# The combinatorial structure of
symmetric strongly shifted ideals

Alessandra Costantini and Alexandra Seceleanu  Oklahoma State University
<EMAIL_ADDRESS>University of Nebraska-Lincoln<EMAIL_ADDRESS>
###### Abstract.

Symmetric strongly shifted ideals are a class of monomial ideals which come
equipped with an action of the symmetric group and are analogous to the well-
studied class of strongly stable monomial ideals. In this paper we focus on
algebraic and combinatorial properties of symmetric strongly shifted ideals.
On the algebraic side, we elucidate properties that pertain to behavior under
ideal operations, primary decomposition, and the structure of their Rees
algebra. On the combinatorial side, we develop a notion of partition Borel
generators which leads to connections to discrete polymatroids, convex
polytopes, and permutohedral toric varieties.

2020 Mathematics Subject Classification: primary 13C70, 05E40, secondary:
13B22, 13F55, 14M25.

Keywords: symmetric strongly shifted ideal, polymatroidal ideal, toric ring,
Rees algebra, permutohedron.

###### Contents

  1. 1 Introduction
  2. 2 Symmetric shifted ideals under ideal operations
  3. 3 Partition Borel generators and combinatorial structure
    1. 3.1 Principal Borel sssi’s and permutohedra
    2. 3.2 The polymatroidal structure of a principal Borel sssi
    3. 3.3 Numerical invariants of symmetric shifted ideals
  4. 4 Integrally closed sssi’s
  5. 5 Associated primes and primary decomposition
    1. 5.1 Stable associated primes of a principal Borel sssi
    2. 5.2 The intersection property of a principal Borel sssi
  6. 6 Toric ideals and Rees algebras
    1. 6.1 Fiber type property
    2. 6.2 The toric ideal of a principal Borel sssi

##  1\. Introduction

Let $R=K[x_{1},\ldots,x_{n}]$ be a polynomial ring over a field $K$ and
consider the set of partitions

$\mathrm{P}_{n}=\\{\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{Z}^{n}:0\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}\\}.$

Orbits of monomials under the natural action of the symmetric group
$\mathfrak{S}_{n}$ can be identified with elements of $P_{n}$. This induces a
bijection between $\mathfrak{S}_{n}$-fixed monomial ideals $I\subset R$ and
sets of partitions $P(I)\subset P_{n}$ given by

$\mathrm{P}(I)=\\{\lambda\in\mathrm{P}_{n}:x^{\lambda}=x_{1}^{\lambda_{1}}\cdots
x_{n}^{\lambda_{n}}\in I\\}.$

The central object of study of this note are the following classes of
$\mathfrak{S}_{n}$-fixed monomial ideals, which were introduced in [BdAG+20].

######  Definition 1.1.

Let $I\subset R$ be an $\mathfrak{S}_{n}$-fixed monomial ideal. We say that
$I$ is a symmetric shifted ideal, or ssi, if, for every
$\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathrm{P}(I)$ and $1\leq i<n$ with
$\lambda_{i}<\lambda_{n}$, one has $x^{\lambda}(x_{i}/x_{n})\in I$. We say
that $I$ is a symmetric strongly shifted ideal, or sssi, if, for every
$\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathrm{P}(I)$ and $1\leq i<j\leq
n$ with $\lambda_{i}<\lambda_{j}$, one has $x^{\lambda}(x_{i}/x_{j})\in I$.
Monomials $x^{\lambda}(x_{i}/x_{n})$ and $x^{\lambda}(x_{i}/x_{j})$ satisfying
the conditions above are referred to as being obtained from $x^{\lambda}$ by a
Borel move.

The definitions of symmetric shifted and strongly shifted ideals are inspired
by the definition of stable and strongly stable ideals. These are the most
important classes of monomial ideals in computational algebra since, e.g., in
characteristic zero generic initial initials are strongly stable. Moreover,
stable and strongly stable ideals have well-understood minimal graded free
resolutions. These were constructed by Eliahou and Kervaire, who also gave a
formula for their graded Betti numbers in terms of the data of their minimal
systems of monomial generators [EK90]. Analogous results for symmetric shifted
ideals were obtained in [BdAG+20].

In this article, we initiate a comprehensive study of the algebraic properties
and combinatorial structure of symmetric strongly shifted ideals. Along the
way we point out numerous items which extend the list of similarities between
symmetric strongly shifted ideals and strongly stable ideals. A key tool we
develop is the notion of _partition Borel generators_ of a symmetric strongly
shifted ideal (see 3.4), which is inspired by the notion of Borel generators
of a strongly stable ideal and allows us to interpret a sssi as the
symmetrization of a strongly stable ideal (we refer the reader to Section 3
for any unexplained terminology). More precisely, our first main result is the
following.

######  Theorem A (cf. 3.8).

An ideal $I$ is symmetric strongly shifted if and only if $I$ is the
symmetrization of a strongly stable ideal $J$ in the following sense

$I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(J).$

Moreover, under this symmetrization process, the partition Borel generators of
$I$ correspond to the Borel generators of $J$.

While symmetric strongly shifted ideals are almost never strongly stable (see
3.2), the symmetrization process described by A sometimes allows to transfer
desirable algebraic properties from the class of strongly stable ideals to the
class of symmetric strongly shifted ideals. As an exemplification of this
principle, in 4.2 we identify classes of normal strongly shifted ideals which
are the symmetrization of normal strongly stable ideals. Moreover, from a
primary decomposition of a _principal Borel ideal_ (i.e., a strongly stable
ideal with exactly one Borel generator) we can determine a primary
decomposition of a _principal Borel sssi’s_ (i.e., a symmetric strongly
shifted ideal with exactly one partition Borel generator).

######  Theorem B (cf. 5.5).

Let $I$ be a principal Borel symmetric strongly shifted ideal, with partition
Borel generator $\lambda=(\lambda_{1},\ldots,\lambda_{n})$. A primary
decomposition of $I$ is given by

$I=\bigcap_{j=1}^{n}\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(x_{1},\ldots,x_{j})^{\sum_{i=1}^{j}\lambda_{i}}.$

The primary decomposition of B can be refined to provide an irredundant
decomposition (5.7), which allows to determine the associated primes of
ordinary powers of principal Borel sssi’s (5.8). The decomposition can also be
rearranged as an intersection of symbolic powers of square-free sssi’s. The
latter ideals coincide with the _square-free Veronese ideals_ (see 2.6) and
are the symmetrizations of the monomial prime ideals of $R$ in the sense of A.
In fact, they dictate the algebraic and combinatorial properties of symmetric
strongly shifted ideals in a similar way as the monomial primes determine the
algebraic and combinatorial properties of strongly stable ideals.

On the algebraic side, these square-free sssi’s determine the radical (2.7)
and the symbolic powers (2.13) of every symmetric strongly shifted ideal.
Moreover, every sssi with exactly one partition Borel generator can be
factored as a product of square-free Veronese ideals (3.15). In turn, the
latter factorization result gives rise to a combinatorial characterization of
principal Borel sssi’s in terms of _discrete polymatroids_ , introduced by
Herzog and Hibi in [HH02] as a generalization of the notion of matroid. A
monomial ideal is called a _polymatroidal ideal_ if the exponent vectors of
its monomial generators form a discrete polymatroid (see 3.18 for more
details). Surprisingly, the class of principal Borel sssi’s coincides with the
class of symmetric polymatroidal ideals.

######  Theorem C (cf. 3.19).

A monomial ideal is symmetric and polymatroidal if and only if it is a
symmetric strongly shifted ideal with exactly one partition Borel generator.

The factorization property of a principal Borel sssi given by 3.15 allows for
a beautiful description of their _toric ideal_ , whose definition we recall in
Section 6. In more detail, our main result is the following.

######  Theorem D (cf. 4.3 and 6.5).

The toric ideal of a symmetric strongly shifted ideal with exactly one
partition Borel generator is generated by quadratic polynomials, namely the
symmetric exchange relations. Moreover, the toric ring is a Cohen-Macaulay
normal domain which has rational singularities in characteristic zero and is
$F$-rational in positive characteristic.

In particular, our result provides supporting evidence for a longstanding
conjecture of White, Herzog and Hibi [Whi80, HH02], which states that, for an
arbitrary polymatroidal ideal $I$, the toric ideal of $I$ is generated by the
symmetric exchange relations. D shows that the conjecture holds for all
_symmetric_ polymatroidal ideals.

From a geometric perspective, a principal Borel sssi defines a normal toric
permutohedral variety, i.e., an algebraic variety associated with a well-
studied convex polytope dubbed the permutohedron (see 3.16 and 6.13).

While the combinatorial structure of symmetric strongly shifted ideals with an
arbitrary number of Borel generators remains more mysterious, the knowledge of
their syzygies from [BdAG+20] offers a valuable complementary source of
information. In particular, it allows us to determine the depths of powers of
an equigenerated symmetric shifted ideal (5.1) and to provide a structure
theorem for the _Rees algebra_ ${\mathcal{R}}(I)\cong\oplus_{k\geq 1}I^{k}$ of
an equigenerated symmetric strongly shifted ideal $I$. Namely, in 6.2 we prove
that ${\mathcal{R}}(I)$ is the quotient of a polynomial ring modulo relations
which are either linear or arise from the toric ideal of $I$. The latter
statement is analogous to a well-known result on the Rees algebra of an
equigenerated strongly stable ideal [HHV05, Theorem 5.1].

#### Structure of the paper

In Section 2 we analyze the behavior of symmetric (strongly) shifted ideals
under various algebraic operations. In Section 3 we introduce partition Borel
generators and prove A. We study principal Borel sssi’s in Section 3.1, while
in Section 3.2 we prove C and identify some distinguished classes of symmetric
polymatroidal ideals (3.21 and 3.22). In Section 3.3 we give combinatorial
formulas for several numerical invariants associated with a symmetric monomial
ideal.

In the rest of the article, we exploit the combinatorial structure of sssi’s
to study their ordinary and symbolic powers. In Section 4 we discuss the
normality property of symmetric strongly shifted ideals, which is used in
Section 5 to study the associated primes of their powers. In Section 5.1 we
prove B and identify irredundant primary decompositions of sufficiently large
powers of principal Borel sssi’s. Finally, we study the Rees algebra of
symmetric strongly shifted ideals in Section 6, where we prove our structure
theorems 6.2 and 6.5 and discuss the geometry of toric varieties associated
with a principal Borel sssi.

#### Notational conventions

We say that a sequence $\lambda=(\lambda_{1},\dots,\lambda_{n})$ of non-
negative integers is a partition of $d$ of length $n$, if
$\lambda_{1}\leq\cdots\leq\lambda_{n}$ and
$|\mathbf{\lambda}|=\lambda_{1}+\dots+\lambda_{n}=d$. We opt for the less
standard convention of ordering the parts nondecreasingly for our conventions
to match those in [BdAG+20], where symetric shifted ideals were originally
introduced. If $\lambda$ has distinct parts $p_{1},\ldots,p_{s}$ which occur
with multiplicities $n_{1},\ldots,n_{s}$ respectively, we sometimes use the
alternate notation $\lambda=(p_{1}^{n_{1}},\ldots,p_{s}^{n_{s}})$. Throughout
the paper the notation $e_{i}$ stands for the $i$-th standard basis vector in
${\mathbb{Z}}^{n}$.

For a monomial $u=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$, we write
$\operatorname{part}(u)\in P_{n}$ for the partition obtained from
$(a_{1},\dots,a_{n})$ by ordering its entries non-increasingly. If a monomial
ideal $I\subset R$ is $\mathfrak{S}_{n}$-fixed, then a monomial $u$ is in $I$
if and only if $x^{\operatorname{part}(u)}$ is in $I$. The set $P(I)$ contains
a partition $\lambda\neq(0^{n})$ if and only if $I\neq R$. Throughout the
paper we assume that $I\neq R$.

#### Acknowledgements

This work was partially supported by an NSF-AWM Mentoring Travel Grant. The
first named author thanks Federico Castillo, Chris Francisco, Jeff Mermin,
Jonathan Montaño, Jay Schweig and Gabriel Sosa for insightful discussions
about strongly stable and polymatroidal ideals. The second author is partially
supported by NSF grant DMS–2101225.

##  2\. Symmetric shifted ideals under ideal operations

In [Cim09, Proposition 1], Cimpoeaş observes that the class of ideals of Borel
type (which generalizes strongly stable ideals) is closed under sum,
intersection, product, and colon operations. The class of strongly stable
ideals is also closed under the same operations as demonstrated in [GW15,
Proposition 1.2] and also under taking integral closure [GW15, Theorem 2.1]
and symbolic powers [GW15, Theorem 3.8]. (In the latter work, symbolic powers
are taken by retaining the primary components associated to minimal primes of
$I$.)

In this section we show that a majority of these statements are also true for
symmetric strongly shifted ideals and fewer also hold for symmetric shifted
ideals. Towards this end, it will be convenient to simplify 1.1 slightly.
Denoting by $G(I)$ the set of minimal monomial generators of $I$, allows to
single out the partition generators of $I$, namely

$\mathrm{\Lambda}(I)=\\{\lambda\in P_{n}:x^{\lambda}\in G(I)\\}.$

In view of [BdAG+20, Lemmas 2.2 and 2.3] it suffices to check the conditions
of 1.1 for the partition generators $\lambda\in\mathrm{\Lambda}(I)$ rather
than for arbitrary elements $\lambda$ in $P(I)$.

We begin by proving that the class of symmetric shifted ideals is closed under
sums and intersections.

######  Proposition 2.1.

Let $I,J$ be symmetric (strongly) shifted ideals. Then $I+J$ and $I\cap J$ are
symmetric (strongly) shifted ideals.

###### Proof.

We prove the statement assuming that $I$ and $J$ are symmetric strongly
shifted. The proof for symmetric shifted ideals is analogous, hence left to
the diligent reader.

First, notice that $I+J$ and $I\cap J$ are $\mathfrak{S}_{n}$-fixed, by the
symmetry of $I$ and $J$. Now, let $\lambda\in P_{n}$ be such that
$\lambda_{i}<\lambda_{j}$. If $x^{\lambda}\in I+J$, then $x^{\lambda}\in I$ or
$x^{\lambda}\in J$. In the first case, $x^{\lambda}x_{i}/x_{j}\in I$ by the
strongly shifted property of $I$, while in the second case
$x^{\lambda}x_{i}/x_{j}\in J$ because $J$ is strongly shifted. In either case,
$x^{\lambda}x_{i}/x_{j}\in I+J$, which proves that $I+J$ is a sssi. Similarly,
if $x^{\lambda}\in I\cap J$, then $x^{\lambda}x_{i}/x_{j}\in I$ and
$x^{\lambda}x_{i}/x_{j}\in J$ by the strongly shifted property of $I$ and $J$.
Hence, $x^{\lambda}x_{i}/x_{j}\in I\cap J$ and $I\cap J$ is symmetric strongly
shifted. ∎

The following example shows that the class of symmetric shifted ideals is not
closed under taking products.

######  Example 2.2.

In $k[x_{1},x_{2},x_{3},x_{4}]$, consider the symmetric shifted ideal $I$ with

$\Lambda(I)=\\{(1,1,2,2),(0,2,2,2),(0,1,2,3)\\}$

Notice that $I$ is not a sssi, since $(0,1,2,3)\in P(I)$ but $(1,1,1,3)\notin
P(I)$; see [BdAG+20, Example 2.5]. Moreover, the maximal ideal
${\mathfrak{m}}=(x_{1},x_{2},x_{3},x_{4})$ is symmetric strongly shifted, with
$\Lambda({\mathfrak{m}})=\\{(0,0,0,1)\\}$. Then, the monomial ideal
$I{\mathfrak{m}}$ is $\mathfrak{S}_{n}$-invariant with

$\Lambda(I{\mathfrak{m}})=\\{(1,2,2,2),(1,1,2,3),(0,2,2,3),(0,1,2,4),(0,1,3,3)\\}$

but is not symmetric shifted, since $(0,1,2,4)\in P(I{\mathfrak{m}})$ but
$(1,1,1,4)\notin P(I{\mathfrak{m}})$.

Nevertheless, the subclass of symmetric strongly shifted ideals is closed
under products.

######  Proposition 2.3.

A product of symmetric strongly shifted ideals is symmetric strongly shifted.

###### Proof.

It suffices to prove the statement for the product of two symmetric strongly
shifted ideals $I,J$. It is clear that the ideal

$IJ=\left(\\{\sigma(x^{\lambda})\tau(x^{\mu}):\sigma,\tau\in\mathfrak{S}_{n},\lambda\in
P(I),\mu\in P(J)\\}\right)$

is fixed by $\mathfrak{S}_{n}$. It remains to show $IJ$ is strongly shifted.
Towards this end let $p\in P_{n}$ be such that
$x^{p}=\sigma(x^{\lambda})\tau(x^{\mu})\in IJ$ be as above and denote the
relevant monomials by $\sigma(x^{\lambda})=x^{\sigma(\lambda)}\in I$ and
$\tau(x^{\mu})=x^{\tau(\mu)}\in J$, respectively. Now assume that
$p_{i}=(\sigma(\lambda)+\tau(\mu))_{i}<(\sigma(\lambda)+\tau(\mu))_{j}=p_{j}$.
This implies that $\sigma(\lambda)_{i}<\sigma(\lambda)_{j}$ or
$\tau(\mu)_{i}<\tau(\mu)_{j}$.

Suppose the former case holds. The inequality
$\sigma(\lambda)_{i}<\sigma(\lambda)_{j}$ can be written equivalently as
$\lambda_{\sigma^{-1}(i)}<\lambda_{\sigma^{-1}(j)}$. Since $I$ is symmetric
and $\sigma(x^{\lambda})\in I$ we have that $x^{\lambda}\in I$. Since $I$ is
additionally strongly shifted and
$\lambda_{\sigma^{-1}(i)}<\lambda_{\sigma^{-1}(j)}$, one deduces that
$x^{\lambda}x_{\sigma^{-1}(i)}/x_{\sigma^{-1}(j)}\in I$. Finally, applying
$\sigma$ yields
$\sigma\left(x^{\lambda}x_{\sigma^{-1}(i)}/x_{\sigma^{-1}(j)}\right)=\sigma(x^{\lambda})x_{i}/x_{j}\in
I$ and thus $x^{p}x_{i}/x_{j}=\sigma(x^{\lambda})\tau(x^{\mu})\in IJ$. The
latter case is identical, hence omitted. ∎

From 2.3 we deduce that the class of symmetric strongly shifted ideals is
closed under taking powers.

######  Corollary 2.4.

Powers of symmetric strongly shifted ideals are symmetric strongly shifted.

2.2 shows that 2.3 cannot be extended to symmetric shifted ideals. However, we
do not know of an example that would show that 2.4 cannot be extended to
symmetric shifted ideals. Thus we are left with the following:

######  Question 2.5.

Is the class of symmetric shifted ideals closed under taking powers?

We next analyze the behavior of symmetric (strongly) shifted ideals under
taking radicals and saturations. The following description of the square-free
symmetric monomial ideals of $R$ will be crucial.

######  Remark 2.6.

Any square-free symmetric ideal is the ideal generated by all the square-free
monomials of a fixed degree $d\in{\mathbb{N}}$. Such an ideal is referred to
in the literature as the _square-free Veronese ideal_ of degree $d$. It can
also be interpreted as the defining ideal of a monomial star configuration,
introduced in [GHM13] and denoted $I_{n,c}$, since it can be described as

$I_{n,c}=\bigcap_{1\leq i_{1}<\cdots<i_{c}\leq
n}(x_{i_{1}},\ldots,x_{i_{c}}),$ (2.1)

where $c=n-d+1$ is the height of the ideal. The ideals $I_{n,c}$ are symmetric
strongly shifted, so in particular all square-free symmetric ideals are in
fact symmetric strongly shifted.

######  Proposition 2.7.

The radical of a symmetric ideal $I$ is symmetric strongly shifted and is
described as

$\sqrt{I}=\bigcap_{1\leq i_{1}<\cdots<i_{c}\leq
n}(x_{i_{1}},\ldots,x_{i_{c}})=I_{n,c\,},\quad\text{ where }\quad
c=\operatorname{ht}(I).$

###### Proof.

Notice that $\sqrt{I}$ is symmetric since $I$ is. As $\sqrt{I}$ is also
square-free, the conclusion follows from 2.6. ∎

2.14 below shows that the class of symmetric shifted ideals is not closed
under colons or saturations. However, in 2.12 we prove that symmetric strongly
shifted ideals are closed under saturation with respect to any symmetric
monomial ideal. To prove this result, we first observe that taking colons
preserves symmetry.

######  Lemma 2.8.

If $I,J$ are symmetric ideals then $I:J$ is symmetric.

###### Proof.

Let $f\in I\colon J$. Since $I$ and $J$ are $\mathfrak{S}_{n}$-invariant, from
$fJ\subset I$ it follows that
$\sigma(f)J=\sigma(f)\sigma(J)\subseteq\sigma(I)=I$ for all
$\sigma\in\mathfrak{S}_{n}$, so $I\colon J$ is symmetric. ∎

Our next observation is the fact that, unlike arbitrary symmetric shifted
ideals, sssi’s can be characterized combinatorially in terms of the so called
dominance order.

######  Definition 2.9.

Partitions $\lambda,\mu\in P_{n}$ are compared in the dominance order $\lhd$
by setting

$\mu\unlhd\lambda\,\text{ iff }\,\Sigma_{k}(\mu)\leq\Sigma_{k}(\lambda)\text{
for all }1\leq k\leq n.$

######  Remark 2.10.

An $\mathfrak{S}_{n}$-fixed monomial ideal $I$ is strongly shifted if and only
if, for every $\lambda,\mu\in\mathrm{P}_{n}$ with $|\lambda|=|\mu|$,
$\lambda\in P(I)$ and $\mu\unlhd\lambda$ imply $\mu\in\mathrm{P}(I)$. This is
because for $\lambda,\mu$ satisfying $|\lambda|=|\mu|$, the inequality
$\mu\unlhd\lambda$ is equivalent to $x^{\mu}$ being obtained from
$x^{\lambda}$ by a sequence of Borel moves; see [DN99, Lemma 1.3] for a proof.

We can now describe the saturation of a symmetric strongly shifted ideal with
respect to a square-free symmetric monomial ideal. Recall that the saturation
operation on ideals is defined by $I:J^{\infty}=\bigcup_{i\geq 0}I:J^{i}$ and
has the property that $I:J^{\infty}=I:J^{N}$ for $N\gg 0$.

######  Proposition 2.11.

Let $I$ be a sssi and let $c$ be a natural number so that $1\leq c\leq n$. For
$\lambda\in P_{n}$ define the truncated partition
$\lambda_{<c}=(\lambda_{1},\ldots,\lambda_{c-1})\in P_{c-1}$. Then we have

  1. (1)

$I:I_{n,c}^{\infty}=(\sigma(x^{\mu})\mid\sigma\in\mathfrak{S}_{n},\mu_{<c}=\lambda_{<c}\text{
for some }\lambda\in P(I))$;

  2. (2)

$I:I_{n,c}^{\infty}$ is symmetric strongly shifted.

###### Proof.

Denote
$J=(\sigma(x^{\mu})\mid\sigma\in\mathfrak{S}_{n},\mu_{<c}=\lambda_{<c}\text{
for some }\lambda\in P(I))$.

(1) Since 2.8 yields that $I:I_{n,c}^{\infty}$ is a symmetric monomial ideal,
it suffices to show

$P(I:I_{n,c}^{\infty})=P(J)=\\{\mu\in P_{n}\mid\mu_{<c}=\lambda_{<c}\text{ for
some }\lambda\in P(I)\\}.$

If $x^{\mu}\in I:I_{n,c}^{\infty}$, since $x_{c}x_{c+1}\cdots x_{n}\in
I_{n,c}$, we have $x^{\mu}(x_{c}x_{c+1}\cdots x_{n})^{N}=x^{\lambda}$ for some
$\lambda\in P(I)$ and $N\gg 0$. This implies that $\mu_{<c}=\lambda_{<c}$ and
establishes the containment $I:I_{n,c}^{\infty}\subseteq J$.

Conversely, take $\mu\in P(J)$ and $\lambda\in P(I)$ with the property that
$\mu_{<c}=\lambda_{<c}$. Take also $N\geq\lambda_{n}$. Then we have
$x^{\lambda}\mid x^{\mu}(x_{c}x_{c+1}\cdots x_{n})^{N}$ and therefore
$x^{\mu}(x_{c}x_{c+1}\cdots x_{n})^{N}\in I$; thus we conclude
$\mu+(0^{c-1},N^{n-c+1})\in P(I)$. Now consider an arbitrary monomial
$x^{\alpha}\in I_{n,c}^{N}$. We aim to show that $x^{\mu}x^{\alpha}\in I$ and
equivalently that $\operatorname{part}(x^{\mu+\alpha})\in P(I)$. To do so it
suffices to show
$\operatorname{part}(x^{\mu+\alpha})\unlhd\mu+(0^{c-1},N^{n-c+1})$ and use the
fact that $I$ is symmetric strongly shifted. Set
$p=\operatorname{part}(x^{\mu+\alpha})$ and observe that $x^{\alpha}\in
I_{n,c}^{N}$ implies $\alpha_{i}\leq N$ for each $1\leq i\leq n$, so that $p$
satisfies inequalities $p_{i}\leq\mu_{i}+N$ for each $1\leq i\leq n$.
Consequently, we have

$\sum_{i=k}^{n}p\leq\sum_{i=k}^{n}(\mu_{i}+N)=\sum_{i=k}^{n}\left(\mu+(0^{c-1},N^{n-c+1})\right)_{i}\text{
for }k\geq c.$

As for the case $k<c$, note that $|\alpha|=N(n-c+1)$, which yields

$\sum_{i=k}^{n}p_{i}=\sum_{i=k}^{n}\operatorname{part}(x^{\mu+\alpha})_{i}\leq\left(\sum_{i=k}^{n}\mu_{i}\right)+|\alpha|=\sum_{i=k}^{n}\left(\mu+(0^{c-1},N^{n-c+1})\right)_{i}\text{
for }k<c.$

As discussed above, this yields $p\in P(I)$, concluding the proof of the
containment $J\subseteq I:I_{n,c}^{\infty}$.

For (2) we will prove the equivalent assertion that $J$ is a sssi. Towards
this goal, we identify the minimal generators of $J$: these are given by the
partitions

$\Lambda(J)=\\{\mu\in P_{n}\mid\mu_{<c}=\lambda_{<c}\text{ for some
}\lambda\in\Lambda(I),\mu_{i}=\mu_{c-1}\text{ for }i\geq c\ \\}.$

Let $\mu\in\Lambda(J)$ and consider $1\leq i<j\leq n$ so that
$\mu_{i}<\mu_{j}$. By the description of $\Lambda(J)$ this yields the
inequality $1\leq i<c-1$. There exists $\lambda\in P(I)$ with
$\mu_{<c}=\lambda_{<c}$. Set $\mu^{\prime}=\mu+e_{i}-e_{j}$ and
$\lambda^{\prime}=\lambda+e_{i}-e_{j}$. Since $\mu_{<c}=\lambda_{<c}$ and
$1\leq i<c-1$ it follows that $\lambda_{i}=\mu_{i}<\mu_{j}\leq\lambda_{j}$ as
either $j<c$ and $\mu_{j}=\lambda_{j}$ or $j\geq c$ and thus
$\mu_{j}=\mu_{c-1}=\lambda_{c-1}\leq\lambda_{j}$. Since $I$ is symmetric
strongly shifted we have $\lambda^{\prime}\in P(I)$ and
$\mu^{\prime}_{<c}=\lambda^{\prime}_{<c}$, so we obtain $\mu^{\prime}\in
P(J)$, as desired. ∎

######  Corollary 2.12.

Let $I$ be a sssi and let $J$ be a symmetric monomial ideal with
$\operatorname{ht}(J)=c$. Then the ideal $I:J^{\infty}$ is symmetric strongly
shifted and is described as $I:J^{\infty}=I:I_{n,c}^{\infty}$.

###### Proof.

First, notice that $I:J^{\infty}=I:\sqrt{J}^{\,\infty}$, since every ideal
contains a power of its radical. Moreover, by 2.7 it follows that
$\sqrt{J}=I_{n,c}$. Therefore, $I:J^{\infty}=I:I_{n,c}^{\infty}$ is symmetric
strongly shifted by 2.11(2). ∎

Thanks to 2.11, we can understand the symbolic powers of a sssi. In the
literature two notions of symbolic powers make an appearance. They are defined
in terms of the associated primes of $I$, the set of which we denote
$\operatorname{Ass}(I)$ and the minimal primes of $I$, the set of which we
denote ${\operatorname{Min}}(I)$, respectively. The symbolic powers of an
ideal $I$ are defined by some authors as

$I^{(m)_{{\operatorname{Min}}}}=\bigcap_{P\in{\operatorname{Min}}(I)}(I^{m}R_{P}\cap
R),$ (2.2)

and by others as

$I^{(m)_{\operatorname{Ass}}}=\bigcap_{P\in\operatorname{Ass}(I)}(I^{m}R_{P}\cap
R).$ (2.3)

The two definitions agree for ideals without embedded primes, in which case we
will denote symbolic powers simply as $I^{(m)}$. The second definition has the
advantage that it satisfies $I^{(1)_{\operatorname{Ass}}}=I$ for all ideals
$I$, while the first definition is more easily handled and more relevant in
geometric contexts. Both notions of symbolic powers can be computed by
saturation, a point that proves relevant in 2.13 below.

In [BdAG+20, Theorem 4.3] it is proven that the symbolic powers of a square-
free sssi are symmetric strongly shifted. In the next proposition we show that
more generally the symbolic powers of any sssi are symmetric strongly shifted.

######  Proposition 2.13.

The symbolic powers of a symmetric strongly shifted ideal are symmetric
strongly shifted.

###### Proof.

Let $I$ be a sssi. Let ${\operatorname{Min}}(I)$ denote the set of minimal
associated prime ideals of $I$ and $\operatorname{Ass}^{*}(I)$ the union of
the associated prime ideals of $I^{n}$ for all $n\geq 0$. It is known that
$\operatorname{Ass}^{*}(I)$ is a finite set [Bro79b].

The $m$-th symbolic powers $I^{(m)_{{\operatorname{Min}}}}$ and
$I^{(m)_{\operatorname{Ass}}}$ of $I$ can both be described as saturations:

$\displaystyle
I^{(m)_{{\operatorname{Min}}}}=\bigcap_{P\in{\operatorname{Min}}(I)}(I^{m}R_{P}\cap
R)=I^{m}:J^{\infty}$ $\displaystyle\quad\text{ for
}J=\bigcap_{P\in\operatorname{Ass}^{*}(I)\setminus{\operatorname{Min}}(I)}P$
(2.4a) $\displaystyle
I^{(m)_{\operatorname{Ass}}}=\bigcap_{P\in\operatorname{Ass}(I)}(I^{m}R_{P}\cap
R)=I^{m}:J^{\infty}$ $\displaystyle\quad\text{ for
}J=\bigcap_{P\in\operatorname{Ass}^{*}(I)\setminus\operatorname{Ass}(I)_{\subseteq}}P$
(2.4b)

Here we denote

$\operatorname{Ass}(I)_{\subseteq}=\\{P:P\subseteq P^{\prime}\text{ for some
}P^{\prime}\in\operatorname{Ass}(I)\\}.$

Observe that both ideals termed $J$ above are square-free. We now show they
are symmetric. Indeed, consider the sets
$S=\operatorname{Ass}(I),{\operatorname{Min}}(I),\operatorname{Ass}(I^{m}),\operatorname{Ass}^{*}(I)$.
Each of them are closed under the action of the symmetric group, that is,
$P\in S$ if and only if $\sigma(P)\in S$ for all $\sigma\in\mathfrak{S}_{n}$.
It follows that the set differences
$\operatorname{Ass}^{*}(I)\setminus{\operatorname{Min}}(I)$ and
$\operatorname{Ass}^{*}(I)\setminus\operatorname{Ass}(I)_{\subseteq}$ are also
closed under the action of $\mathfrak{S}_{n}$ which implies that in both cases
$J$ is symmetric. Since $J$ is symmetric and square-free, 2.6 yields that $J$
is a square-free Veronese ideal.

To finish the proof, it then suffices to invoke 2.11(2). ∎

We emphasize that 2.11 and 2.13 do not extend to symmetric shifted ideals
which are not strongly shifted, as shown by the following example.

######  Example 2.14.

Let $I$ be the symmetric shifted ideal with $\Lambda(I)=\\{(1,2,2),(0,2,3)\\}$
and let ${\mathfrak{m}}=(x_{1},x_{2},x_{3})$. Note that ${\mathfrak{m}}$ is
symmetric strongly shifted, but $I$ is not, since $(0,2,3)\in P(I)$ but
$(1,1,3)\notin P(I)$. Then the ideal
$I^{(1)_{{\operatorname{Min}}}}=I:{\mathfrak{m}}=I:{\mathfrak{m}}^{\infty}=(x_{1}^{2}x_{2}^{2},\,x_{1}^{2}x_{3}^{2},\,x_{2}^{2}x_{3}^{2})$
is not symmetric shifted and nor is the second symbolic power
$I^{(2)_{{\operatorname{Min}}}}=(x_{1}^{2}x_{2}^{2}x_{3}^{2},\,x_{1}^{4}x_{2}^{4},\,x_{1}^{4}x_{3}^{4},\,x_{2}^{4}x_{3}^{4})$
in the sense of (2.2).

We do not currently know of any ssi $I$ admitting a symbolic power
$I^{(m)_{\operatorname{Ass}}}$ which is not symmetric shifted.

Our last result in this section concerns the integral closure of symmetric
shifted ideals.

######  Proposition 2.15.

The integral closure $\overline{I}$ of a symmetric (strongly) shifted ideal
$I$ is also symmetric (strongly) shifted.

###### Proof.

First note that the integral closure of a (not necessarily monomial) symmetric
ideal is symmetric. Indeed if $f$ is a solution of a monic polynomial

$x^{n}+r_{1}x^{n-1}+\cdots+r_{n}=0$

with $r_{i}\in I^{i}$ and if $\sigma\in\mathfrak{S}_{n}$ then $\sigma(f)$ is a
solution of the equation

$x^{n}+\sigma(r_{1})x^{n-1}+\cdots+\sigma(r_{n})=0$

in which each $\sigma(r_{i})\in\sigma(I^{i})=I^{i}$ because $I^{i}$ is stable
under the action of $\mathfrak{S}_{n}$.

Second, recall that since $I$ is a monomial ideal, $x^{\alpha}\in\overline{I}$
if and only if it satisfies an equation of the form

$x^{s\alpha}=\prod_{k=1}^{s}x^{\beta^{(k)}}\text{ for some }s\in{\mathbb{N}},$
(2.5)

where $x^{\beta^{(k)}}\in I$ for each $k$; see [HS06, p. 9]. Now assume that
$\alpha\in P(\overline{I})$ and take $i<j$ so that $\alpha_{i}<\alpha_{j}$ (if
$I$ is shifted, assume additionally that $j=n$). Set
$S=\left\\{k\in\\{1,\ldots,s\\}\mid\beta^{(k)}_{i}<\beta^{(k)}_{j}\right\\}$
and $\alpha^{\prime}=\alpha+e_{i}-e_{j}$. From (2.5) it follows that

$\sum_{k\in S}(\beta^{(k)}_{j}-\beta^{(k)}_{i})\geq
s(\alpha_{j}-\alpha_{i})\geq s.$

Therefore there exist nonnegative integers $c_{k}$, one for each $k\in S$, so
that $\sum_{k\in S}c_{k}=s$ and $c_{k}\leq\beta^{(k)}_{j}-\beta^{(k)}_{i}$ for
each $k\in S$. Define $\alpha^{\prime}:=\alpha+e_{i}-e_{j}$ and

$\gamma^{(k)}=\begin{cases}\beta^{(k)}+c_{k}(e_{i}-e_{j})&\text{ if }k\in S\\\
\beta^{(k)}&\text{ otherwise}\end{cases}$

so that
$\sum_{k=1}^{n}\gamma^{(k)}=\sum_{k=1}^{n}\beta^{(k)}+s(e_{i}-e_{j})=s\alpha^{\prime}$.
Then $\gamma^{(k)}\in I$ by the (strongly) shifted property and the relation

$x^{s\alpha^{\prime}}=\prod_{k=1}^{s}x^{\gamma^{(k)}}$

shows that $x^{\alpha^{\prime}}\in\overline{I}$, concluding the proof. ∎

##  3\. Partition Borel generators and combinatorial structure

The proofs of 2.11 and 2.13, together with 2.14, suggest that the algebraic
differences between sssi’s and symmetric shifted ideals which are not strongly
shifted might be due to the combinatorial characterization of sssi’s in terms
of the dominance order (2.10).

In this section, we further investigate the role of the dominance order in
determining the algebraic structure of symmetric strongly shifted ideals, by
introducing the key notion of _partition Borel generators_ of a sssi (see
3.4). As this definition is inspired by the notion of Borel generators of a
strongly stable ideal, we begin by recalling relevant background information
on strongly stable ideals which we will need throughout this section.

######  Definition 3.1.

A monomial ideal $I\subset R$ is said to be a strongly stable ideal if, for
every $\alpha\in{\mathbb{N}}^{n}$ with $x^{\alpha}\in I$ and for every pair
$i<j$ so that $\alpha_{j}\neq 0$ one has $x^{\alpha}x_{i}/x_{j}\in I$.

A monomial ideal $I\subset R$ is said to be a stable ideal if, for every
$\alpha\in{\mathbb{N}}^{n}$ with $x^{\alpha}\in I$, setting
$\max(\alpha)=\max\\{j\colon\alpha_{j}\neq 0\\}$, for every $i<\max(\alpha)$
one has $x^{\alpha}x_{i}/x_{\max(\alpha)}\in I$.

The monomials $x^{\alpha}(x_{i}/x_{j})$ and
$x^{\alpha}(x_{i}/x_{\max(\alpha)})$ occurring above are referred to as being
obtained from $x^{\alpha}$ via a Borel move.

Despite the formal analogy between 1.1 and 3.1, symmetric strongly shifted
ideals are almost never strongly stable.

######  Remark 3.2.

The only $\mathfrak{S}_{n}$-fixed ideals which are also strongly stable are
the powers of the homogeneous maximal ideal. Indeed, if $I$ is strongly stable
in $K[x_{1},\ldots,x_{n}]$, then it must contain $x_{1}^{d}$, where
$d\in{\mathbb{N}}$ is the least degree of a nonzero element of $I$. Hence by
symmetry $I$ must contain $x_{n}^{d}$. Now the strongly stable property yields
$(x_{1},\ldots,x_{n})^{d}\subseteq I$, which must be an equality since $d$ was
the initial degree of $I$.

Given a set of monomials $M$, the smallest strongly stable monomial ideal
which contains $M$ is called the Borel ideal generated by $M$ and is denoted
by $\operatorname{Borel}(M)$. The following well-known result conveniently
describes the set of monomials obtained from $x^{\alpha}$ by performing Borel
moves, that is, the monomials in the Borel ideal
$\operatorname{Borel}(\\{x^{\alpha}\\})$.

######  Remark 3.3 ([DN99, Lemma 1.3]).

For each $\alpha=(\alpha_{1},\dots,\alpha_{n})\in{\mathbb{N}}^{n}$, define
$\Sigma_{k}(\alpha)=\alpha_{k}+\cdots+\alpha_{n}$. Monomials $x^{\alpha}$ and
$x^{\beta}$ with $|\alpha|=|\beta|$ satisfy
$x^{\alpha}\in\operatorname{Borel}(\\{x^{\beta}\\})$ if and only if
$\Sigma_{k}(\alpha)\leq\Sigma_{k}(\beta)$ for all $1\leq k\leq n$. We denote
this condition by $x^{\alpha}\prec_{B}x^{\beta}$. This defines a partial order
on the set of $n$-tuples of nonnegative integers which restricts to the
dominance order on $P_{n}$.

The relationship between strongly stable ideals and symmetric strongly shifted
ideals is best understood in terms of the following notion, which can be
interpreted as a symmetric analogue to the notion of Borel generators of a
strongly stable ideal.

######  Definition 3.4.

Let $B\subseteq P_{n}$ be a set of partitions and set
$x^{B}=\\{x^{\lambda}:\lambda\in B\\}$.

The symmetric shifted ideal generated by $B$, denoted $\operatorname{Ss}(B)$,
is defined to be the smallest symmetric shifted ideal which contains $x^{B}$.
Similarly, define the symmetric strongly shifted ideal generated by $B$,
denoted $\operatorname{Sss}(B)$, to be the smallest symmetric strongly shifted
ideal which contains $x^{B}$.

Conversely, for a symmetric strongly shifted (resp. shifted) ideal $I$ we
define the set of partition Borel generators of $I$, denoted $B(I)$, to be the
smallest $B\subseteq P_{n}$ so that $I=\operatorname{Sss}(B)$ (resp.
$I=\operatorname{Ss}(B)$).

In analogy with 3.3 for strongly stable ideals, the partition Borel generators
of a symmetric strongly shifted ideal coincide with the maximal elements in
each degree of $\mathrm{\Lambda}(I)$ with respect to the dominance order. More
precisely, if for a set $C\subseteq P_{n}$ $\max_{\unlhd}\\{C\\}$ denotes the
set of maximal elements of $C$ in dominance order, we have the following
characterization.

######  Proposition 3.5.

  1. (1)

Let $B\subseteq P_{n}$ be a set of partitions, then

$\operatorname{Sss}(B)=\left(\\{\sigma(x^{\mu}):\sigma\in\mathfrak{S}_{n},\
\mu\in P_{n},\ \exists\lambda\in B\text{ such that }|\lambda|=|\mu|\text{ and
}\ \mu\unlhd\lambda\\}\right).$

  2. (2)

Let $I$ be a symmetric strongly shifted ideal. Then,
$B(I)=\max_{\unlhd}\\{\Lambda(I)\\}$, that is,

$B(I)=\\{\lambda\in\mathrm{\Lambda}(I):\mu\in\Lambda(I)\text{ with
}|\lambda|=|\mu|,\ \lambda\unlhd\mu\text{ implies }\lambda=\mu\\}.$

are the partition Borel generators of $I$.

In particular, for any symmetric strongly shifted ideal $I$ one has
$I=\operatorname{Sss}(B(I))$.

###### Proof.

(1) Denote $I=\left(\\{\sigma(x^{\mu}):\sigma\in\mathfrak{S}_{n},\ \mu\in
P_{n},\ \exists\lambda\in B\text{ such that }|\lambda|=|\mu|\text{ and }\
\mu\unlhd\lambda\\}\right)$. It is clear that $I$ is symmetric and
$x^{B}\subseteq I$. To show that $I$ is strongly shifted, notice that
$\mathrm{\Lambda}(I)\subseteq S$, where

$S=\\{\mu:\mu\in P_{n},\ \exists\lambda\in B\text{ such that
}|\lambda|=|\mu|\text{ and }\ \mu\unlhd\lambda\\}\subset I.$

and $S$ is closed under Borel moves. By 2.10 this ensures that $I$ is a sssi.
Finally, $I$ is indeed the smallest symmetric strongly shifted ideal
containing $x^{B}$, since any sssi $I^{\prime}$ with $I^{\prime}\supseteq
x^{B}$ must contain $x^{S}$ by 2.10, hence it must contain $I$ by symmetry.

(2) Set $B=\max_{\unlhd}\\{\Lambda(I)\\}$. Since $B\subseteq\Lambda(I)$, it is
clear that $I\supseteq x^{B}$, whence $I\supseteq\operatorname{Sss}(B)$ since
$I$ is an sssi. Moreover, by the definition of $B$, for each
$\mu\in\Lambda(I)$ we have that $\mu\unlhd\lambda$ for some $\lambda\in B(I)$,
hence $I\subseteq\operatorname{Sss}(B)$ by 2.10, so equality holds. That $B$
is the smallest set of Borel generators of $I$ follows by noting that
$\lambda\in B(I)$ and $B^{\prime}\subset B\setminus\\{\lambda\\}$ yields
$\operatorname{Sss}(B^{\prime})\subset I\setminus\\{x^{\lambda}\\}$. Thus we
conclude that $B=B(I)$. ∎

For a set of partitions $B\subseteq P_{n}$, let $\operatorname{Borel}(x^{B})$
be the strongly stable ideal generated by $x^{B}$. The next result clarifies
how the ideal $\operatorname{Sss}(B)$ relates with
$\operatorname{Borel}(x^{B})$. It provides a containment between these ideals,
which becomes an equality when one restricts to monomials whose exponents are
partitions.

######  Proposition 3.6.

Let $B\subseteq P_{n}$ be a set of partitions. Then,
$\operatorname{Sss}(B)\subseteq\operatorname{Borel}(x^{B})$ and

$P\left(\operatorname{Sss}(B)\right)=\\{\lambda\in
P_{n}:x^{\lambda}\in\operatorname{Borel}(x^{B})\\}.$ (3.1)

###### Proof.

Let $\sigma(x^{\lambda})\in\operatorname{Sss}(B)$ for some
$\sigma\in\mathfrak{S}_{n},\lambda\in P_{n}$. Since $\lambda$ is a partition,
we have
$\displaystyle{\Sigma_{k}(\sigma(\lambda))=\sum_{i=k}^{n}\lambda_{\sigma^{-1}(i)}\leq\sum_{i=k}^{n}\lambda_{i}=\Sigma_{k}(\lambda)}$,
which means that $\sigma(x^{\lambda})\prec_{B}x^{\lambda}$ by 3.3.
Furthermore, since $x^{\lambda}\in\operatorname{Sss}(B)$, 2.10 together with
3.3 then imply that
$\sigma(x^{\lambda})\in\operatorname{Borel}(\\{x^{\lambda}\\})\subseteq\operatorname{Borel}(x^{B})$
for each $\sigma\in\mathfrak{S}_{n}$, which justifies the ideal containment.

The first part of the proof yields the containment
$P\left(\operatorname{Sss}(B)\right)\subseteq\\{\lambda\in
P_{n}:x^{\lambda}\in\operatorname{Borel}(x^{B})\\}$.

For the converse, let $\lambda\in P_{n}$ with
$x^{\lambda}\in\operatorname{Borel}(x^{B})$. Then there exists a minimal
generator of $\operatorname{Borel}(x^{B})$, $x^{\alpha}$, so that
$x^{\alpha}|x^{\lambda}$ and $x^{\alpha}\prec_{B}x^{\beta}$ for some $\beta\in
B$ with $|\alpha|=|\beta|$. This yields $\lambda_{i}\geq\alpha_{i}$ for all
$1\leq i\leq n$ and $\sum_{i=k}^{n}\alpha_{i}\leq\sum_{i=k}^{n}\beta_{i}$ for
all $1\leq k\leq n$. Note that this implies

$\sum_{i=1}^{k}\lambda_{i}\geq\sum_{i=1}^{k}\alpha_{i}\geq\sum_{i=1}^{k}\beta_{i}\text{
for each }1\leq k\leq n.$ (3.2)

If $|\lambda|=|\alpha|$ then $\lambda=\alpha$, which yields
$\lambda\unlhd\beta$ and thus $x^{\lambda}\in\operatorname{Sss}(B)$.

Assume now that $|\lambda|>|\alpha|=|\beta|$ and set
$q=\max\\{k:\sum_{i=1}^{k}\lambda_{i}\leq|\beta|\\}$ and
$t=|\beta|-\sum_{i=1}^{q}\lambda_{i}$. By the assumption we have $q<n$.
Moreover, since $\sum_{i=1}^{q+1}\lambda_{i}>|\beta|$ we deduce that
$\lambda_{q+1}>t$. Now consider the vector

$\gamma=(\beta_{1},\ldots,\beta_{q},\beta_{q+1}+\lambda_{q+1}-t,\beta_{q+2}+\lambda_{q+2},\ldots,\beta_{n}+\lambda_{n}).$

Note that $\gamma$ is a partition since $\beta,\lambda\in P_{n}$ and since
$\lambda_{q+1}-t>0$. Moreover, observe that $x^{\beta}\mid x^{\gamma}$, which
yields $x^{\gamma}\in\operatorname{Sss}(B)$ since
$x^{\beta}\in\operatorname{Sss}(B)$. The sum of entries of $\gamma$ is

$|\gamma|=\sum_{i=1}^{n}\beta_{i}+\sum_{i=q+1}^{n}\lambda_{i}-t=|\beta|+\sum_{i=q+1}^{n}\lambda_{i}-|\beta|+\sum_{i=1}^{q}\lambda_{i}=|\lambda|.$

We claim that $\lambda\unlhd\gamma$. To see this, compute

$\sum_{i=k}^{n}\gamma_{i}=\begin{cases}\sum_{i=k}^{n}\beta_{i}+\sum_{i=k}^{n}\lambda_{i}&\text{
if }k>q+1\\\ \sum_{i=k}^{n}\beta_{i}+\sum_{i=q+1}^{n}\lambda_{i}-t&\text{ if
}k\leq q+1.\end{cases}$

If $k>q+1$ it is clear that
$\sum_{i=k}^{n}\gamma_{i}\geq\sum_{i=k}^{n}\lambda_{i}$. If $k\leq q+1$, then
inequality (3.2) yields

$\sum_{i=k}^{n}\gamma_{i}=\sum_{i=k}^{n}\beta_{i}+\sum_{i=q+1}^{n}\lambda_{i}-|\beta|+\sum_{i=1}^{q}\lambda_{i}=|\lambda|-\sum_{i=1}^{k-1}\beta_{i}\geq|\lambda|-\sum_{i=1}^{k-1}\lambda_{i}=\sum_{i=k}^{n}\lambda_{i}.$

Since we have shown $\lambda\unlhd\gamma$ and
$x^{\gamma}\in\operatorname{Sss}(B)$, we deduce that
$x^{\lambda}\in\operatorname{Sss}(B)$ by 2.10 and thus $\lambda\in
P(\operatorname{Sss}(B))$, as desired.

∎

The conclusion of 3.6 does not hold if one replaces $\operatorname{Sss}(B)$
with $\operatorname{Ss}(B)$ and $\operatorname{Borel}(x^{B})$ with the
smallest stable ideal containing $x^{B}$, denoted $\operatorname{St}(x^{B})$.

######  Example 3.7.

Let $B=\\{(0,0,1,1)\\}$,
$J=\operatorname{St}(x^{B})=(x_{3}x_{4},x_{3}^{2},x_{2}x_{3},x_{1}x_{3})$ and
let $I=\operatorname{Ss}(\\{\lambda\in P_{n}:x^{\lambda}\in J\\})$. Observe
that $I\nsubseteq J$, since $x_{1}x_{2}\in I\setminus J$.

Building upon the relationship between $\operatorname{Sss}(B)$ and
$\operatorname{Borel}(x^{B})$, 3.8 shows that an ideal $I$ is an sssi if and
only if it can be obtained from a strongly stable ideal $J$ by symmetrization,
i.e., $I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(J)$.

######  Theorem 3.8.

  1. (1)

Let $J$ be a strongly stable ideal. Then the ideal

$I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(J)$ (3.3)

is symmetric strongly shifted, with $P(I)=\\{\lambda\in P_{n}:x^{\lambda}\in
J\\}$.

  2. (2)

Conversely, every symmetric strongly shifted ideal $I$ has the form in (3.3)
for some strongly stable ideal $J$. In detail, let $B=B(I)\subset P_{n}$.
Then, the ideal $J=\operatorname{Borel}(x^{B})$ satisfies

$I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(J).$

Moreover, $J$ is the smallest strongly stable ideal with this property.

###### Proof.

(1) It is clear that $I$ is symmetric and that $P(I)\subseteq\\{\lambda\in
P_{n}:x^{\lambda}\in J\\}$. Moreover, since $J$ is strongly stable, it follows
from 3.6 that for each $\sigma\in\mathfrak{S}_{n}$,
$\sigma(x^{\lambda})\in\operatorname{Borel}(\\{x^{\lambda}\\})\subseteq J$ and
consequently $x^{\lambda}\in I$ whenever $\lambda\in P_{n}$ and
$x^{\lambda}\in J$. Therefore we have $\\{\lambda\in P_{n}:x^{\lambda}\in
J\\}\subseteq P(I)$ and thus equality holds. Since $J$ is closed under Borel
moves, it then follows that $P(I)$ is also closed under Borel moves, i.e. $I$
is a sssi.

(2) From 3.6 we know that
$\operatorname{Sss}(B)\subseteq\operatorname{Borel}(x^{B})$. Applying
permutations to this containment, we obtain that
$\operatorname{Sss}(B)=\sigma(\operatorname{Sss}(B))\subseteq\sigma(\operatorname{Borel}(x^{B}))$
for every $\sigma\in\mathfrak{S}_{n}$. Therefore,
$\operatorname{Sss}(B)\subseteq\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(\operatorname{Borel}(x^{B}))$.
For the opposite containment, notice that
$x^{\alpha}\in\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(\operatorname{Borel}(x^{B}))$
implies that $\sigma(x^{\alpha})\in\operatorname{Borel}(x^{B})$ for all
$\sigma\in\mathfrak{S}_{n}$ and in particular that
$x^{\operatorname{part}(x^{\alpha})}\in\operatorname{Borel}(x^{B})$. Setting
$\lambda=\operatorname{part}(x^{\alpha})$ we deduce by 3.6 that $\lambda\in
P\left(\operatorname{Sss}(B)\right)$. It follows by the
$\mathfrak{S}_{n}$-invariance of $\operatorname{Sss}(B)$ that also
$x^{\alpha}\in\operatorname{Sss}(B)$, thus finishing the proof. ∎

In light of 3.8, it is natural to expect that algebraic properties which are
well-behaved under taking intersections are preserved under symmetrization.
4.2 and 5.5 will present instances when this is indeed the case.

Analogously to [FMS11, Propositions 2.15, 2.16 and 2.17] for strongly stable
ideals, our next goal is to describe sums, intersections and products of
symmetric strongly shifted ideals in terms of partition Borel generators. To
do so we must consider the lattice structure of $P_{n}$.

The following lemma is inspired by a similar description in [Bry73] for the
lattice structure on the set of partitions of fixed sum $P(d)=\\{\lambda$ is a
partition and $|\lambda|=d\\}$. We include the proof here for lack of a
specific reference which treats the case of $P_{n}$, the set of partitions
with $n$ parts.

######  Lemma 3.9.

For every $n$, the set $P_{n}$ forms a lattice with respect to the dominance
order.

###### Proof.

Notice that there is a one-to-one correspondence between partitions
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in P_{n}$ and non-decreasing
sequences $\hat{\lambda}\in\mathbb{N}^{n}$ so that
$\hat{\lambda}_{k}+\hat{\lambda}_{k-2}\leq 2\hat{\lambda}_{k-1}$ for every
$k$. Indeed, given $\lambda=(\lambda_{1},\ldots,\lambda_{n})\in P_{n}$ one
defines
$\hat{\lambda}=(\sum_{n}\lambda,\sum_{n-1}\lambda,\ldots,\sum_{1}\lambda)$,
where for every $1\leq k\leq n$ we denote
$\sum_{k}\lambda=\lambda_{k}+\ldots+\lambda_{n}$. Conversely, given a vector
$\hat{\lambda}\in\mathbb{N}^{n}$, one defines a partition
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in P_{n}$ by setting
$\lambda_{n}=\hat{\lambda}_{1}$ and
$\lambda_{k}=\hat{\lambda}_{n-k+1}-\hat{\lambda}_{n-k}$ for $1\leq k\leq n-1$.
Under this correspondence, one has that $\eta\unlhd\lambda$ if and only if
$\hat{\eta}_{k}\leq\hat{\lambda}_{k}$ for each $k$.

Now, let $\lambda,\mu\in P_{n}$. We need to prove that $\lambda$ and $\mu$
have a meet and a join in the dominance order. For each $1\leq k\leq n$, let
$v_{k}=\min\\{\hat{\lambda}_{k},\hat{\mu}_{k}\\}$ and let
$v=(v_{1},\ldots,v_{n})$. Observe that for every $k$, $v_{k}+v_{k-2}\leq
2v_{k-1}$, so there exists a partition $\eta\in P_{n}$ corresponding to $v$
under the identification above. Moreover, $v=\hat{\lambda}\wedge\hat{\mu}$
with respect to the componentwise order, hence $\eta=\lambda\wedge\mu$ with
respect to the dominance order. To prove that $\lambda$ and $\mu$ have a meet,
recall that for partitions $\lambda,\eta\in P_{n}$ one has that
$\lambda\unlhd\eta$ if and only if $\eta^{T}\unlhd\lambda^{T}$, where
$\eta^{T}$ and $\lambda^{T}$ denote the partitions corresponding to the
transpose of the Young diagrams of $\eta$ and $\lambda$ respectively.
Therefore, the join of $\lambda$ and $\mu$ is
$\lambda\vee\mu=(\lambda^{T}\wedge\mu^{T})^{T}$. ∎

In the next two propositions, $\max_{\unlhd}\\{C\\}$ denotes the set of
maximal elements of a set $C\subset P_{n}$ in dominance order.

######  Proposition 3.10.

Let $A,B\subseteq P_{n}$. Then

  1. (1)

$\operatorname{Sss}(A)+\operatorname{Sss}(B)=\operatorname{Sss}(A\cup B)$ and

  2. (2)

$\operatorname{Sss}(A)\cap\operatorname{Sss}(B)=\operatorname{Sss}(A\wedge
B)$, where $A\wedge B=\\{\lambda\wedge\mu:\lambda\in A,\mu\in B\\}$ and
$\lambda\wedge\mu$ denotes the meet of $\lambda$ and $\mu$ in the dominance
order.

For any two symmetric strongly shifted ideals $I$ and $J$ one has

$B(I+J)=\max_{\unlhd}\\{B(I)\cup B(J)\\}\text{ and }B(I\cap
J)=\max_{\unlhd}\\{B(I)\wedge B(J)\\}.$

###### Proof.

For (1), it is clear that
$x^{\lambda}\in\operatorname{Sss}(A)+\operatorname{Sss}(B)$ whenever
$\lambda\in A\cup B$. Hence, since symmetric strongly shifted ideals are
closed under sums by 2.1, we deduce that $\operatorname{Sss}(A\cup
B)\subset\operatorname{Sss}(A)+\operatorname{Sss}(B)$. Conversely, let $p\in
P_{n}$ be so that $x^{p}\in\operatorname{Sss}(A)+\operatorname{Sss}(B)$. Since
$x^{p}$ is a monomial, we must then have $x^{p}\in\operatorname{Sss}(A)$ or
$x^{p}\in\operatorname{Sss}(B)$. In either case there exists a partition
$\lambda\in A\cup B$ with $p\unlhd\lambda$. This implies that
$x^{p}\in\operatorname{Sss}(A\cup B)$.

For (2), notice that
$x^{\lambda\wedge\mu}\in\operatorname{Sss}(A)\cap\operatorname{Sss}(B)$
whenever $\lambda\in A$ and $\mu\in B$, since $\lambda\wedge\mu\unlhd\lambda$
and $\lambda\wedge\mu\unlhd\mu$ and since
$\operatorname{Sss}(A)\cap\operatorname{Sss}(B)$ is symmetric strongly shifted
by 2.1. We deduce that $\operatorname{Sss}(A\wedge
B)\subseteq\operatorname{Sss}(A)\cap\operatorname{Sss}(B)$. Conversely, let
$p\in P_{n}$ be so that
$x^{p}\in\operatorname{Sss}(A)\cap\operatorname{Sss}(B)$. Then, there exist
partitions $\lambda\in A,\mu\in B$ so that $p\unlhd\lambda$ and $p\unlhd\mu$.
Since $\lambda$ and $\mu$ have a meet by 3.9, it then follows that
$p\unlhd\lambda\wedge\mu$, so $x^{p}\in\operatorname{Sss}(A\wedge B)$.

The remaining statement follows from the other two, since for any sssi’s $I$
and $J$

$\operatorname{Sss}(B(I)\cup
B(J))=\operatorname{Sss}(B(I))+\operatorname{Sss}(B(J))=I+J$

and

$\operatorname{Sss}(B(I)\wedge
B(J))=\operatorname{Sss}(B(I))\cap\operatorname{Sss}(B(J))=I\cap J.$

This implies that $\max_{\unlhd}\\{B(I)\cup
B(J)\\})=\max_{\unlhd}\\{\mathrm{\Lambda}(I+J)\\}=B(I+J)$ and
$\max_{\unlhd}\\{B(I)\wedge B(J)\\})=\max_{\unlhd}\\{\mathrm{\Lambda}(I\cap
J)\\}=B(I\cap J)$, which completes the proof. ∎

######  Proposition 3.11.

For $A,B,C\subseteq P_{n}$ set $A+B=\\{\lambda+\mu:\lambda\in A,\mu\in B\\}$.
Then we have

  1. (1)

$\operatorname{Sss}(A)\cdot\operatorname{Sss}(B)=\operatorname{Sss}(A+B)$

  2. (2)

$B(IJ)=\max_{\unlhd}\\{B(I)+B(J)\\}$ for any symmetric strongly shifted ideals
$I,J$.

###### Proof.

For (1), it is clear that
$x^{\lambda+\mu}\in\operatorname{Sss}(A)\cdot\operatorname{Sss}(B)$ whenever
$\lambda\in A$ and $\mu\in B$. Therefore, using the fact that
$\operatorname{Sss}(A)\cdot\operatorname{Sss}(B)$ is symmetric strongly
shifted by 2.3, we deduce that
$\operatorname{Sss}(A+B)\subseteq\operatorname{Sss}(A)\cdot\operatorname{Sss}(B)$.
Conversely, let $p\in P_{n}$ be so that $x^{p}$ is a monomial generator of
$\operatorname{Sss}(A)\cdot\operatorname{Sss}(B)$. Then, by 3.5 (1) there
exist partitions $\lambda\in A$, $\mu\in B$ and
$\lambda^{\prime},\mu^{\prime}\in P_{n}$ and permutations
$\sigma,\tau\in\mathfrak{S}_{n}$ so that $\lambda^{\prime}\unlhd\lambda$,
$\mu^{\prime}\unlhd\mu$, and $p=\sigma(\lambda^{\prime})+\tau(\mu^{\prime})$.
Since $\lambda^{\prime},\mu^{\prime}$ are ordered increasingly, for all $k$
one has that

$\displaystyle p_{k}+\cdots+p_{n}$ $\displaystyle=$
$\displaystyle\lambda^{\prime}_{\sigma^{-1}(k)}+\ldots+\lambda^{\prime}_{\sigma^{-1}(n)}+\mu^{\prime}_{\tau^{-1}(k)}+\ldots+\mu^{\prime}_{\tau^{-1}(n)}$
$\displaystyle\leq$
$\displaystyle\lambda^{\prime}_{k}+\ldots+\lambda^{\prime}_{n}+\mu^{\prime}_{k}+\ldots+\mu^{\prime}_{n}$
$\displaystyle\leq$
$\displaystyle\lambda_{k}+\ldots+\lambda_{n}+\mu_{k}+\ldots+\mu_{n},$

where the last inequality follows from the fact that
$\lambda^{\prime}\unlhd\lambda$ and $\mu^{\prime}\unlhd\mu$. Thus
$p\unlhd\lambda+\mu$, whence $x^{p}\in\operatorname{Sss}(A+B)$, which
completes the proof.

For (2), notice that part (1) and the fact that $I,J$ are strongly shifted
imply the identity

$\displaystyle\operatorname{Sss}(B(I)+B(J))=\operatorname{Sss}(B(I))\cdot\operatorname{Sss}(B(J))=IJ.$

This yields the desired conclusion by ensuring that
$\max_{\unlhd}\\{B(I)+B(J)\\})=\max_{\unlhd}\\{\mathrm{\Lambda}(IJ)\\}=B(IJ)$.
∎

###  3.1. Principal Borel sssi’s and permutohedra

Strongly stable ideals with only one Borel generator are called _principal
Borel ideals_. They play a special role in the theory of strongly stable
ideals, due to their rich combinatorial structure and to the fact that every
strongly stable ideal is a sum of principal Borel ideals. Analogously, we
introduce the following notion.

######  Definition 3.12.

A principal Borel symmetric shifted ideal is any ideal of the form
$\operatorname{Sss}(\\{\lambda\\})$ for some $\lambda\in P_{n}$, i.e., an
ideal whose set of partition Borel generators is a singleton.

Note that principal Borel sssi’s are necessarily generated in a single degree
(equigenerated).

######  Example 3.13.

Examples of principal Borel sssi’s include the following:

  * •

Powers of the maximal ideal
$(x_{1},\ldots,x_{n})^{d}=\operatorname{Sss}(\\{(0^{n-1},d)\\})$.

  * •

The square-free Veronese ideal of degree $n-c+1$ in $n$ variables; see 2.6

$I_{n,c}=\operatorname{Sss}(\\{(0^{c-1},1^{n-c+1})\\}).$

  * •

Powers $I_{n,c}^{m}=\operatorname{Sss}(\\{(0^{c-1},m^{n-c+1})\\})$ of a
square-free Veronese ideal. The formula for the unique partition Borel
generator follows from 3.11.

It is clear from 3.10(1) that every sssi is a sum of principal Borel sssi’s
and that the class of principal Borel sssi’s is not closed under sums.
However, it is closed under intersections and products.

######  Corollary 3.14.

Let $I,J$ be principal Borel sssi’s. Then $I\cap J$ and $IJ$ are principal
Borel sssi’s. In particular, principal Borel sssi’s are closed under taking
powers.

###### Proof.

Let $I=\operatorname{Sss}(\\{\lambda\\})$ and $J=\operatorname{Sss}(\\{\mu\\}$
for some $\lambda,\mu\in P_{n}$. Then, 3.10(2) and 3.11(1) imply that
$\operatorname{Sss}(\\{\lambda\\})\cap\operatorname{Sss}(\\{\mu\\})=\operatorname{Sss}(\\{\lambda\wedge\mu\\})$
and
$\operatorname{Sss}(\\{\lambda\\})\operatorname{Sss}(\\{\mu\\})=\operatorname{Sss}(\\{\lambda+\mu\\})$
are principal Borel sssi’s. ∎

A remarkable consequence of 3.11 is that a principal Borel sssi decomposes as
a product of square-free Veronese ideals. Recall that the transpose of
$\lambda$, denoted by $\lambda^{T}$, is defined as the partition corresponding
to the transpose of the Young diagram of $\lambda$, so that the parts of
$\lambda^{T}$ record the number of boxes in each row of the Young diagram of
$\lambda$. More precisely $\lambda^{T}\in P_{\lambda_{n}}$ is defined by

$\lambda^{T}_{i}=|\\{j:\lambda_{j}\geq\lambda_{n}-i+1\\}|\leq n.$

######  Proposition 3.15.

Let $\lambda\in P_{n}$ be a partition and set $\lambda_{0}=0$. The principal
Borel sssi $\operatorname{Sss}(\\{\lambda\\})$ decomposes as

$\operatorname{Sss}(\\{\lambda\\})=\prod_{i=1}^{n}I_{n,i}^{\lambda_{i}-\lambda_{i-1}}.$

###### Proof.

By the definition of $\lambda^{T}$, $\lambda$ can be decomposed as the sum of
vectors corresponding to the rows of its Young diagram, that is,
$\lambda=\sum_{j=1}^{\lambda_{n}}(0^{n-\lambda^{T}_{j}},1^{\lambda^{T}_{j}}).$
This yields an ideal decomposition
$\,\operatorname{Sss}(\\{\lambda\\})=\prod_{j=1}^{\lambda_{n}}\,\operatorname{Sss}(\\{(0^{n-\lambda^{T}_{j}},1^{\lambda^{T}_{j}})\\})=\prod_{j=1}^{\lambda_{n}}\,I_{n,n-\lambda^{T}_{j}+1}\,$
by 3.11(1). It remains to observe that, by definition of $\lambda^{T}$, the
number of parts of $\lambda^{T}$ of size $n-i+1$, that is, the number of rows
for the Young diagram of $\lambda$ which contain exactly $n-i+1$ boxes, is
$\lambda_{i}-\lambda_{i-1}$. Thus combining the repeated factors of the
previous identity yields the claim. ∎

The notion of partition Borel generators has an interesting geometric
application, as it allows to describe the Newton polytope of a symmetric
strongly shifted ideal in terms of well-studied convex polytopes. The Newton
polytope of a monomial ideal $I$ is defined as the convex hull of the
exponents of its minimal generators, that is,

$\operatorname{np}(I)=\operatorname{conv}\\{(a_{1},\ldots,a_{n})\mid x^{a}\in
G(I)\\}.$

In convex geometry a permutohedron is a convex body defined as follows

${\mathbf{P}}(a_{1},\ldots,a_{n})=\operatorname{conv}\\{(a_{\sigma(1)},\ldots,a_{\sigma(n)})\mid\sigma\in\mathfrak{S}_{n}\\},$

where $\operatorname{conv}$ denotes taking the convex hull of a set of points
in $\mathbb{R}^{n}$. Permutohedra have emerged as objects of recent interest
in combinatorics [Pos09] and in algebraic geometry [Huh14]. An important
example of permutohedron is the hypersimplex
$\Delta_{n,d}={\mathbf{P}}(0^{n-d},1^{d})$. We now show that the convex
geometry of sssi’s is governed by permutohedra. Some geometric implications of
this fact will be discussed in Section 6.2.

######  Proposition 3.16.

The Newton polytope of a principal Borel sssi is a permutohedron, namely,

$\operatorname{np}(\operatorname{Sss}(\\{\lambda\\}))={\mathbf{P}}(\lambda)=\operatorname{conv}\\{(\sigma(\lambda)\mid\sigma\in\mathfrak{S}_{n}\\}.$

In general, the Newton polytope of an sssi $I$ is a convex hull of a union of
permutohedra:

$\operatorname{np}(I)=\operatorname{conv}\left(\bigcup_{\lambda\in
B(I)}{\mathbf{P}}(\lambda)\right).$

###### Proof.

It is clear from the definitions that
${\mathbf{P}}(\lambda)\subseteq\operatorname{np}(\operatorname{Sss}(\\{\lambda\\}))$
since the vertices of the permutohedron are exponent vectors for some of the
monomials in $G(\operatorname{Sss}(\\{\lambda\\})$. For the converse, a
theorem by Rado [Rad52], as transcribed in [Pos09, Proposition 2.5], states
that ${\mathbf{P}}(\lambda)$ is defined by the following (in)equalities

${\mathbf{P}}(\lambda)=\left\\{(t_{1},\ldots,t_{n})\in\mathbb{R}^{n}\mid\begin{cases}t_{1}+\cdots+t_{n}=\lambda_{1}+\cdots\lambda_{n}\\\
t_{i_{k}}+\cdots+t_{i_{n}}\leq\lambda_{k}+\cdots+\lambda_{n},&\forall 1\leq
k\leq n,\forall 1\leq i_{k}\leq\ldots\leq i_{n}\leq n\end{cases}\right\\}.$

Each exponent vector of a monomial in $G(I)$ satisfies the above system by
2.10, thus we obtain
$\operatorname{np}(\operatorname{Sss}(\\{\lambda\\}))\subseteq{\mathbf{P}}(\lambda)$.

To deduce the general statement from that regarding principal Borel sssi’s it
suffices to note that by 3.10 (1) an arbitrary sssi $I$ decomposes as
$I=\sum_{\lambda\in B(I)}\operatorname{Sss}(\\{\lambda\\})$ and that the
Newton polytope of a sum of ideals is the convex hull of the union of the
Newton polytopes of the summands. ∎

The Minkowski sum of polytopes $A,B$ is defined as $A+B=\\{a+b\mid a\in A,b\in
B\\}$. As a corollary to our decomposition formula for principal Borel sssi’s
in 3.15, we recover a well-known decomposition for the permutohedron as a
Minkowski sum of hypersimplices.

######  Corollary 3.17.

For $\lambda\in P_{n}$ and $\lambda_{0}=0$ there is an identity
${\mathbf{P}}(\lambda)=\sum_{i=1}^{n}(\lambda_{i}-\lambda_{i-1})\Delta_{i,n}$.

###### Proof.

3.15 yields the following polyhedral identities as a consequence of the fact
that the Newton polytope of a product of ideals is the Minkowski sum of the
Newton polytopes of the summands

$\operatorname{np}(\operatorname{Sss}(\\{\lambda\\})=\sum_{i=1}^{n}\operatorname{np}(I_{n,i}^{\lambda_{i}-\lambda_{i-1}})=\sum_{i=1}^{n}(\lambda_{i}-\lambda_{i-1})\Delta_{i,n}.$

∎

###  3.2. The polymatroidal structure of a principal Borel sssi

In this subsection, we discuss another combinatorial characterization of
principal Borel sssi’s which is a direct consequence of 3.15. We recall that a
homogeneous ideal is termed equigenerated if all its minimal generators have
the same degree. The following definition is due to Herzog and Hibi [HH02,
Definition 2.1 and Remark 6.4].

######  Definition 3.18.

An equigenerated monomial ideal $I$ is called a _polymatroidal ideal_ if any
two monomial generators $x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$ and
$x_{1}^{v_{1}}\cdots x_{n}^{v_{n}}$ of $I$ satisfy the following exchange
property:

For every $i$ so that $u_{i}>v_{i}$ there exists a $j$ so that $u_{j}<v_{j}$
and $(x_{1}^{u_{1}}\cdots x_{n}^{u_{n}})x_{j}/x_{i}\in I$.

The terminology refers to the fact that the exponent vectors $(u_{1},\ldots
u_{n})\in\mathbb{Z}^{n}$ of the monomials $x_{1}^{u_{1}}\cdots x_{n}^{u_{n}}$
generating a polymatroidal ideal form a set of bases of a _discrete
polymatroid_. The index $j$ in 3.18 can be chosen so that also
$(x_{1}^{v_{1}}\cdots x_{n}^{v_{n}})x_{i}/x_{j}\in I$ [HH02, Theorem 4.1].
This is referred in the literature as the _symmetric exchange property_ , and
is conjectured to determine the algebraic structure of the toric ring of a
polymatroidal ideal [Whi80, HH02] (see Conjecture 6.6 and our discussion
therein).

Examples of polymatroidal ideals include ordinary powers of square-free
Veronese ideals and principal Borel ideals by [HH02, Examples 2.6(c) and 9.4].
Since powers of square-free Veronese ideals are principal Borel sssi’s and
principal Borel ideals become principal Borel sssi’s after symmetrization in
the sense of 3.8, it is then natural to ask whether _every_ principal Borel
sssi is polymatroidal. The following theorem shows that this is indeed the
case.

######  Theorem 3.19.

A principal Borel sssi is a polymatroidal ideal. In fact, a symmetric monomial
ideal is polymatroidal if and only if it is a principal Borel sssi.

###### Proof.

That every principal Borel sssi is a polymatroidal ideal follows from 3.15,
since square-free Veronese ideals are polymatroidal and products of
polymatroidal ideals are polymatroidal by [CH03, Theorem 5.3].

To prove the converse, we first show that every symmetric polymatroidal ideal
is a sssi. Let $I$ be a symmetric polymatroidal ideal. Let
$\lambda\in\Lambda(I)$ be a partition with $\lambda_{j}<\lambda_{i}$ and let
$\sigma=(ij)\in\mathfrak{S}_{n}$. Then, as $I$ is symmetric, the monomials
$f=x^{\lambda}$ and $g=\sigma(f)$ are in $I$. For convenience, in the
following we denote $\deg_{i}(h)$ the exponent of $x_{i}$ in a monomial $h$.
By assumption we have $\lambda_{i}=\deg_{i}(f)>\deg_{i}(g)=\lambda_{j}$ and
$j$ is the only index for which
$\lambda_{j}=\deg_{j}(f)<\deg_{j}(g)=\lambda_{i}$. Since $I$ is polymatroidal,
the exchange property then yields that $fx_{j}/x_{i}\in I$, so I is symmetric
strongly shifted.

We next prove that every polymatroidal sssi must be a principal Borel sssi.
Let I be a polymatroidal sssi and suppose it is not principal Borel, thus
there exist distinct $\lambda,\mu\in B(I)$. After possibly switching the names
of $\lambda$ and $\mu$, we may assume that
$\lambda_{1}=\mu_{1},\lambda_{2}=\mu_{2},\ldots,\lambda_{i-1}=\mu_{i-1}$ and
$\lambda_{i}<\mu_{i}$. Since $I$ is polymatroidal, there exists an index $j$
so that $\lambda_{j}>\mu_{j}$ and $x^{\mu}x_{j}/x_{i}\in I$. Since
$\lambda_{k}=\mu_{k}$ for $k<i$, and $\lambda_{i}<\mu_{i}$, $j$ must satisfy
$i<j$. Setting $\mu^{\prime}:=\operatorname{part}(x^{\mu-e_{i}+e_{j}})$, the
previous considerations yield that $\mu^{\prime}\in P(I)$. Note that
$\mu^{\prime}=\mu-e_{i^{\prime}}+e_{j^{\prime}}$, where if $\mu_{i}<\mu_{j}$
then $i^{\prime}=\max\\{k:\mu_{k}=\mu_{i}\\}$ and
$j^{\prime}=\min\\{k:\mu_{k}=\mu_{j}\\}$ and if $\mu_{i}=\mu_{j}$ then
$i^{\prime}=\min\\{k:\mu_{k}=\mu_{j}\\}$ and
$j^{\prime}=\max\\{k:\mu_{k}=\mu_{j}\\}$ and in both cases $i\leq
i^{\prime}<j^{\prime}\leq j$ . Hence
$x^{\mu}=x^{\mu^{\prime}}x_{i^{\prime}}/x_{j^{\prime}}$ is obtained from
$x^{\mu^{\prime}}$ via a Borel move and consequently
$\mu^{\prime}\in\Lambda(I)$ is larger than $\mu$ in the dominance order. But
this is a contradiction, since $\mu\in B(I)$ is a maximal element of
$\Lambda(I)$ with respect to dominance by 3.5 (2). ∎

######  Remark 3.20.

It follows from work of Postnikov [Pos09] that a discrete polymatroid can be
interpreted as the set of integer points of a generalized permutohedron. The
latter is a convex polytope obtained by parallel translation of the facets of
a permutohedron; see also [CL20, Theorem 3.17]. 3.16 and 3.19 then imply that
a symmetric discrete polymatroid is the set of integer points of a
permutohedron.

A polymatroidal ideal $I$ is said to satisfy the _strong exchange property_
if, for any two distinct monomial generators $x_{1}^{u_{1}}\cdots
x_{n}^{u_{n}}$ and $x_{1}^{v_{1}}\cdots x_{n}^{v_{n}}$ of $I$ and all indices
$i$ and $j$ so that $u_{i}>v_{i}$ and $u_{j}<v_{j}$, then
$(x_{1}^{u_{1}}\cdots x_{n}^{u_{n}})x_{j}/x_{i}$ is in $I$ (see [HH02,
Definition 2.5]).

Notice that square-free Veronese ideals and their ordinary powers satisfy this
property. However, this is not true for arbitrary principal Borel sssi’s.

######  Proposition 3.21.

Let $I=\operatorname{Sss}(\\{\lambda\\})$ be a principal Borel symmetric
strongly shifted ideal. Then $I$ satisfies the strong exchange property if and
only if $\lambda$ is of one of the following types:

  1. (1)

$\lambda=(a,\ldots,a)$ for some $a\neq 0\in{\mathbb{N}}$;

  2. (2)

$\lambda=(a^{s},b^{n-s})$ for some $a<b\in{\mathbb{N}}$, $s>0$;

  3. (3)

$\lambda=(a^{s},b,c^{n-s-1})$ for some $a<b<c\in{\mathbb{N}}$, $s>0$.

###### Proof.

Notice that $I$ satisfies the strong exchange property trivially if (1) holds.
Let $\lambda$ be as in (2) and let $u=x_{1}^{u_{1}}\cdots
x_{n}^{u_{n}},v=x_{1}^{v_{1}}\cdots x_{n}^{v_{n}}$ be distinct monomial
minimal generators of $I$. Then, if $u_{i}>v_{i}$ and $u_{j}<v_{j}$ it must be
that $u_{j}=v_{i}=a$ and $u_{i}=v_{j}=b$. Then, for $u^{\prime}=ux_{j}/x_{i}$,
we have

$\operatorname{part}(u^{\prime})=\begin{cases}(a^{s-1},a+1,b-1,b^{n-s-1})\unlhd\lambda&\text{
if }b>a+1,\\\ (a^{s-1},b-1,a+1,b^{n-s-1})=\lambda&\text{ if }b=a+1\end{cases}$

Thus, in any case $\operatorname{part}(u^{\prime})\in P(I)$, whence
$u^{\prime}\in I$. This shows that $I$ satisfies the strong exchange property.
Similarly, if (3) holds, exchanging variables $x_{i},x_{j}$ appearing with
distinct exponents among distinct monomial generators $u,v$ produces
$u^{\prime}=ux_{j}/x_{i}$, where either
$\operatorname{part}(u^{\prime})=\lambda$, or
$\operatorname{part}(u^{\prime})=(a^{s-1},a+1,b-1,c^{n-s-1})$, or
$\operatorname{part}(u^{\prime})=(a^{s},b+1,c-1,c^{n-s-2})$, or
$\operatorname{part}(u^{\prime})=(a^{s-1},a+1,b,c-1,c^{n-s-2})$. Since
$\operatorname{part}(u^{\prime})\in P(I)$ in each of these cases, the strong
exchange property is satisfied.

We next show that in all other cases the strong exchange property does not
hold. Assume first that $\lambda=(a^{s},b^{t},c^{q},d^{r},\ldots)$ has at
least four distinct parts $a<b<c<d$. Then there exist monomial generators of
$I$ of the form $u=x_{1}^{a}x_{2}^{b}x_{3}^{c}x_{4}^{d}\cdots$ and
$v=x_{1}^{c}x_{2}^{a}x_{3}^{d}x_{4}^{b}\cdots$ respectively. Hence,
$u^{\prime}=ux_{3}/x_{2}$ is such that
$\operatorname{part}(u^{\prime})=(a^{s},b-1,b^{t-1},c^{q},c+1,d^{r},\ldots)$.
Since $\lambda\unlhd\operatorname{part}(u^{\prime})$,
$\lambda\neq\operatorname{part}(u^{\prime})$ and
$I=\operatorname{Sss}(\\{\lambda\\})$, we deduce
$\operatorname{part}(u^{\prime})\notin P(I)$, whence $u^{\prime}\notin I$.
Finally, suppose that $\lambda=(a^{s},b^{t},c^{q})$ with $a<b<c,s,q\geq
1,t\geq 2$ and consider minimal generators for $I$ of the form
$u=x_{1}^{a}x_{2}^{b}x_{3}^{b}x_{4}^{c}\cdots,v=x_{1}^{b}x_{2}^{a}x_{3}^{c}x_{4}^{b}\cdots$.
Then, $u^{\prime}=ux_{3}/x_{2}$ is such that
$\operatorname{part}(u^{\prime})=(a^{s},b-1,b^{t-2},b+1,c^{q},\ldots)$. Since
$\lambda\unlhd\operatorname{part}(u^{\prime})$, it follows that
$\operatorname{part}(u^{\prime})\notin P(I)$, which completes the proof. ∎

The factorization of principal Borel sssi’s given in 3.15 parallels the known
factorization of principal Borel ideals as products of monomial prime ideals;
see, e.g., [FMS13, Propositions 2.7]. Polymatroidal ideals endowed with such a
factorization property are called transversal polymatroidal ideals. In detail,
a _transversal polymatroidal ideal_ is an ideal which can be written as a
product of monomial ideals generated by subsets of the variables
$x_{1},\ldots,x_{n}$, with repeated factors allowed. The following proposition
shows that principal Borel sssi’s need not be transversal. In its statement,
for $\lambda\in P_{n}$ we define the the discrete difference vectors
$\Delta^{i}\lambda$ inductively by $\Delta^{0}\lambda=\lambda$,
$(\Delta\lambda)_{j}=\lambda_{j+1}-\lambda_{j}$, and
$\Delta^{i}\lambda=\Delta(\Delta^{i-1}\lambda)$.

######  Proposition 3.22.

The following are equivalent:

  1. (1)

$I=\operatorname{Sss}(\\{\lambda\\})$ with $\lambda\in P_{n}$ is a transversal
polymatroidal ideal,

  2. (2)

there exist integers $a_{j}\geq 0$ such that
$\displaystyle{\lambda_{i}=\sum_{j=1}^{i}{i-1\choose j-1}a_{j}}$ for $1\leq
i\leq n$,

  3. (3)

for some (equivalently, for each) $1\leq i\leq n-1$ we have
$\Delta^{i}\lambda\in P_{n-i}$ and
$\operatorname{Sss}(\\{\Delta^{i}\lambda\\})$ is transversal polymatroidal.

  4. (4)

$(\Delta^{i}\lambda)_{1}\geq 0$ for all $0\leq i\leq n-1$.

###### Proof.

Notice that a symmetric transversal polymatroidal ideal in $n$ variables is of
the form

$J=\left(\prod_{i=1}^{n}(x_{i})^{a_{1}}\right)\left(\prod_{1\leq i<j\leq
n}(x_{i},x_{j})^{a_{2}}\right)\left(\prod_{1\leq i<j<k\leq
n}(x_{i},x_{j},x_{k})^{a_{3}}\right)\cdots(x_{1},\ldots,x_{n})^{a_{n}}$

for some integers $a_{j}\geq 0$. Indeed, if
$(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{c}})^{a_{c}}$ is a factor in the product
decomposition of $J$ then for each $\sigma\in\mathfrak{S}_{n}$ we have
$(x_{\sigma(i_{1})},x_{\sigma(i_{2})},\ldots,x_{\sigma(i_{c})})^{a_{c}}$ as a
factor in a product decomposition of $\sigma(J)=J$. Thus, it follows from the
uniqueness of the decomposition of $J$ as a product of powers of monomial
prime ideals [HRV13, Lemma 4.1] that
$(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{c}})^{a_{c}}$ is a factor in the
decomposition of $J$ if and only if for all $\sigma\in\mathfrak{S}_{n}$ the
ideal $(x_{\sigma(i_{1})},x_{\sigma(i_{2})},\ldots,x_{\sigma(i_{c})})^{a_{c}}$
is a factor in the decomposition of $J$.

For the equivalence of (1) and (2), if $I=\operatorname{Sss}(\\{\lambda\\})$
is transversal then $I=J$ for some $J$ as described above. The partition Borel
generator $\lambda$ of $I$ is the largest monomial in $G(I)$ with respect to
the monomial order ${\operatorname{antilex}}$ defined by
$\alpha<_{{\operatorname{antilex}}}\beta$ if the leftmost non-zero entry of
$\alpha-\beta$ is positive, since this order refines the dominance order on
partitions. Thus, in order to establish whether $J=I$, we must identify the
largest monomial $u$ in $G(J)$ with respect to this order. We claim that this
monomial $u$ is obtained as follows:

$u=\left(\prod_{i=1}^{n}x_{i}^{a_{1}}\right)\left(\prod_{1\leq i<j\leq
n}x_{j}^{a_{2}}\right)\left(\prod_{1\leq i<j<k\leq
n}x_{k}^{a_{3}}\right)\cdots
x_{n}^{a_{n}}=\prod_{i=1}^{n}x_{i}^{\sum_{j=1}^{i}{i-1\choose j-1}a_{j}}.$
(3.4)

Indeed, if $i_{1}<i_{2}<\cdots<i_{c}$, the largest monomial in
$(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{c}})^{a_{c}}$ with respect to
${\operatorname{antilex}}$ is $x_{i_{c}}^{a_{c}}$ and since this is a monomial
order, hence compatible with products, it follows that $u$ is indeed the
${\operatorname{antilex}}$-largest monomial of $G(J)$. The second equality in
(3.4) follows by observing that for fixed $c$ and $i_{c}$ there are
${i-1\choose j-1}$ ideals $(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{c}})$ with $1\leq
i_{1}<i_{2}<\cdots<i_{c}\leq n$. Thus $\lambda=\operatorname{part}(u)$ yields
the desired description of the $\lambda_{i}$ in statement (2). Conversely, if
the $\lambda_{i}$ can be described as in statement (2) of our result, then we
note that $J$ is a symmetric polymatroidal ideal and thus
$J=\operatorname{Sss}(\operatorname{part}(u))$ by 3.19. Since
$\operatorname{part}(u)=\lambda$, we have $I=J$ and so $I$ is transversal.

Observe that the system of equations
$\,\displaystyle{\lambda_{i}=\sum_{j=1}^{i}{i-1\choose j-1}a_{j}}\,$ for
$1\leq i\leq n$ is equivalent to the system
$\,\displaystyle{\lambda_{i+1}-\lambda_{i}=\sum_{j=2}^{i-1}{i-1\choose
j-2}a_{j}}\,$ for $1\leq i\leq n$ in the unknowns $a_{2},\ldots,a_{n}$,
together with the equation $\lambda_{1}=a_{1}$. If the latter system admits
nonnegative solutions then $\Delta\lambda$ is a partition, since the functions
${i-1\choose j-2}$ are nondecreasing in the argument $i$. Moreover, this
system admits nonnegative solutions if and only if
$\operatorname{Sss}(\\{\Delta\lambda\\})$ is transversal polymatroidal, by the
equivalence of (1) and (2). It follows by induction on $i$ that $(2)$ implies
that, for each $1\leq i\leq n-1$, $\Delta^{i}\lambda\in P_{n-i}$ and
$\operatorname{Sss}(\\{\Delta^{i}\lambda\\})$ is transversal polymatroidal.
Thus $(2)$ implies the stronger form of $(3)$ and hence also the weaker form.

The equivalence of the two systems of equations mentioned above shows that if
$\Delta\lambda\in P_{n-1}$ and $\operatorname{Sss}(\\{\Delta\lambda\\})$ is
transversal polymatroidal then $\operatorname{Sss}(\\{\lambda\\})$ is
transversal polymatroidal. Thus, it follows by induction on $i$ that if for
some $1\leq i\leq n$ we have $\Delta^{i}\lambda\in P_{n-i}$ and
$\operatorname{Sss}(\\{\Delta^{i}\lambda\\})$ is transversal polymatroidal
then $\operatorname{Sss}(\\{\lambda\\})$ is transversal polymatroidal. Thus
the weaker (hence also the stronger) form of $(3)$ implies $(1)$.

Since (3) clearly implies (4), we end by showing $(4)$ implies $(2)$. For this
note the solution to the system of identities listed in $(2)$ is
$a_{i}=(\Delta^{i}\lambda)_{1}\geq 0$. This can be seen by induction on $i$
using the fact that taking the difference of consecutive identities replaces
the system in $(2)$ for $\lambda$ with the similar system for $\Delta\lambda$
and the equation $a_{i}=\lambda_{1}$, as noted above. ∎

A special class of transversal polymatroidal ideals, which includes all
principal Borel ideals, is that of _lattice path polymatroidal ideals_. These
ideals were introduced by Schweig in [Sch10] and are characterized by the
property that their minimal generators correspond to certain planar lattice
paths. The next corollary shows that lattice path polymatroidal ideals are
rarely symmetric. In particular, the polymatroidal structure of principal
Borel sssi’s is usually different than that of principal Borel ideals.

######  Proposition 3.23.

Let $I$ be a lattice path polymatroidal ideal. $I$ is symmetric if and only if
either $I=\operatorname{Sss}(\\{(a,...,a)\\})$ for some $a\in{\mathbb{N}}$, or
$I$ is a power of the homogeneous maximal ideal.

###### Proof.

In [Lu17, Proposition 2.9] it is shown that $I$ is a lattice path
polymatroidal ideal if and only if $I$ is the product of monomial prime
ideals, each of which is generated by consecutive variables:

$I=(x_{s_{1}},\ldots,x_{s_{1}+k_{1}})\cdots(x_{s_{n}},\ldots,x_{s_{n}+k_{n}}),$
(3.5)

with $s_{1}\leq\ldots\leq s_{n}$ and $s_{1}+k_{1}\leq\ldots\leq s_{n}+k_{n}$.
In particular, for every $a\in{\mathbb{N}}$, the symmetric ideals
$\operatorname{Sss}(\\{(a,...,a)\\})$ and ${\mathfrak{m}}^{a}$ are lattice
path polymatroidal ideals.

Conversely, if $I$ is a lattice path polymatroidal ideal, (3.5) implies that
$I$ is transversal. If $I$ is also symmetric, then from the proof of 3.22 it
follows that, for every $i$, every permutation of
$(x_{s_{i}},....,x_{s_{i}+k_{i}})$ must be a factor of $I$ too. Thus, the only
admissible factors are of the form $P_{i}=(x_{i})$ for all $i$, or
${\mathfrak{m}}=(x_{1},\ldots,x_{n})$. The first case yields
$I=\operatorname{Sss}\\{(a,...,a)\\}$ while the second case yields a power of
the maximal ideal ${\mathfrak{m}}^{b}$, where $a$ and $b$ respectively denote
the multiplicity of each $P_{i}$ or of ${\mathfrak{m}}$ in the factorization
(3.5). ∎

3.21 and 3.22 suggest that the combinatorial properties of principal Borel
sssi’s might be very different from those of polymatroidal ideals which
satisfy the strong exchange property or are transversal. In fact, one can
easily construct principal Borel sssi’s which do not belong to any of these
classes.

######  Example 3.24.

Let $I=\operatorname{Sss}(\\{(1,3,4,5)\\})$. By 3.21, $I$ does not satisfy the
strong exchange property. Moreover, $I$ cannot be transversal, since the
equations for the $a_{j}$’s in 3.22 have no integer solutions for the given
values $\lambda_{1}=1,\lambda_{2}=3,\lambda_{3}=4,\lambda_{4}=5$.
Alternatively $\Delta\lambda=(2,1,1)$ is not a partition.

Nevertheless, we will see in later sections that ordinary powers of principal
Borel sssi’s share similar algebraic properties as powers of polymatroidal
ideals that are either transversal or satisfy the strong exchange property.

###  3.3. Numerical invariants of symmetric shifted ideals

Inspired by the case of stable and strongly stable ideals, in this subsection
we use partition generators to calculate numerical invariants associated with
symmetric shifted and strongly shifted ideals. In fact, most formulas will
hold more generally for arbitrary symmetric ideals.

Denote $\min(a)=\min\\{i\colon a_{i}\neq 0\\}$, $\max(a)=\max\\{i\colon
a_{i}\neq 0\\}$. If $I$ is strongly stable, then

$\operatorname{ht}(I)=\max\\{\min(a)\colon x^{a}\in
G(I)\\}\quad\mathrm{and}\quad\operatorname{pd}(R/I)=\max\\{\max(a)\colon
x^{a}\in G(I)\\}$

The latter formula follows from the Eliahou-Kervaire resolution [EK90].
Moreover, in both cases it is enough to only consider the Borel generators of
$I$; see [FMS11, Proposition 2.14].

Replacing exponent vectors with partitions, we derive an analogous formula for
the codimension of a symmetric strongly shifted ideal, which is actually
satisfied by an arbitrary symmetric monomial ideal.

######  Proposition 3.25.

Let $I$ be a symmetric monomial ideal. For each
$\lambda\in\mathrm{\Lambda}(I)$ denote
$\min(\lambda)=\min\\{i:\lambda_{i}>0\\}$. The height of $I$ is given by

$\operatorname{ht}(I)=\max_{\lambda\in\mathrm{\Lambda}(I)}\\{\min(\lambda)\\}.$

Moreover, if $I$ is symmetric strongly shifted, then
$\operatorname{ht}(I)=\max_{\lambda\in B(I)}\\{\min(\lambda)\\}$.

###### Proof.

From 2.7 it follows that $\sqrt{I}=I_{n,c}$, where $c=\operatorname{ht}(I)$.
It remains to determine the height of $I$. For this notice that a monomial
$m\in\sqrt{I}$ if and only if $|\operatorname{supp}(m)|>n-c$. Therefore one
has the chain of equalities

$\displaystyle c$ $\displaystyle=$ $\displaystyle n-\min_{m\in
I}\\{|\operatorname{supp}(m)|\\}+1=n-\min_{\lambda\in\mathrm{\Lambda}(I)}\\{|\operatorname{supp}(\lambda)|\\}+1$
$\displaystyle=$ $\displaystyle
n-\min_{\lambda\in\mathrm{\Lambda}(I)}\\{n-\min(\lambda)+1\\}-1=\max_{\lambda\in\mathrm{\Lambda}(I)}\\{\min(\lambda)\\}.$

In the first line of the displayed equalities we use the fact that
$\operatorname{supp}(m)=\operatorname{supp}(\operatorname{part}(m))$.

Suppose now that $I$ is a symmetric strongly shifted ideal. We prove that we
can replace $\mathrm{\Lambda}(I)$ with $B(I)$ in the formula above by showing
that each Borel move can only increase the size of the support. This is
because if $\mu_{j}>\mu_{i}$ and $\mu^{\prime}=\mu-e_{j}+e_{i}$ then either
$\mu_{j}-1>0$ and then
$\operatorname{supp}(\mu)\subseteq\operatorname{supp}(\mu^{\prime})$ or
$\mu_{j}=1$ and $\mu_{i}=0$ in which case
$\operatorname{part}(\mu^{\prime})=\mu$ and thus
$|\operatorname{supp}(\mu^{\prime})|=|\operatorname{supp}(\mu)|$. ∎

On the other hand, the projective dimension of a symmetric strongly shifted
ideal cannot be expressed in terms of min/max of the partition Borel
generators. In 3.27 below, we provide a formula for the projective dimension
of a symmetric shifted ideal in terms of its partition generators using a
different partition statistic we call $\operatorname{med}$. Recall from
[BdAG+20, Theorem 3.1] that the Betti numbers of a symmetric shifted ideal are
given by

$\beta_{i}(I)=\sum_{u\in G(I)}\binom{|G(C(u))|}{i},$ (3.6)

where for a monomial $u=\sigma(x^{\lambda})\in G(I)$ with $\lambda\in P_{n}$
and $\sigma\in\mathfrak{S}_{n}$, the ideal $C(u)$ is constructed as follows
(see [BdAG+20, proof of Theorem 3.2]). First, one defines a total order on the
set of monomials in $S=k[x_{1},\ldots,x_{n}]$.

######  Definition 3.26.

Let $\lambda,\mu\in\mathrm{P}_{n}$ and let $v=\tau(x^{\mu})$ and
$u=\sigma(x^{\lambda})$ be distinct monomials in $S$, for some
$\sigma,\tau\in\mathfrak{S}_{n}$. Define $v\prec u$ if one of the following
conditions holds

  1. (1)

$\mu<_{\operatorname{antilex}}\lambda$, that is, either $|\mu|<|\lambda|$, or
$|\mu|=|\lambda|$ and the leftmost non-zero entry of $\mu-\lambda$ is positive

  2. (2)

$\mu=\lambda$ and $v<_{\operatorname{antilex}}u$, that is, the leftmost non-
zero entry of $\tau(\mu)-\sigma(\lambda)$ is positive.

Next, for a symmetric shifted ideal $I$ and a monomial
$u=\sigma(x^{\lambda})\in G(I)$, one defines $J=(v\in G(I):v\prec u)$ and

$C(u):=J:u=(x_{\sigma(1)},\dots,x_{\sigma(p)})+(x_{\sigma(k)}:p+1\leq k\leq
n-r,\ \sigma(k)<u_{\max}),$ (3.7)

where

$\begin{split}&p=p(\lambda)=\\#\\{k:\lambda_{k}<\lambda_{n}-1\\},\\\
&r=r(\lambda)=\\#\\{k:\lambda_{k}=\lambda_{n}\\},\\\
&{u_{\max}=\max\\{\sigma(k):\lambda_{k}=\lambda_{n}\\}.}\end{split}$

We are now ready to calculate the projective dimension of a symmetric shifted
ideal.

######  Proposition 3.27.

Let $I$ be a symmetric shifted ideal and define for every $\lambda\in P_{n}$
the integer $\operatorname{med}(\lambda)=|\\{i:\lambda_{i}<\lambda_{n}\\}|$.
Then the projective dimension of $I$ is given by

$\operatorname{pd}(R/I)=\max\limits_{\lambda\in\mathrm{\Lambda}(I)}\\{\operatorname{med}(\lambda)\\}+1.$

###### Proof.

Utilizing the notation in (3.7), from the formula for the Betti numbers given
by (3.6) we deduce that

$\operatorname{pd}(I)=\max\\{i:\exists u\in G(I)\text{ with }|G(C(u))|\geq
i\\}=\max\\{|G(C(u))|:u\in G(I)\\}.$

From (3.7) it follows that $|G(C(u))|\leq G(C(x^{\operatorname{part}(u)}))$.
Moreover, it can also be seen from the definitions that
$|G(C(x^{\lambda}))|=\operatorname{med}(\lambda)$ for all $\lambda\in P_{n}$.
Putting together the considerations above, we have shown that

$\operatorname{pd}(I)=\max\\{\operatorname{med}(\lambda):\lambda\in\mathrm{\Lambda}(I)\\}.$

∎

Unlike for height, when calculating the projective dimension of a symmetric
strongly shifted ideal we cannot replace $\mathrm{\Lambda}(I)$ with the
partition Borel generators of $I$.

######  Example 3.28.

Let $I=\operatorname{Sss}(\\{(1,5,5)\\})$. Notice that
$(2,4,5)\in\mathrm{\Lambda}(I)$, however
$\operatorname{med}((2,4,5))=2>1=\operatorname{med}((1,5,5))$.

When studying powers of an ideal $I$, a useful invariant is the _analytic
spread_ , $\ell(I)$, of $I$ (i.e, the Krull dimension of the fiber cone
$\mathcal{F}(I)$ of $I$, which we define in Section 6). For instance,
$\ell(I)$ controls the asymptotic growth of $\operatorname{depth}(R/I^{k})$ as
a function of $k$, thanks to Brodmann’s formula [Bro79a]:

$\lim_{k\to\infty}\operatorname{depth}(R/I^{k})\leq n-\ell(I),$

The analytic spread of an arbitrary ideal is usually difficult to calculate.
However, if $I$ is an equigenerated monomial ideal, $\ell(I)$ coincides with
the rank of the matrix whose rows are the exponent vectors of the monomial
generators of $I$ (see [HS06, Exercise 8.21]). Using this fact, we can compute
the analytic spread of any equigenerated symmetric monomial ideal.

######  Proposition 3.29.

Let $I$ be an equigenerated symmetric monomial ideal. The analytic spread of
$I$ is given by

$\ell(I)=\begin{cases}n&\text{ if
}I\neq\operatorname{Sss}(\\{(a,\ldots,a)\\})\text{ for any
}a\in{\mathbb{N}},\\\ 1&\text{ if
}I=\operatorname{Sss}(\\{(a,\ldots,a)\\})\text{ for some
}a\in{\mathbb{N}}.\end{cases}$

###### Proof.

By [HS06, Exercise 8.21], the analytic spread of an equigenerated monomial
ideal $I$ is the rank of the matrix $M$ whose rows are the exponent vectors of
the monomial generators of $I$. Hence, we only need to show that $M$ contains
$n$ linearly independent rows.

Let $V$ be the row space of $M$ viewed as a ${\mathbb{Q}}$-vector space. Since
$I$ is a symmetric ideal, $V\subseteq{\mathbb{Q}}^{n}$ is a representation of
$\mathfrak{S}_{n}$ acting naturally on ${\mathbb{Q}}^{n}$. Recall that the
natural permutation representation of $\mathfrak{S}_{n}$ on ${\mathbb{Q}}^{n}$
decomposes into two irreducible representations: the trivial representation
$T=\\{(a,\ldots,a):a\in{\mathbb{Q}}\\}$ and the standard representation
$S=\\{(q_{1},\ldots,q_{n}):q_{i}\in{\mathbb{Q}},\sum_{i=1}^{n}q_{i}=0\\}$.
Consider the projections $V_{T}$ and $V_{S}$ of $V$ onto $T$ and $S$
respectively. Since $V_{T}$ and $V_{S}$ are subrepresentations of the
irreducible representations $T$ and $S$ respectively, we have $V_{T}=0$ or
$V_{T}=T$ and $V_{S}=0$ or $V_{S}=S$ respectively, which yields four
possibilities for $V$: $V=0$ or $V=T$ or $V=S$ or $V=S\oplus
T={\mathbb{Q}}^{n}$. Since $\Lambda(I)\subseteq V$, $\Lambda(I)$ contains at
least one nonzero vector, and no element of $\Lambda(I)$ is in $S$ (since
every nonzero element of $S$ must have at least one negative coordinate), we
are left with the possibilities $V=T$ or $V=S\oplus T={\mathbb{Q}}^{n}$.

The case $V=T$ corresponds to $I=\operatorname{Sss}(\\{(a,\ldots,a)\\})$ for
some $a\in{\mathbb{N}}$, which gives $\ell(I)=\dim(V)=\dim(S)=1$. The case
$V={\mathbb{Q}}^{n}$ corresponds to $\Lambda(I)\cap S\neq\emptyset$ and gives
$\ell(I)=\dim(V)=n$. ∎

For $I\neq\operatorname{Sss}(\\{(a,\ldots,a)\\})$, 3.29 implies that
$\ell(I)=\max_{\lambda\in\mathrm{\Lambda}(I)}\\{\max(\lambda)\\}$, where for
each $\lambda\in\mathrm{\Lambda}(I)$ we denote
$\max(\lambda)=\max\\{i:\lambda_{i}>0\\}$. A similar formula for the analytic
spread of an equigenerated strongly stable ideal, with partitions replaced by
exponent vectors of the monomial generators, was proved by Montaño in [Mn15,
Proposition 5.2.6], using methods from convex geometry.

For a monomial ideal $I$, the Newton polyhedron of $I$ is defined as the
Minkowski sum

$NP(I):=\operatorname{np}(I)+\mathbb{R}^{n}_{\geq 0}.$

It is well known that the integral closure $\overline{I}$ of $I$ can be
determined via the formula $NP(\overline{I})=NP(I)\cap{\mathbb{N}}^{n}$.
Moreover, by [BA03, Theorem 2.3] the analytic spread of $I$ is

$\ell(I)=\max\\{\dim F\mid F\text{ is a compact face of }NP(I)\\}+1.$ (3.8)

If $I$ is an equigenerated monomial ideal, the Newton polyhedron $NP(I)$ has a
unique compact face of maximal dimension, which coincides with the Newton
polytope $\operatorname{np}(I)$. As a consequence of 3.29, we then have the
following description of the Newton polytope of an arbitrary equigenerated
symmetric monomial ideal.

######  Corollary 3.30.

Let $I$ be an equigenerated symmetric monomial ideal and let
$\operatorname{np}(I)$ denote the Newton polytope of $I$. Then

$\dim(\operatorname{np}(I))=\begin{cases}n-1&\text{ if
}I\neq\operatorname{Sss}(\\{(a,\ldots,a)\\})\text{ for any
}a\in{\mathbb{N}},\\\ 0&\text{ if
}I=\operatorname{Sss}(\\{(a,\ldots,a)\\})\text{ for some
}a\in{\mathbb{N}}.\end{cases}$

###### Proof.

This is immediate from (3.8) and 3.29. ∎

In [HQ15, DHQ21], the analytic spread of an equigenerated monomial ideal $I$
has been characterized in terms of the _linear relation graph_
$\Gamma=\Gamma(I)$ of $I$. This graph is defined as follows. If
$G(I)=\\{u_{1},\ldots,u_{m}\\}$ denotes a minimal monomial generating set for
$I$, the edge set of $\Gamma$ is given by

$E(\Gamma)=\\{\\{i,j\\}:\text{ there exist }u_{k},u_{l}\in G(I)\text{ such
that }x_{i}u_{k}=x_{j}u_{l}\\},$ (3.9)

while the vertex set of $\Gamma$ is given by
$\,\displaystyle{V(\Gamma)=\bigcup_{i:\\{i,j\\}\in E(\Gamma)}}\\{i\\}$.

Assume further that $I$ is equigenerated and let $r$ and $s$ denote the number
of vertices and the number of connected components of $\Gamma(I)$
respectively. By [HQ15, Lemma 4.2] one has

$\ell(I)\geq r-s+1,$ (3.10)

with equality holding if $I$ is a polymatroidal ideal, or, more generally, if
$I$ has linear syzygies [DHQ21, Lemma 4.3]. Since equigenerated symmetric
shifted ideals have a linear resolution by [BdAG+20, Theorem 3.2], we then
deduce the following immediate corollary.

######  Corollary 3.31.

Let $I$ be an equigenerated symmetric shifted ideal. Then, $\ell(I)=r-s+1$,
where $r$ is the number of vertices in $\Gamma(I)$ and $s$ is the number of
its connected components.

In fact, we can give a full description of the linear relation graph of an
equigenerated ssi, which we will use in Section 5 to study the depths of
powers of an ideal of this kind.

######  Proposition 3.32.

Let $I$ be an equigenerated symmetric shifted ideal. If
$I=\operatorname{Sss}(\\{(a,\ldots,a)\\})$ for some $a\in{\mathbb{N}}$, then
the linear relation graph $\Gamma=\Gamma(I)$ is the graph with $n$ vertices
and no edges. Otherwise, $\Gamma$ is connected, with
$V(\Gamma)=\\{1,\ldots,n\\}$.

###### Proof.

The first claim is clear, since
$I=\operatorname{Sss}(\\{(a,\ldots,a)\\}=(x_{1}^{a}\cdots x_{n}^{a})$ is a
principal ideal, so no linear relations occur.

Suppose that $I\neq\operatorname{Sss}(\\{(a,\ldots,a)\\})$. Then, there exists
a $\lambda\in\Lambda(I)$ with $\lambda_{1}<\lambda_{n}$. Let
$\mu:=\lambda+e_{1}-e_{n}$, then $\mu\in\Lambda(I)$ since $I$ is symmetric
shifted. Moreover, $x_{1}x^{\lambda}=x_{n}x^{\mu}$, that is, $\\{1,n\\}\in
E(\Gamma)$. We next show that for any $1<i<n$ either $\\{i,n\\}\in E(\Gamma)$
or $\\{1,i\\}\in E(\Gamma)$, whence $\Gamma$ is a connected graph on $n$
vertices. If $\lambda_{i}<\lambda_{n}$, by the symmetric shifted property of
$I$ we have that $\eta:=\lambda+e_{i}-e_{n}\in\mathrm{\Lambda}(I)$. Moreover,
$x_{i}x^{\lambda}=x_{n}x^{\eta}$, that is, $\\{i,n\\}\in E(\Gamma)$. On the
other hand, if $\lambda_{i}=\lambda_{n}$, let $\tau=(i,n)\in\mathfrak{S}_{n}$.
Then, applying $\tau$ to the equality $x_{1}x^{\lambda}=x_{n}x^{\mu}$ we get
$x_{1}x^{\lambda}=x_{i}\tau(x^{\mu})$. Since $\tau(x^{\mu})\in G(I)$, this
means that $\\{1,i\\}\in E(\Gamma)$ and the proof is complete. ∎

##  4\. Integrally closed sssi’s

In this section and the next, we use the results from Section 3 to study
ordinary powers of symmetric strongly shifted ideals. As it turns out, the
symmetrization result of 3.8 provides deep insight on this matter.

We begin by identifying classes of symmetric strongly shifted ideals which are
well-behaved with respect to integral closure. An ideal is said to be
_integrally closed_ if it coincides with its integral closure, and is called
_normal_ if all its powers are integrally closed. To understand normality of
an ideal $I$, a useful object is the _Rees ring_ of $I$, which is defined as
the subring ${\mathcal{R}}(I)=\bigoplus_{i\geq 0}I^{i}t^{i}\subseteq R[It]$.
Indeed, since $R=K[x_{1},\ldots,x_{n}]$ is a normal domain, $I$ is normal if
and only if the Rees algebra ${\mathcal{R}}(I)$ is a normal domain [HS06,
Propositions 5.2.1 and 5.2.4].

The following example shows that a sssi need not be integrally closed or
normal.

######  Example 4.1.

Consider $I=\operatorname{Sss}(\\{(2,2,8),(0,6,6)\\})$ and observe that for
$\lambda=(1,4,7)$ one has $2\lambda=(2,2,8)+(0,6,6)$. Hence, $x^{2\lambda}\in
I$ and $x^{\lambda}\in\overline{I}$. However $\lambda\not\in P(I)$ so
$x^{\lambda}\not\in I$. Hence, $I$ is not integrally closed and therefore not
normal.

However, 3.8 indicates a strategy to construct sssi’s which are integrally
closed or normal. In particular, we can identify two classes of normal
symmetric strongly shifted ideals, obtained by symmetrization of certain
normal strongly stable ideals.

A monomial ideal is called a _lex-segment ideal_ if, for every degree $d$, the
set of monomials of degree $d$ in $I$ forms a lex-segment; namely, for each
degree $d$, there exists a monomial $u\in I$ of degree $d$ so that $I$
contains every degree-$d$ monomial $v$ which is larger than $u$ in the
lexicographic order. It is well-known that lex-segment ideals are strongly
stable, e.g. [HH11, p. 103].

######  Proposition 4.2.

Let $J$ be an integrally closed strongly stable ideal, and let
$\displaystyle{I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(J)}$ be its
symmetrization in the sense of 3.8. Then, $I$ is an integrally closed sssi.

Moreover, if $I$ is a sssi such that $\operatorname{Borel}(x^{B(I)})$ is
normal, then $I$ is normal. In particular, $I$ is normal if either of the
following conditions hold

  1. (1)

$B(I)=\\{\lambda\\}$ for some $\lambda\in P_{n}$, i.e., $I$ is a principal
Borel sssi; or

  2. (2)

$\operatorname{Borel}(x^{B(I)})$ is an equigenerated lex-segment ideal.

###### Proof.

First, suppose that $J$ is integrally closed. Since every
$\sigma\in\mathfrak{S}_{n}$ acts as an isomorphism on $R$, then $\sigma(J)$ is
integrally closed for every $\sigma\in\mathfrak{S}_{n}$. As the intersection
of integrally closed ideals is integrally closed [HS06, Remark 1.1.3], it
follows that $I$ is integrally closed.

Now let $I$ be a sssi and $J=\operatorname{Borel}(x^{B(I)})$. Then, for every
$k\geq 1$, $J^{k}=\operatorname{Borel}(x^{kB(I)})$ is strongly stable by
[GW15, Proposition 1.2]. Moreover, by 2.4 and 3.11 we have that $I^{k}$ is
symmetric strongly shifted, with $I^{k}=\operatorname{Sss}(kB(I))$. Hence, 3.8
(2) implies that
$\displaystyle{I^{k}=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(J^{k})}$.
Therefore, from the first part of the proof it follows that $I^{k}$ is
integrally closed whenever $J^{k}$ is, that is, $I$ is normal whenever $J$ is.

Finally, to establish the two particular cases it remains to note that if $I$
is a principal Borel sssi then $J$ is a principal Borel ideal. Furthermore,
principal Borel ideals and equigenerated lex-segment ideals are normal as
follows from [DN99, Proposition 2.14] (see also [BC17, p. 3] for principal
Borel ideals and [Mn15, Theorem 5.3.2] for equigenerated lex-segment ideals).
∎

The normality of an ideal $I$ as in 4.2 provides information on the
singularities of the Rees ring ${\mathcal{R}}(I)$ and of the _special fiber
ring_ ${\mathcal{F}}(I):={\mathcal{R}}(I)/{\mathfrak{m}}{\mathcal{R}}(I)$.
These rings naturally appear in algebraic geometry in the context of blow-up
constructions, which are a fundamental tool in the study of singularities.

######  Proposition 4.3.

Let $I=\operatorname{Sss}(B)$ be an equigenerated, normal sssi (e.g., $I$
satisfies one of the conditions in 4.2). Then, the Rees ring
${\mathcal{R}}(I)$ and the special fiber ring ${\mathcal{F}}(I)$ are Cohen-
Macaulay normal domains. Moreover, ${\mathcal{F}}(I)$ has rational
singularities if $K$ is of characteristic 0, and is $F$-rational if $K$ is of
positive characteristic

###### Proof.

Since $R$ is a normal domain and $I$ is a normal ideal, ${\mathcal{R}}(I)$ is
normal by [HS06, Propositions 5.2.1 and 5.2.4]. Moreover, [HH02, Corollary
6.2] implies that ${\mathcal{F}}(I)$ is also normal. Since
${\mathcal{F}}(I)=[{\mathcal{R}}(I)]_{(0,*)}$ is a subring of
${\mathcal{R}}(I)$ and ${\mathcal{R}}(I)$ is a subring of the integral domain
$R[t]$, both ${\mathcal{R}}(I)$ and ${\mathcal{F}}(I)$ are integral domains.
Their Cohen-Macaulay property now follows from their normality, thanks to a
well-known theorem of Hochster (see for instance [HH11, Theorem B.6.2]).

By [Hoc72, Proposition 1], the normality of ${\mathcal{F}}(I)$ is equivalent
to ${\mathcal{F}}(I)$ being a direct summand of a polynomial ring. Thus by
[Bou87] it follows that ${\mathcal{F}}(I)$ has rational singularities if the
characteristic of $K$ is zero. If $K$ has positive characteristic
${\mathcal{F}}(I)$ is strongly $F$-regular, and in particular is $F$-rational
by [HH89]. ∎

In Section 6 we will investigate the algebraic structure of the Rees ring and
special fiber ring of a symmetric strongly shifted ideal, in relation with the
geometry of algebraic toric varieties. In particular, 6.13 will describe
interesting geometric consequences of the normality property of
${\mathcal{R}}(I)$ for a principal Borel sssi $I$. In the rest of this section
and in Section 5, we instead use 4.3 to analyze the asymptotic behavior of the
ordinary powers of a sssi.

An ideal $L\subseteq I$ is called a _reduction_ of $I$ if
$\overline{L}=\overline{I}$; equivalently, if there exists an integer $r$ so
that $LI^{r}=I^{r+1}$. If this is the case, it follows that $LI^{k}=I^{k+1}$
for every $k\geq r$. Hence, the reductions of an ideal $I$ give information on
the growth of powers of $I$. A reduction $L$ of $I$ is called a _minimal
reduction_ if it is minimal with respect to inclusion. The smallest integer
$r$ so that $LI^{r}=I^{r+1}$ for a minimal reduction $L$ of $I$ is called the
reduction number of $I$ with respect to $L$, and is denoted by $r_{L}(I)$. The
_reduction number_ of $I$ is

$r(I)=\min\\{r_{L}(I):L\text{ is a minimal reduction of }I\\}.$

The reduction number of a normal symmetric strongly shifted ideal can be
estimated as follows.

######  Corollary 4.4.

Let $I\subseteq R=K[x_{1},\ldots,x_{n}]$ be equigenerated, normal sssi (e.g.,
$I$ satisfies either one of the assumptions in 4.2) and assume that $K$ is an
infinite field. Then $r(I)\leq n-1$, and $r(I)=0$ if
$I=\operatorname{Sss}(\\{(a,\ldots,a)\\})$ for some $a\in{\mathbb{N}}$.

###### Proof.

By 4.3 ${\mathcal{R}}(I)$ is Cohen-Macaulay, whence [JK95, Theorem 2.3]
implies that $r(I)\leq\ell(I)-1$. The claim now follows from 3.29. ∎

While the reductions of a monomial ideal need not be monomial, in [Sin07,
Proposition 2.1] Singla showed that every monomial ideal $I$ admits a unique
minimal monomial reduction, that is, a reduction which is monomial and
contains no other monomial reduction of $I$. We next determine the unique
monomial reduction of a principal Borel sssi.

######  Proposition 4.5.

For $\lambda\in P_{n}$ define the monomial ideal generated by the
$\mathfrak{S}_{n}$-orbit of $x^{\lambda}$ as

$L(\lambda)=\left(\sigma(\lambda)\mid\sigma\in\mathfrak{S}_{n}\right).$

Then $L=L(\lambda)$ is the minimal monomial reduction of
$\operatorname{Sss}(\\{\lambda\\})$ and we have
$\operatorname{Sss}(\\{\lambda\\})=\overline{L}$.

###### Proof.

Let $I=\operatorname{Sss}(\\{\lambda\\})$. In [Sin07, Proposition 2.1 and
Remark 1.3] it is shown that the exponents for the monomial generators of the
minimal monomial reduction of $I$ correspond to the vertices of
$NP(I)=\operatorname{np}(I)+\mathbb{R}^{n}_{\geq 0}$, which coincide with the
vertices of $\operatorname{np}(I)$, as the vertices of a Minkowski sum are
obtained as sums of vertices from each summand. Therefore, we only need to
show that $V=\\{\sigma(\lambda)\mid\sigma\in\mathfrak{S}_{n}\\}$ is the set of
vertices of $\operatorname{np}(I)$. The latter form a subset of $V$, because
$\operatorname{np}(I)={\mathbf{P}}(\lambda)$ by 3.16. Since $V$ forms an orbit
under the $\mathfrak{S}_{n}$ action on $\mathbb{R}^{n}$ and since
$\operatorname{np}(I)$ and hence its vertex set are
$\mathfrak{S}_{n}$-invariant, we conclude that $V$ coincides with the set of
vertices of $\operatorname{np}(I)$. Hence, $L$ is indeed the minimal monomial
reduction of $I$. Now, the fact that $L$ is a reduction of $I$ implies that
$\overline{L}=\overline{I}$, while $\overline{I}=I$ follows by 4.2(1),
finishing the proof. ∎

##  5\. Associated primes and primary decomposition

When studying the ordinary powers of an ideal $I$, one often wishes to
understand how the depths and associated primes of $I^{k}$ depend on the
exponent $k$. In turn, thanks to (2.2) and (2.3), this process sometimes gives
information on the symbolic powers of $I$.

For an ideal $I$ in a Noetherian ring, Brodmann [Bro79a, Bro79b] proved that,
for $k\gg 0$, $\operatorname{Ass}(I^{k})=\operatorname{Ass}(I^{k+1})$ and
$\operatorname{depth}(R/I^{k})=\operatorname{depth}(R/I^{k+1})$; also,
$\displaystyle{\lim_{k\to\infty}\operatorname{depth}(R/I^{k})\leq n-\ell(I)}$.
The smallest number $k_{0}$ for which equality holds is called the _index of
stability_ of $I$ and is denoted as $\operatorname{astab}(I)$;
$\operatorname{Ass}(I^{k_{0}})$ is called the _stable set of associated
primes_ and is denoted as $\operatorname{Ass}^{\infty}(I)$. Similarly, the
smallest integer so that
$\operatorname{depth}(R/I^{k})=\operatorname{depth}(R/I^{k+1})$ is called the
_index of depth stability_ of $I$, $\operatorname{dstab}(I)$.

An ideal $I$ is said to satisfy the _persistence property_ if
$\operatorname{Ass}(I^{k})\subseteq\operatorname{Ass}(I^{k+1})$ for every
$k\geq 1$.

For monomial ideals in a polynomial ring, various combinatorial techniques
have been used to study these notions by several authors; see, e.g., [HRV13,
FMS13, HV13, HQ15, KM19]. In [HQ15, Theorem 3.3], Herzog and Qureshi provide
bounds for the depths of powers of an arbitrary equigenerated monomial ideal
$I$ in terms of its linear relation graph $\Gamma=\Gamma(I)$; see (3.9) for
the definition of this graph. If $r$ and $s$ denote the number of vertices and
the number of connected components of $\Gamma$ respectively, they show that

$\operatorname{depth}(R/I^{k})\leq n-k-1\,\text{ for }\,1\leq k\leq r-s,$
(5.1)

as long as $r-s\geq 1$. Using theirs result and our understanding of the
linear relation graph of an equigenerated ssi from 3.32, we next determine the
depths of powers of ideals in this class.

######  Proposition 5.1.

Let $I$ be an equigenerated symmetric shifted ideal of a polynomial ring in
$n$ variables. Then,
$\operatorname{depth}(R/I^{k})\geq\operatorname{depth}(R/I^{k+1})$ for all
$k\geq 1$. Moreover

  1. (1)

if $I=\operatorname{Sss}(\\{(a,\ldots,a)\\})$ for some $a\in{\mathbb{N}}$,
then $\operatorname{depth}(R/I^{k})=n-1$ for all $k\geq 1$, and
$\operatorname{dstab}(I)=1$;

  2. (2)

otherwise, $\operatorname{depth}(R/I^{k})\leq n-k-1$ for $1\leq k\leq n-1$ and
$\operatorname{depth}(R/I^{k})=0$ for $k\geq n-1$; in particular,
$\operatorname{dstab}(I)\leq n-1$ and
${\mathfrak{m}}\in\operatorname{Ass}(I^{k})$ for every $k\geq n-1$.

###### Proof.

Since equigenerated symmetric shifted ideals have a linear free resolution by
[BdAG+20, Theorem 3.2], it follows from [HRV13, Proposition 2.2] that

$\operatorname{depth}(R/I^{k})\geq\operatorname{depth}(R/I^{k+1})\text{ for
all }k\geq 1.$ (5.2)

If $I=\operatorname{Sss}(\\{(a,\ldots,a)\\})$ for some $a\in{\mathbb{N}}$,
then for all $k\geq 1$ one has $I=\operatorname{Sss}(\\{(ka,\ldots,ka)\\})$ by
3.11(2). Hence, for all $k\geq 1$,
$\operatorname{depth}(R/I^{k})=n-\operatorname{pd}(R/I^{k})=n-1$, where the
latter equality follows from 3.27. This implies that
$\operatorname{dstab}(I)=1$.

Assume now that $I\neq\operatorname{Sss}(\\{(a,\ldots,a)\\})$ for any
$a\in{\mathbb{N}}$. Then, by 3.32 the linear relation graph $\Gamma$ of $I$ is
connected, with $V(\Gamma)=\\{1,\ldots,n\\}$. Thus, (5.1) implies that
$\operatorname{depth}(R/I^{k})\leq n-k-1$ for $1\leq k\leq n-1$. In
particular, $\operatorname{depth}(R/I^{n-1})=0$, whence from (5.2) we obtain
that $\operatorname{depth}(R/I^{k})=0$ for all $k\geq n-1$. The remaining
claims now follow immediately (see also [HQ15, Corollary 3.4]). ∎

With a different method, in [HQ15, Theorem 1.3] Herzog and Qureshi also prove
that an equigenerated graded ideal $I$ satisfies the persistence property if
$I^{k+1}:I=I^{k}$ for all $k\geq 1$. By work of Ratliff [Rat76, Proposition
4.7] the latter condition is satisfied whenever $I$ is a normal ideal. 4.2
then implies the following result for symmetric strongly shifted ideals that
either have exactly one partition Borel generators, or are the symmetrization
of a lex-segment ideal in the sense of 3.8.

######  Proposition 5.2.

Let $I=\operatorname{Sss}(B)$ be an equigenerated, normal symmetric strongly
shifted ideal (e.g. $I$ satisfies one of the assumptions in 4.2). Then

  1. (1)

For every $k\geq 1$, $I^{k+1}:I=I^{k}$.

  2. (2)

For every $k\geq 1$,
$\operatorname{Ass}(I^{k})\subseteq\operatorname{Ass}(I^{k+1})$ for every
$k\geq 1$. That is, $I$ satisfies the persistence property.

  3. (3)

$\displaystyle{\lim_{k\to\infty}\operatorname{depth}(R/I^{k})=n-\ell(I)=\begin{cases}0&\text{
if }\lambda\neq(a,\ldots,a)\text{ for any }a\in{\mathbb{N}},\\\ n-1&\text{ if
}\lambda=(a,\ldots,a)\text{ for some }a\in{\mathbb{N}}.\end{cases}}$

###### Proof.

Since $I$ is a normal ideal, [Rat76, Proposition 4.7] implies (1), whence
[HQ15, Theorem 1.3] follows from (2). Moreover, since the Rees ring
${\mathcal{R}}(I)$ is Cohen-Macaulay by 4.3, a well-known result of Huneke
implies that equality holds in Brodmann’s inequality
$\displaystyle{\lim_{k\to\infty}\operatorname{depth}(R/I^{k})\leq n-\ell(I)}$
(see, e.g., [HH11, Proposition 10.3.2]). This implies the first equality in
(3), whence the second equality follows from 3.29. ∎

While the techniques used to prove [HQ15, Theorem 1.3] do not seem to apply to
arbitrary equigenerated symmetric strongly shifted ideals, we currently do not
know of any sssi failing to satisfy 5.2(1). This motivates the following
questions.

######  Question 5.3.

Is it true that $I^{k+1}:I=I^{k}$ for every $k\geq 1$ if $I$ is an arbitrary
symmetric strongly shifted ideal? Does any sssi satisfy the persistence
property?

###  5.1. Stable associated primes of a principal Borel sssi

When $I$ is a principal Borel sssi, its polymatroidal nature complements the
information provided by 5.1 and 5.2, allowing to estimate the indices of
stability and depth stability of $I$.

######  Proposition 5.4.

Let $I=\operatorname{Sss}(\\{\lambda\\})$ be a principal Borel symmetric
strongly shifted ideal of a polynomial ring in $n$ variables. Then

  1. (1)

$\operatorname{dstab}(I)=1$ if and only if either $\lambda=(a,\ldots,a)$ for
some $a\in{\mathbb{N}}$, or $\lambda\neq(a,\ldots,a)$ for any
$a\in{\mathbb{N}}$ and
${\mathfrak{m}}=(x_{1},\ldots,x_{n})\in\operatorname{Ass}(I)$.

  2. (2)

$\operatorname{dstab}(I)\leq\operatorname{astab}(I)\leq n-1$, provided $n\geq
2$.

###### Proof.

From 5.1 it follows that, for all $k\geq 1$,

$\operatorname{depth}(R/I)\geq\operatorname{depth}(R/I^{k})\geq\lim_{k\to\infty}\operatorname{depth}(R/I^{k}).$
(5.3)

Then, $\operatorname{dstab}(I)=1$ if and only if
$\operatorname{depth}(R/I)=\lim_{k\to\infty}\operatorname{depth}(R/I^{k})$. By
5.2(3), the latter equality is equivalent to the condition that
${\mathfrak{m}}\in\operatorname{Ass}(I)$ if $\lambda\neq(a,\ldots,a)$ for any
$a\in{\mathbb{N}}$, and follows from 5.1(1) otherwise. This proves (1).

To prove (2), assume first that $\lambda\neq(a,\ldots,a)$ for any
$a\in{\mathbb{N}}$. Then, from the persistence property and (5.3) it follows
that

$\operatorname{dstab}(I)=\min\\{k:\operatorname{depth}(R/I^{k})=0\\}=\min\\{k:{\mathfrak{m}}\in\operatorname{Ass}(I^{k})\\}.$

Hence, 5.1(2) implies that
$\operatorname{dstab}(I)\leq\operatorname{astab}(I)\leq n-1$ (see also [KM19,
Lemma 2.20] and [HQ15, Theorem 4.1]). Finally, if $\lambda=(a,\ldots,a)$ for
some $a\in{\mathbb{N}}$, then $I$ is a transversal polymatroidal ideal, whence
$\operatorname{astab}(I)=\operatorname{dstab}(I)=1\leq n-1$ by [HRV13,
Corollaries 4.6 and 4.14]. ∎

The inequality $\operatorname{dstab}(I)\leq\operatorname{astab}(I)$ in 5.4(2)
is remarkable, as for an arbitrary monomial ideal $I$ either of the integers
$\operatorname{astab}(I)$ and $\operatorname{dstab}(I)$ might be smaller than
the other; see [HRV13, p. 295]. Moreover, it is known that
$\operatorname{dstab}(I)=\operatorname{astab}(I)$ if $I$ is a transversal
polymatroidal ideal [HRV13, Corollaries 4.6 and 4.14], an ideal of Veronese
type [HRV13, Corollary 5.7], or a polymatroidal ideal with the strong exchange
property [KM19, Proposition 2.15]. Since these classes include principal Borel
ideals and ordinary powers of square-free Veronese ideals, it then makes sense
to ask whether $\operatorname{dstab}(I)=\operatorname{astab}(I)$ for all
principal Borel symmetric strongly shifted ideals.

Although this is not true in general (see 5.6 below), in this section we prove
that several principal Borel sssi’s satisfy
$\operatorname{dstab}(I)=\operatorname{astab}(I)=1$. In fact, in 5.8 we
determine the stable set of associated primes for every principal Borel sssi.
As a preliminary result, we prove that every principal Borel sssi can be
decomposed as a finite intersection of symbolic powers of square-free sssi’s.

######  Proposition 5.5.

Let $\lambda=(\lambda_{1},\ldots,\lambda_{n})$ be a partition. For $1\leq
j\leq n$, let $P_{j}=(x_{1},\ldots,x_{j})$ and denote
$a_{j}=\sum_{i=1}^{j}\lambda_{i}$. Then, the principal Borel sssi
$\operatorname{Sss}(\\{\lambda\\})$ can be decomposed as

$\operatorname{Sss}(\\{\lambda\\})=\bigcap_{j=1}^{n}\left(\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(P_{j})^{a_{j}}\right)=\bigcap_{j=1}^{n}I_{n,j}^{(a_{j})}.$

###### Proof.

By 3.8 we know that
$\displaystyle{I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(\operatorname{Borel}(x^{\lambda}))}$.
Moreover, from [FMS13, Proposition 2.7 and Theorem 3.1] it follows that

$\operatorname{Borel}(x^{\lambda})=\prod_{j=1}^{n}P_{j}^{\lambda_{j}}=\bigcap_{j=1}^{n}P_{j}^{a_{j}},$

where $\displaystyle{a_{j}=\sum_{i:P_{i}\subseteq
P_{j}}\lambda_{i}}=\sum_{i=1}^{j}\lambda_{i}$. Therefore, we deduce the
identity

$\operatorname{Sss}(\\{\lambda\\})=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(\operatorname{Borel}(x^{\lambda}))=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma\left(\bigcap_{j=1}^{n}P_{j}^{a_{j}}\right)=\bigcap_{j=1}^{n}\left(\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(P_{j})^{a_{j}}\right).$

For every $j$ we can rewrite
$\displaystyle{\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(P_{j})^{a_{j}}=I_{n,j}^{(a_{j})}}$,
which proves the last equality. ∎

Notice that the first equality in 5.5 gives a (possibly redundant) primary
decomposition of $I=\operatorname{Sss}(\\{\lambda\\})$, since for every $j$
and every $\sigma\in\mathfrak{S}_{n}$ the ideals $\sigma(P_{j})^{a_{j}}$ are
$P_{j}$-primary. Since the powers of a principal Borel sssi are still
principal Borel sssi’s, 3.11 and 5.5 together imply that, for every integer
$k\geq 1$,

$I^{k}=\operatorname{Sss}(\\{\lambda\\})^{k}=\operatorname{Sss}(\\{k\lambda\\})=\bigcap_{j=1}^{n}I_{n,j}^{(ka_{j})}.$
(5.4)

In particular, as detailed in the proof of 5.7 below, for every $k\geq 1$, we
have
$\operatorname{Ass}(I^{k})\subseteq\\{\sigma(P_{1}),\ldots,\sigma(P_{n}):\sigma\in\mathfrak{S}_{n}\\}$,
and $\sigma(P_{j})\in\operatorname{Ass}(I^{k})$ if and only if
$I_{n,j}^{(ka_{j})}$ is needed in the decomposition given by (5.4).

######  Example 5.6.

Let $I=\operatorname{Sss}(\\{1,2,2,4,4\\})$. Irredundant primary
decompositions of $I$ and $I^{2}$ computed using Macaulay2 [GS] are the
following:

$I=I_{5,1}\cap I_{5,2}^{(3)}\cap I_{5,4}^{(9)}\cap I_{5,5}^{(13)},$
$I^{2}=I_{5,1}^{(2)}\cap I_{5,2}^{(6)}\cap I_{5,3}^{(10)}\cap
I_{5,4}^{(18)}\cap I_{5,5}^{(26)}.$

Thus $\operatorname{astab}(I)=2$ follows from (5.4) and the persistence
property 5.2(1). Moreover, $\operatorname{dstab}(I)=1$ by 5.2(3), since
${\mathfrak{m}}=I_{5,5}\in\operatorname{Ass}(I)$.

To determine $\operatorname{Ass}^{\infty}(I)$, $\operatorname{astab}(I)$ and
$\operatorname{dstab}(I)$ for an arbitrary principal Borel sssi, one needs to
be able to predict which components in (5.4) are irredundant in a systematic
way. We identify sufficient conditions in 5.7 below. Our proof relies on the
following fact from [BdAG+20, Proposition 4.1]): for all $j\geq 1$,

$\displaystyle P\left(I_{n,j}^{(ka_{j})}\right)$ $\displaystyle=\\{\mu\in
P_{n}:\sum_{i=1}^{j}\mu_{i}\geq ka_{j}\\},$ (5.5a)
$\displaystyle\Lambda\left(I_{n,j}^{(ka_{j})}\right)$ $\displaystyle=\\{\mu\in
P_{n}:\sum_{i=1}^{j}\mu_{i}=ka_{j}\text{ and }\mu_{i}=\mu_{j}\text{ for all
}i>j\\}.$ (5.5b)

######  Theorem 5.7.

Let $I=\operatorname{Sss}(\\{\lambda\\})$ and $j^{\prime}:=\min(\lambda)$.
Adopt the notation of 5.5. Then, for a fixed $k\geq 1$,

  1. (1)

all components $I_{n,j}^{(ka_{j})}$ with $j<j^{\prime}$ are redundant in
(5.4);

  2. (2)

$I_{n,j^{\prime}}^{(ka_{j^{\prime}})}$ is not redundant in (5.4) and moreover
$I^{(k)_{\operatorname{Min}}}=I_{n,j^{\prime}}^{(k\lambda_{j^{\prime}})}$;

  3. (3)

if $j>j^{\prime}$ and $\lambda_{j-1}<\lambda_{j}$ then the component
$I_{n,j}^{(ka_{j})}$ is not redundant in (5.4);

  4. (4)

if $\lambda_{1}<\lambda_{j}$ and either $k>j(j-1)$ or
$\displaystyle{\lambda_{j}>\frac{q+j-r}{k}}$ where
$k\sum_{i=1}^{j-1}\lambda_{i}=(j-1)q+r$ and $0\leq r\leq j-2$, then the
component $I_{n,j}^{(ka_{j})}$ is not redundant in (5.4) if and only if $j\geq
j^{\prime}$.

  5. (5)

if $\lambda_{1}=\lambda_{j}$ for some $j>1$ then the component
$I_{n,j}^{(ka_{j})}$ is redundant in (5.4).

In particular, for $k\gg 0$ a minimal primary decomposition of $I^{k}$ is

$I^{k}=\bigcap_{j\in\mathcal{J}}\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(P_{j})^{ka_{j}},$
(5.6)

where

$\mathcal{J}=\begin{cases}\\{j:j\geq j^{\prime}\\},&\text{ if }j^{\prime}>1\\\
\\{1\\}\cup\\{j:j>1\text{ and }\lambda_{j}\neq\lambda_{1}\\},&\text{ if
}j^{\prime}=1.\end{cases}$

###### Proof.

For simplicity of notation, in the rest of the proof we denote
$\sum_{s}\lambda:=\sum_{r=1}^{s}\lambda_{r}$ for any partition $\lambda\in
P_{n}$.

(1) For $j<j^{\prime}$ one has $a_{j}=0$, so the corresponding component
$I_{n,j}^{(ka_{j})}=R$ of the decomposition (5.4) is redundant.

(2) Since by 3.25 one has $\operatorname{ht}(I)=\min(\lambda)=j^{\prime}$, the
component $I_{n,j^{\prime}}^{(ka_{j^{\prime}})}$ is the intersection of the
primary ideals $\sigma(P_{j^{\prime}}^{ka_{j}^{\prime}})$ which belong to
minimal primes of $I^{k}$, that is,
$I^{(k)_{\operatorname{Min}}}=I_{n,j^{\prime}}^{(ka_{j^{\prime}})}$. It
remains to note that $a_{j^{\prime}}=\lambda_{j^{\prime}}$ by definition.

(3) We now establish irredundancy of the components $I_{n,j}^{(a_{j})}$ so
that $j\geq j^{\prime}$ and $\lambda_{j-1}<\lambda_{j}$. This amounts to
showing that $\bigcap_{i\neq j}I_{n,i}^{(ka_{i})}\not\subseteq
I_{n,j}^{(ka_{j})}$. Equation (5.5a) implies that for $\mu\in P_{n}$,
$x^{\mu}\in I_{n,i}^{(ka_{i})}$ if and only if $\sum_{i}\mu\geq
ka_{i}=k\sum_{i}\lambda$. Set

$\mu_{i}=\begin{cases}k\lambda_{i}&\text{ for }1\leq i<j\\\
k\lambda_{j-1}&\text{ for }i=j\\\
k(\lambda_{j+1}+\lambda_{j}-\lambda_{j-1})&\text{ for }i=j+1,\text{ provided
}j+1\leq n\\\ k\lambda_{i}&\text{ for }j+2\leq i\leq n.\end{cases}$

Then $\mu\in P_{n}$ and $x^{\mu}\in I_{n,i}^{(ka_{i})}$ for each $i\neq j$
since $\sum_{i}\mu=k\sum_{i}\lambda$. By contrast, we have
$\sum_{j}\mu<k\sum_{j}\lambda$ since $\lambda_{j-1}<\lambda_{j}$ and thus
$x^{\mu}\not\in I_{n,j}^{(ka_{j})}$. We conclude that the component
$I_{n,j}^{(ka_{j})}$ is not redundant in the decomposition (5.4).

(4) Now suppose $j>j^{\prime}$ and in particular $j>1$. We construct $\mu\in
P_{n}$ so that $x^{\mu}\in\bigcap_{i\neq j}I_{n,i}^{(ka_{i})}$. By (5.5a), it
must satisfy $\sum_{i}\mu\geq k\sum_{i}\lambda$ for each $i\neq j$. Set

$k\sum_{j-1}\lambda=(j-1)q+r\text{\, with \,}0\leq r\leq j-2$ (5.7)

and consider the partition given by

$\mu_{i}=\begin{cases}q+1&\text{ for }1\leq i\leq j\\\ N&\text{ for }j+1\leq
i\leq n,\end{cases}$ (5.8)

where $N\gg 0$ is a sufficiently large integer such that $N\geq q+1$ and
$j(q+1)+N(i-j)\geq k\sum_{i}\lambda$ for each $i>j$. By construction this
satisfies $\sum_{i}\mu\geq k\sum_{i}\lambda$ for $i>j$. If $p\in P_{n}$ and
$a,b\in{\mathbb{N}}$ with $a\leq b$, then

$\sum_{b}p\geq\sum_{a}p+(b-a)p_{a}\geq\sum_{a}p+\frac{b-a}{a}\sum_{a}p=\frac{b}{a}\sum_{a}p.$
(5.9)

For the $\mu$ defined in (5.8) and for $i<j$ we have

$\displaystyle\sum_{i}\mu$ $\displaystyle=$ $\displaystyle i(q+1)$
$\displaystyle=$ $\displaystyle
i\left[\frac{k\sum_{j-1}\lambda-r}{j-1}+1\right]\qquad\text{by
}\eqref{eq:j-1}$ $\displaystyle\geq$ $\displaystyle
k\sum_{i}\lambda+\frac{i(j-r-1)}{j-1}\qquad\text{by
}\eqref{eq:ij}(a=i,b=j-1,p=\lambda)$ $\displaystyle\geq$ $\displaystyle
k\sum_{i}\lambda\qquad\text{since }j-r-1\geq 0.$

Similarly we have

$\displaystyle\sum_{j}\mu$ $\displaystyle=$ $\displaystyle
j(q+1)=\frac{jk\sum_{j-1}\lambda}{j-1}+\frac{j(j-r-1)}{j-1}$ $\displaystyle=$
$\displaystyle
k\sum_{j-1}\lambda+k\left[\frac{\sum_{j-1}\lambda}{j-1}+\frac{j(j-r-1)}{k(j-1)}\right],$

whence $\sum_{j}\mu\geq k\sum_{j}\lambda$ if and only if

$\lambda_{j}\leq\frac{\sum_{j-1}\lambda}{j-1}+\frac{j(j-r-1)}{k(j-1)}=\frac{q+j-r}{k}.$
(5.10)

Thus, $x^{\mu}\in I_{n,j}^{(ka_{j})}$ if and only if (5.10) is satisfied.

Whenever the inequality (5.10) fails, we have $\bigcap_{i\neq
j}I_{n,i}^{(ka_{i})}\not\subseteq I_{n,j}^{(ka_{j})}$ and hence
$I_{n,j}^{(ka_{j})}$ is needed in (5.4). Assume $k>j(j-1)$. Then we see that

$\left\lfloor\frac{\sum_{j-1}\lambda}{j-1}+\frac{j(j-r-1)}{k(j-1)}\right\rfloor\leq\frac{\sum_{j-1}\lambda}{j-1}.$

If (5.10) holds, it implies $\lambda_{j}\leq\frac{\sum_{j-1}\lambda}{j-1}$.
But this is possible if and only if
$\lambda_{1}=\cdots=\lambda_{j-1}=\lambda_{j}$, in which case equality holds
in (5.10). Since by assumption $\lambda_{1}<\lambda_{j}$, (5.10) must then
fail, so $I_{n,j}^{(ka_{j})}$ is not redundant.

(5) If $\lambda_{1}=\lambda_{j}$ for some $j>1$, then
$\lambda_{1}=\cdots=\lambda_{j-1}=\lambda_{j}$, whence
$\lambda_{j}=\frac{\sum_{j-1}\lambda}{j-1}$. Let $\mu\in P_{n}$ be defined as
in (5.8). If $x^{\mu}\in I_{n,j-1}^{(ka_{j-1})}$, (5.9) with
$a=j-1,\,b=j,\,p=\mu$ then yields

$\sum_{j}\mu\geq\frac{j}{j-1}\sum_{j-1}\mu\geq\frac{j}{j-1}k\sum_{j-1}\lambda\geq
k\sum_{j}\lambda.$

This shows that $I_{n,j-1}^{(ka_{j-1})}\subseteq I_{n,j}^{(a_{j})}$ and thus
the latter ideal is redundant in (5.4).

The formula for the primary decomposition follows by substituting
$I_{n,j}^{(ka_{j})}=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(P_{j})^{ka_{j}}$
in (5.4) and removing redundant components. In detail,
$\sigma(P_{j})^{ka_{j}}$ is irredundant in

$I^{k}=\bigcap_{j=1}^{n}\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(P_{j})^{ka_{j}}$
(5.11)

if and only if $\sigma(P_{j})\in\operatorname{Ass}(I^{k})$. As $I^{k}$ is
symmetric, $P_{j}\in\operatorname{Ass}(I^{k})$ if and only if
$\sigma(P_{j})\in\operatorname{Ass}(I^{k})$ for all
$\sigma\in\mathfrak{S}_{n}$, so we see that the primary components
$\sigma(P_{j})^{ka_{j}}$ are either all simultaneously redundant in (5.11), in
which case $I_{n,j}^{(ka_{j})}$ is redundant in (5.4), or simultaneously
irredundant in (5.11), in which case $I_{n,j}^{(ka_{j})}$ is iredundant in
(5.4). Formula (5.6) follows from (5.11) by means of statements (1)–(5) and
the previous considerations provided $k>j(j-1)$ is satisfied for every $j$. ∎

Combining the previous results together, we can determine the stable set of
associated primes of a principal Borel sssi.

######  Theorem 5.8.

Let $I=\operatorname{Sss}(\\{\lambda\\})$. Adopt the notation of 5.5 where
$j^{\prime}=\min(\lambda)$. Then

$\operatorname{Ass}^{\infty}(I)=\begin{cases}\\{\sigma(P_{j}):j^{\prime}\leq
j\leq n,\,\sigma\in\mathfrak{S}_{n}\\},&\text{ if }j^{\prime}>1,\\\
\\{\sigma(P_{j}):j=1\text{ or }(j>1\text{ and
}\lambda_{j}\neq\lambda_{1}),\,\sigma\in\mathfrak{S}_{n}\\},&\text{ if
}j^{\prime}=1.\end{cases}$

Moreover,

  1. (1)

if either $\lambda=(a,\ldots,a)$ for some $a\in{\mathbb{N}}$, or $\lambda\in
P_{n}$ has no repeated parts other than possibly allowing for repetitions of
$\lambda_{1}$, then $\operatorname{astab}(I)=\operatorname{dstab}(I)=1$ and
for each $k\in{\mathbb{N}}$, $I^{k}=I^{(k)_{\operatorname{Ass}}}$;

  2. (2)

otherwise, setting $s=\max\\{j:\lambda_{1}<\lambda_{j-1}=\lambda_{j}\\}$ we
have
$\operatorname{dstab}(I)\leq\operatorname{astab}(I)\leq\min\\{n-1,s(s-1)+1\\}$.

###### Proof.

The claim regarding the stable set of associated primes follows from the
primary decomposition (5.6) in 5.7.

For (1), from 5.7 (3) and (5) it follows that, in either cases,
$\operatorname{Ass}(I)=\operatorname{Ass}(I^{k})$ for every $k\geq 1$, whence
$\operatorname{astab}(I)=1$. Moreover, $\operatorname{dstab}(I)=1$ by 5.4(1),
while the claim about the symbolic powers $I^{(k)_{\operatorname{Ass}}}$
follows from (2.4b).

For (2), notice that ${\mathfrak{m}}\in\operatorname{Ass}(I^{k})$ for all
$k>s(s-1)$ by 5.7 (3) and (4). Hence, the conclusion follows from the
persistence property and 5.4(2). ∎

######  Remark 5.9.

From 5.8 it follows that $\operatorname{astab}(I)=\operatorname{dstab}(I)=1$
for many principal Borel sssi’s. While this equality is known to hold for
transversal polymatroidal ideals by [HRV13, Corollaries 4.6 and 4.14], 3.24
shows that there exist principal Borel sssi’s with
$\operatorname{astab}(I)=\operatorname{dstab}(I)=1$ which are not transversal.
Hence, 5.8 produces examples of monomial ideals with
$\operatorname{astab}(I)=\operatorname{dstab}(I)=1$ which were not previously
included in [HRV13, Corollaries 4.6 and 4.14].

######  Remark 5.10.

It follows from (2.4b) that, for an ideal $I$, the equalities
$I^{k}=I^{(k)_{\operatorname{Ass}}}$ for $1\leq k\leq\operatorname{astab}(I)$
imply that $I^{k}=I^{(k)_{\operatorname{Ass}}}$ for all $k\geq 1$.

In particular, by 5.2(2) and 5.4(2), a principal Borel symmetric strongly
shifted ideal satisfies $I^{k}=I^{(k)_{\operatorname{Ass}}}$ for all $k\geq 1$
if and only if $I^{k}=I^{(k)_{\operatorname{Ass}}}$ for $1\leq k\leq n-1$.
While the latter statement is true for every polymatroidal ideal by [HQ15,
Theorem 4.1], 5.8 shows that for a principal Borel sssi it suffices to check
equality of powers and symbolic powers in a potentially smaller range, for
$1\leq k\leq\min\\{n-1,s(s-1)+1\\}$. This addresses a question of Huneke,
which is open for arbitrary monomial ideals.

In [HV13], Herzog and Vladoiu define a monomial ideal to be of _intersection
type_ if it can be decomposed as an intersection of powers of monomial primes.
Principal Borel sssi’s satisfy this property by 5.5. In fact, in [HV13,
Proposition 2.1] it is shown that every polymatroidal ideal is of intersection
type. Exploiting this fact, in [HV13, Corollary 3.5] the authors show that for
every polymatroidal ideal $I$ generated in degree $d$, one has
$I^{(dk)_{{\operatorname{Min}}}}\subseteq I^{k}$ for any $k\geq 1$. In
particular, this applies to principal Borel sssi’s. Our contribution below is
to strengthen this containment.

######  Proposition 5.11.

Let $I=\operatorname{Sss}(\lambda)$ be a principal Borel sssi, with
$d=|\lambda|$ and $c=\min(\lambda)$. Then,
$I^{(m)_{{\operatorname{Min}}}}\subseteq I^{k}$ whenever $m/k\geq
d/\lambda_{c}$.

###### Proof.

Recall from 5.7 (2) that
$I^{(m)_{{\operatorname{Min}}}}=I_{n,c}^{(\lambda_{c}m)}$. Let
$\mu\in\mathrm{\Lambda}(I_{n,c}^{(\lambda_{c}m)})$ and assume that $m/k\geq
d/\lambda_{c}$, i.e., $\lambda_{c}m\geq dk$. Then, by (5.5b) for each $j\geq
c$ we have

$\sum_{i=1}^{j}\mu_{i}=\sum_{i=1}^{c}\mu_{i}+\sum_{i=c+1}^{j}\mu_{i}\geq\lambda_{c}m+(j-c)\frac{\lambda_{c}m}{c}=\frac{\lambda_{c}mj}{c}\geq\frac{dkj}{c}\geq
dk\geq\left(\sum_{i=1}^{j}\lambda_{i}\right)k,$

since $d=\sum_{i=1}^{n}\lambda_{i}$. The above inequality shows in view of
(5.5b) that $I_{n,c}^{(\lambda_{c}m)}\subseteq I_{n,j}^{(a_{j}k)}$ for each
$c\leq j\leq n$ and for $a_{j}=\sum_{i=1}^{j}\lambda_{i}$. It follows by 5.5
that $I_{n,c}^{(\lambda_{c}m)}\subseteq I^{k}$. ∎

The above is particularly relevant to the Containment Problem, which asks for
an ideal $I$ to determine the pairs $m,k$ so that $I^{(m)}\subseteq I^{k}$.
This is an important and well studied problem in commutative algebra, which is
open in its full generality. We refer the reader to [CEHH17, GH19, MnNnB21]
for known results in the case of monomial ideals.

###  5.2. The intersection property of a principal Borel sssi

The decomposition formula in 5.5 mimics analogous decompositions for principal
Borel ideals or transversal polymatroidal ideals; see [FMS13, Theorem 3.1] and
[HRV13, Corollary 4.10]. Inspired by these results, in [BC17], Bruns and Conca
defined an ideal $I$ to be $P$-adically closed if

$I=\bigcap_{P\in\operatorname{Ass}(I)}P^{(v(P))},\text{ where
}v(P)=\max\\{t:I\subseteq P^{(t)}\\}.$

Since the associated primes of monomial ideals are generated by a subset of
the variables, that is, a regular sequence, one obtains that a monomial ideal
$I$ is $P$-adically closed if

$I=\bigcap_{P\in\operatorname{Ass}(I)}P^{v(P)},\text{ where
}v(P)=\max\\{t:I\subseteq P^{t}\\}.$ (5.12)

It follows from 5.5 that principal Borel sssi’s are $P$-adically closed, since
for each $j$ the exponent $a_{j}$ coincides with $v(\sigma(P_{j}))$ for every
$\sigma\in\mathfrak{S}_{n}$.

We now seek to characterize $P$-adically closed symmetric monomial ideals.

######  Proposition 5.12.

A symmetric monomial ideal is $P$-adically closed if and only if it can be
decomposed as

$I=\bigcap_{i=1}^{t}I_{n,c_{i}}^{(v_{i})},$

where $v_{i}=v(x_{1},\ldots,x_{c_{i}})$. Moreover, all such ideals $I$ are
symmetric strongly shifted.

###### Proof.

It is clear from the definition that an ideal admitting such a decomposition
is $P$-adically closed. Moreover, since each $I_{n,c_{i}}^{(v_{i})}$ is
symmetric strongly shifted by [BdAG+20, Theorem 4.3], then $I$ is symmetric
strongly shifted by 2.1.

To prove the converse, we first show that a $P$-adically closed symmetric
monomial ideal $I$ that is height-unmixed with $\operatorname{ht}(I)=c$ is a
symbolic power ideal of a square-free Veronese ideal. This is because
$\operatorname{Ass}(I)$ is closed under the action of $\mathfrak{S}_{n}$ and
the monomial primes of height equal to $\operatorname{ht}(I)$ form a single
orbit under the action of $\mathfrak{S}_{n}$. Moreover the symmetry of $I$
yields that for $P=(x_{1},\ldots,x_{c})$ and each $\sigma\in\mathfrak{S}_{n}$

$v(P)=\max\\{t:I\subseteq
P^{t}\\}=\max\\{t:I\subseteq\sigma(P)^{t}\\}=v(\sigma(P)).$

Thus equations (5.12) and (2.1) yield for $I$ P-adically closed, symmetric and
height-unmixed

$I=\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(x_{1},\ldots,x_{c})^{v}=\left(\bigcap_{\sigma\in\mathfrak{S}_{n}}\sigma(x_{1},\ldots,x_{c})\right)^{(v)}=I_{n,c}^{(v)}$
(5.13)

with $v=v(x_{1},\ldots,x_{c})$.

Next, suppose that $I$ is a $P$-adically closed symmetric monomial ideal that
is not necessarily height-unmixed, with associated primes of distinct heights
$c_{1},\ldots,c_{t}$. Observe that for each $i$ the ideal
$\displaystyle{\,\bigcap_{P\in\operatorname{Ass}(I):\,\operatorname{ht}(P)=c_{i}}P^{v(P)}}$
is a $P$-adically closed symmetric ideal that is height-unmixed. Hence,
equation (5.13) implies that setting $v_{i}=v(x_{1},\ldots,x_{c_{i}})$ we
obtain

$I=\bigcap_{i=1}^{t}\left(\bigcap_{P\in\operatorname{Ass}(I):\,\operatorname{ht}(P)=c_{i}}P^{v(P)}\right)=\bigcap_{i=1}^{t}I_{n,c_{i}}^{(v_{i})}.$

∎

Comparing the previous result with 5.5, it would be natural to ask wether a
$P$-adically closed sssi must be a principal Borel sssi. However, the
following example shows that this is not necessarily true. This is in sharp
contrast with the case of strongly stable ideals, which can only be written as
intersections of powers of monomial prime ideals if they are principal Borel;
see [HV13, Proposition 2.8].

######  Example 5.13.

The ideal $I=I_{5,1}\cap I_{5,3}^{(4)}$ is $P$-adically closed by 5.12 but it
has partition Borel generating set $B(I)=\\{(1,1,3,3,3),(1,2,2,2,2)\\}$.

We conclude this section by stating another interesting consequence of 5.5.
Recall that an ideal $I$ is said to be _sequentially Cohen-Macaulay_ if there
exists a filtration of $R$-modules

$D_{0}=0\subsetneq D_{1}\subsetneq\ldots\subsetneq D_{s}=R/I$

so that, for all $1\leq i\leq s$, $\dim(D_{i-1})<\dim(D_{i})$ and the quotient
modules $C_{i}=D_{i}/D_{i-1}$ are Cohen-Macaulay $R$-modules. In particular, a
Cohen-Macaulay ideal is sequentially Cohen-Macaulay, as it suffices to
construct the $D_{i}$’s by going modulo a maximal regular sequence, one
element at a time. The following result can be interpreted as a generalization
of the fact that ideals of monomial star configurations are Cohen-Macaulay
[GHM13, Proposition 2.9].

######  Proposition 5.14.

Let $I$ be a principal Borel sssi. Then, $I$ is sequentially Cohen-Macaulay.

###### Proof.

By 5.5, $I$ can be decomposed as an intersection of symbolic powers of
squarefreeVeronese ideals (also known as ideals defining monomial star
configurations). The conclusion now follows from [LS22, Proposition 3.1]. ∎

##  6\. Toric ideals and Rees algebras

Given a finite set of monomials $G=\\{m_{1},\ldots,m_{s}\\}\subset R$, the
toric ring of $G$ is the subring $K[G]=K[m_{1},\ldots,m_{s}]$ of $R$. Let
$A=K[T_{1},\ldots,T_{s}]$ denote a polynomial ring in $s$ new indeterminates
over K and define a surjective homomorphism $\pi:A\twoheadrightarrow K[G]$ by
$\pi(T_{i})=m_{i}$. The toric ideal of $G$ is ${\mathcal{I}}_{G}=\ker(\pi)$.
In other words, the toric ideal of $G$ is the defining ideal of the toric ring