig3} both relaxations appear for the different surface treatments.
Importantly, we find that the intrinsic bulk properties \textcircled{\footnotesize{3}}, as well as the first relaxation \textcircled{\footnotesize{1}} step, are -- as expected for the bulk properties and an internal BLC -- independent of the electrode material and the thickness of the sample. In contrast, the enhanced wetting of the sputtered electrode increases the upper plateau of $\varepsilon'$ by a factor of about two. Furthermore, reducing the sample thickness gives rise to an increase in the CDC feature.
This leads us to the conclusion that the second relaxation \textcircled{\footnotesize{2}} is due to the formation of an insulating depletion layer due to a Schottky barrier at the sample surface. Critical, and in contrast, the first relaxation is not affected by these changes to the surface, corroborating the hypothesis of an internal BLC mechanism related to insulating DWs\cite{Ruff2017ConductivityYMnO3}.

\subsection{Insulating DW barrier layer capacitors}

From the local probe analysis of the sample surface (Fig.\,\ref{fig2}), in combination with bulk dielectric measurements, we calculate an approximate volume fraction (in \%) of the insulating DWs in our sample: $V_{DW} = n_{DW} d_{DW}$, where $n_{DW}$ denotes the number of insulating DWs per \textmu m and $d_{DW}$ the electronic thickness of the DWs referred to as electrical dressing in \textcite{Meier2012}. Note, this electronic width is much larger than the structural width of about 5\,Å\cite{Holtz2017}, reaching values in the order of 100 to 150\,nm, which was related to a delocalisation of charge carriers\cite{Meier2012}.
As the contributions of the ionic and electronic polarizability to $\varepsilon_{\infty}$ for both bulk and DWs are almost the same\cite{Schaab2018ElectricalWalls}, we can estimate the value of the dielectric constant $\varepsilon_{1}$. Within the framework of the Maxwell-Wagner model, $\varepsilon_{1}$ is given by the relation: $\varepsilon_{1} \approx \varepsilon_{\infty}/V_{DW}$.

\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{Figures/Fig4.pdf}
\caption{Frequency-dependence of $\varepsilon$' \textbf{a} and $\sigma$' \textbf{b} for two representative temperatures. In this figure the measurement data for sputtered gold and silver paint as contacting material are shown and, for the latter, also the thickness dependence ($d_{\text{thick}}$ = 1\,mm and $d_{\text{thin}}$ = 0.6\,mm). Numbers indicate the dielectric features: \textcircled{\footnotesize{1}}$\rightarrow$ BLC one (internal), \textcircled{\footnotesize{2}}$\rightarrow$ BLC two (surface),
\textcircled{\footnotesize{3}}$\rightarrow$ bulk properties.}
\label{fig4}
\end{figure}

In table \ref{tab1}, the calculated values for the dielectric response of different sample are compared.
The estimated and measured values for all samples listed in table \ref{tab1} are almost in the same order, providing a strong indication for a BLC-DW based mechanism. However, we note that a smeared-out relaxation due to a distribution of relaxation times of the BLC, an inhomogeneous polar pattern, as well as possible variations of the intrinsic dielectric constant cannot be excluded, adding to the uncertainty of the presented values in table \ref{tab1}.

Finally, we address the question, why in-plane polarized samples seem to be suited better than out-of-plane polarized specimens for detecting BLC-DW features. This can be explained employing the Maxwell-Wagner model: The relaxation-time $\tau_{BLC-DW}$ of the BLC-DW $RC$-circuit connected in series to the bulk is: $\tau_{BLC-DW} \propto \varepsilon'_{1}/\sigma_{dc}$. Thus, $\varepsilon'$ -- mainly based on the number of insulating DWs and their effective thickness -- and the temperature-dependent bulk $\sigma_{dc}$ are strongly affecting the frequency range in which the relaxation occurs. Especially, for the samples h-ErMnO$_3$ and h-(Er$_{0.99}$,Ca$_{0.01}$)MnO$_3$, we measured a strongly anisotropic dc-conductivity, which is enhanced for the samples with in-plane polarisation by up to a factor of 100 -- depending on the temperature -- compared to the samples with out-of-plane polarisation (Figs.\,S1 \& S2). An enhanced dc-conductivity results in a decrease in $\tau_{BLC-DW}$ and, thus in an increase in the corresponding frequency $\nu_{BLC-DW} = 1/(2\pi\tau_{BLC-DW})$. According to this, the anisotropy in the bulk $\sigma_{dc}$ seems to be the most likely reason, which allows us to disentangle the contributions of surface and internal BLCs, especially in the case of samples of h-ErMnO$_3$ with in-plane polarisation.

\section{SUMMARY}

In this study, we investigate the contribution of insulating DWs in h-ErMnO$_3$ to the overall dielectric response up to the MHz regime. Depending on the temperature range of the dielectric measurements, two distinct relaxation-like processes are revealed, especially for samples with in-plane polarisation. One of these relaxation-like features originates from a Schottky barrier, which we proof by target-oriented manipulation of the sample-electrode interface. For the other feature we conclude, corroborated by PFM and cAFM measurements, that an internal barrier layer is formed by insulating DWs. To proof this hypothesis, we first use an equivalent-circuit model to quantify the bulk dielectric properties of this internal barrier layer. Secondly, we compare these values to a barrier layer calculated with the density and the electronic thickness of insulating DWs measured by PFM and cAFM. Based on this data, we confirm that internal barrier layer capacitors are formed at insulating DWs, which is corroborated by the comparison of different h-YMnO$_3$ and h-ErMnO$_3$ samples.

As both the density\cite{PhysRevX.7.041014, PhysRevX.2.041022}
and electrical characteristic of the insulating DWs\cite{Holstad2018, Hassanpour_2016} can be tuned, the engineering the macroscopic dielectric response is feasible. This may pave the way to generate high and even colossal dielectric constants by robust internal barrier layers in h-ErMnO$_3$ and h-YMnO$_3$, which makes these improper ferroelectrics promising for the use as dielectrics in multilayer ceramic capacitors.

\section*{Author contributions}
S.K. initiated and coordinated the project. L.P. and A.S. conducted the dielectric experiments, and L.P., I.K. and S.K. analyzed the data. J.S., L.P., M.A. and D.E. performed and analyzed the local-probe experiments supervised by D.M. The single crystals were prepared by E.B. and Z.Y. All authors participated in the discussion and interpretation of the results. D.E., L.P., J.S., D.M. and S.K. wrote the manuscript.

\section*{Acknowledgments}
J.S. acknowledges the support from the Alexander von Humboldt Foundation through the Feodor-Lynen fellowship. D.M. acknowledges support by NTNU through the Onsager Fellowship Program and the Outstanding Academic Fellows Program, and funding from the European Research Council (ERC) under the European Union`s Horizon 2020 Research and Innovation Programme (Grant Agreement No. 86691). S.K., M.A., A.S., L.P., and I.K. acknowledge the funding of the German Science foundation via the Collaborative Research Center TRR80.

\section*{Data Availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.

\section*{References}

\subsection{Parameters of equivalent circuit analysis}
Table I lists the fit parameters of the equivalent-circuit analysis performed for the data partially presented in Fig.\,3. The equivalent circuit, depicted as inset in Fig.\,3\,\textbf{b}, consists of three $RC$-elements, one for the bulk and two for barrier-layer contributions. The bulk $RC$-circuit also takes into account contributions of a frequency-dependent conductivity, the UDR behaviour\cite{Jonscher1977}. Details of the fitting routine can be found in \textcite{Lunkenheimer2010} and \textcite{Ruff2017ConductivityYMnO3}. Here, we list only the values for the dielectric properties. The additional quantities within the fitting model, i.e., the broadening-parameters $\alpha$, which comprises the distribution of relaxation times, and the UDR behaviour remain almost constant for the selected temperatures.

\begin{table*}[htb]
    \caption{Fit parameters of the equivalent circuit analysis of the frequency-dependent dielectric properties of h-ErMnO$_3$}
    \centering
\begin{tabular}{c|c|c|c|c|c|c}

Temperature (K) &   $\varepsilon_{\infty}$  &   $\varepsilon_1$ &   $\varepsilon_2$ &   $\sigma_{dc}$ ($\Omega^{-1}\text{cm}^{-1}$)  &   $\sigma_{1}$ ($\Omega^{-1}\text{cm}^{-1}$) &   $\sigma_{2}$ ($\Omega^{-1}\text{cm}^{-1}$)\\
    \hline
290 &   35   &  200 &   14000  &   $1.45\cdot10^{-4}$   &   $1.56\cdot10^{-5}$  &   $7.29\cdot10^{-9}$\\

270 &   35   &  230 &   11941  &   $2.90\cdot10^{-5}$   &   $5.50\cdot10^{-6}$  &   $3.3\cdot10^{-9}$\\

250 &   35   &  231 &   11585  &   $7.10\cdot10^{-6}$   &   $1.14\cdot10^{-6}$  &   $6.13\cdot10^{-10}$\\

230 &   35   &  200 &   13000  &   $2.37\cdot10^{-6}$   &   $2.22\cdot10^{-7}$  &   $6.21\cdot10^{-11}$\\

210 &   35   &  200 &   13000  &   $2.34\cdot10^{-7}$   &   $3.26\cdot10^{-8}$  &   $2.66\cdot10^{-11}$\\

190 &   35.3   &  300 &   13000  &   $9.84\cdot10^{-9}$   &   $2.85\cdot10^{-9}$  &   $2.22\cdot10^{-14}$\\

170 &   35.8   &  300 &   13000  &   $7.53\cdot10^{-10}$   &   $2.15\cdot10^{-10}$  &   $3.66\cdot10^{-15}$\\
\hline
\end{tabular}
\end{table*}

\subsection{Dielectric and local probe measurements of h-(Er$_{0.99}$,Ca$_{0.01}$)MnO$_3$}

Beside the undoped h-ErMnO$_3$ we performed the bulk dielectric and local-probe analyses on two 1\% Calcium doped ErMnO$_3$ samples of the same batch with out-of-plane and in-plane polarisation. Details of sample preparation can be found in \textcite{Hassanpour_2016}.
Figure S\ref{SFigure1} shows the frequency dependent dielectric properties for samples with out-of-plane (closed symbols) and in-plane (open symbols) polarisation. Again, in analogy to the discussion of Fig.\,3, for the sample with in-plane polarisation two distinct relaxation-like features show up, a smaller one reaching $\varepsilon_1$ of about 220 and a more prominent one reaching a CDC for $\varepsilon_2$ of about $2\times 10^3$. Interestingly, for both samples a further increase at low frequencies is revealed, which could be originating from hopping conductivity\cite{Lunkenheimer2010, Jonscher1977}.

\begin{figure}[htb]
    \centering
    \includegraphics[width=0.7\linewidth]{Figures/SFig2.pdf}
    \caption{Frequency-dependent dielectric constant $\varepsilon$' \textbf{a} and conductivity $\sigma$' \textbf{b} h-(Er$_{0.99}$,Ca$_{0.01}$)MnO$_3$ sample with out-of-plane (closed symbols) and in-plane (open symbols) polarisation at various temperatures.}
    \label{SFigure1}
\end{figure}

Figure S\ref{SFigure2} provides the characterization of the ferroelectric domain pattern measured by PFM \textbf{a} and cAFM \textbf{c}. The density of the domain walls with enhanced and reduced conductance in comparison to the bulk is measured by several line-scans. For the DW BLC model we only take into account the number of the DW with reduced conductance, which is for \textbf{b} 0.46 per \textmu m.

\begin{figure}[htb]
    \centering
    \includegraphics[width=0.7\linewidth]{Figures/SFig4.png}
    \caption{\textbf{a} Calibrated in-plane PFM scan on h-(Er$_{0.99}$,Ca$_{0.01}$)MnO$_3$ where the yellow (blue) represents ferroelectric domains pointing to the left (right). These form the characteristic domain pattern present in hexagonal manganite single crystals, where the ferroelectric $180^{\circ}$ domains come together at sixfold vertex points. \textbf{b} A representative line-profile from \textbf{a} where high values are from domains with \textit{$\leftarrow$P} and low values are from \textit{$\rightarrow$P} domains, used to get an approximate density of DWs per length. \textbf{c} A cAFM image from the area marked be the red box in \textbf{a}. Light colors indicated areas of enhanced conductance (tail-to-tail walls) while dark areas indicate areas of lower conductance (head-to-head walls).}
    \label{SFigure2}
\end{figure}

\subsection{Quantifying the DW density of ErMnO$_3$}

Figure S\ref{SFigure3} shows the PFM measurement of h-ErMnO$_3$ as well as the corresponding lines for the evaluation of the DW density.

\begin{figure}[htb]
    \centering
    \includegraphics[width=0.7\linewidth]{Figures/SFig5.png}
    \caption{\textbf{a} In-plane PFM scan on h-ErMnO$_3$ with in-plane polarisation (c.f. Fig.\,2). \textbf{b} Line-scans (coloured lines in \textbf{a}) are extracted, from which we deduce an average amount of domain walls with reduced conductance of 10$\pm$1 per 20\,\textmu m. High values are from domains with $\leftarrow$\textit{P} and low values are from $\rightarrow$\textit{P} domains.}
    \label{SFigure3}
\end{figure}