1 Introduction
Schubert polynomials are classical and important objects in algebraic combinatorics. They were introduced by Lascoux and Schützenberger [Reference Lascoux and Schützenberger17] to study the cohomology classes of Schubert varieties. Since then, Schubert polynomials have played a fundamental role in algebraic combinatorics (see, e.g., [Reference Fink, Mészáros and Dizier8, Reference Monical, Tokcan and Yong19, Reference Bergeron and Billey1, Reference Billey, Jockusch and Stanley2, Reference Knutson and Miller13, Reference Knutson and Miller14, Reference Lam, Lee and Shimozono16] and the references therein).
We first recall the definition of Schubert polynomials. Let
$\mathscr {S}_p$
be the symmetric group on the set
$[p] = \{1, \ldots , p\}$
. For every
$i\in [p-1] = \{1,\ldots ,p-1\}$
, we have the transposition
$\sigma _i=(i,i+1)\in \mathscr {S}_p$
. Recall that the set
$\mathfrak {T} = \{ \sigma _i \mid 1\leq i< p \}$
generates
$\mathscr {S}_p$
. The length
$\ell (\pi )$
of a permutation
$\pi $
is the least amount of elements in
$\mathfrak {T}$
counting repetitions needed to obtain
$\pi $
from the identity permutation. The permutation
$\pi _0=(p,p-1,\ldots ,2,1)$
(in one-line notation) is the longest permutation and has length
$\frac {p(p-1)}{2}$
. We follow the notation of [Reference Knutson and Miller14] and [Reference Miller and Sturmfels18, Chapter 15] to present permutations and Schubert polynomials.
Definition 1.1. The double Schubert polynomial
${\mathfrak {S}}_\pi ({ \mathbf {t}}, { \mathbf {s}}) \in \mathbb {Z}[t_1,\ldots ,t_p,s_1,\dots ,s_p]$
of a permutation
$\pi \in \mathscr {S}_p$
is defined recursively in the following way. First, we define
${\mathfrak {S}}_{\pi _0}=\prod _{i+j\leq p}(t_i-s_j)$
, and for any permutation
$\pi $
and transposition
$\sigma _i$
with
$\ell (\sigma _i\pi )<\ell (\pi )$
we set
$$\begin{align*}{\mathfrak{S}}_{\sigma_i\pi} \,=\, \dfrac{{\mathfrak{S}}_{\pi}-\sigma_i{\mathfrak{S}}_{\pi}}{t_i-t_{i+1}}, \end{align*}$$
where
$\mathscr {S}_p$
acts only on
$ \mathbb {Z}[t_1,\ldots ,t_p]$
by permutation of variables. The (ordinary) Schubert polynomial
${\mathfrak {S}}_\pi ({ \mathbf {t}}, \mathbf {0}) \in \mathbb {Z}[t_1,\ldots ,t_p]$
is obtained from
${\mathfrak {S}}_\pi $
by setting each variable
$s_j$
equal to
$0$
.
The monomial expansion of ordinary Schubert polynomials has been combinatorially analyzed using different objects such as compatible sequences [Reference Billey, Jockusch and Stanley2], reduced pipe dreams [Reference Fomin and Kirillov9, Reference Bergeron and Billey1] and Kohnert diagrams [Reference Kohnert15]. The description using pipe dreams also works for the double Schubert polynomials [Reference Miller and Sturmfels18, Corollary 16.30]. We also have a formula for double Schubert polynomials using bumpless pipe dreams [Reference Lam, Lee and Shimozono16].
Following [Reference Monical, Tokcan and Yong19], we say that a polynomial
$f = \sum _{{ \mathbf {n}}}c_{{ \mathbf {n}}}{ \mathbf {x}}^{ \mathbf {n}}\in \mathbb {Z}[x_1,\ldots ,x_n]$
has the saturated Newton polytope property (SNP property for short) if the support
${ \mathrm {supp}}(f)=\{{ \mathbf {n}}\in \mathbb {N}^n\mid c_{ \mathbf {n}} \neq 0\}$
of f is equal to
$\mathrm {Newton}(f)\cap \mathbb {N}^n$
, where
$\mathrm {Newton}(f) = \mathrm {ConvexHull}\{{ \mathbf {n}}\in \mathbb {N}^n\mid c_{ \mathbf {n}} \neq 0\}$
denotes the Newton polytope of f; in other words, if the support of f consists of the integer points of a polytope.
The main goal of this paper is to confirm the following challenging conjecture by Monical, Tokcan and Yong that appeared in [Reference Monical, Tokcan and Yong19, Conjecture 5.2].
Conjecture 1.2 [Reference Monical, Tokcan and Yong19].
Double Schubert polynomials have the SNP property.
We confirm the conjecture by proving a stronger result that the support of each double Schubert polynomial is a discrete polymatroid. A discrete polymatroid
$\mathcal {P}$
on
$[n]=\{1,\ldots ,n\}$
is a collection of points in
$ \mathbb {N}^n$
of the following form
$$\begin{align*}\mathcal{P=}\Big\{(x_1,\ldots, x_n)\in \mathbb{N}^n \;\mid\; \sum_{j\in{\mathfrak{J}}} x_j\leq r({\mathfrak{J}}), \;\forall {\mathfrak{J}}\subsetneq [n], \;\sum_{i\in [n]} x_i=r([n])\Big\} \end{align*}$$
with r being a rank function on
$[n]$
. A rank function on
$[n]$
is a function
$r : 2^{[n]}\rightarrow \mathbb {N}$
satisfying the following three properties: (i)
$r(\emptyset ) = 0$
, (ii)
$r({\mathfrak {J}}_1)\leq r({\mathfrak {J}}_2)$
if
${\mathfrak {J}}_1\subseteq {\mathfrak {J}}_2 \subseteq [n]$
and (iii)
$r({\mathfrak {J}}_1\cap {\mathfrak {J}}_2)+r({\mathfrak {J}}_1\cup {\mathfrak {J}}_2)\leq r({\mathfrak {J}}_1)+r({\mathfrak {J}}_2)$
if
${\mathfrak {J}}_1,{\mathfrak {J}}_2 \subseteq [n]$
.
The following is the main theorem of this article.
Theorem A. Let
$\pi \in \mathscr {S}_p$
be a permutation and
${\mathfrak {S}}_\pi ({ \mathbf {t}}, { \mathbf {s}}) \in \mathbb {Z}[t_1,\ldots ,t_p,s_1,\dots ,s_p]$
be the corresponding double Schubert polynomial. Then, the support
${ \mathrm {supp}}({\mathfrak {S}}_\pi ) \subset \mathbb {N}^{2p}$
of
${\mathfrak {S}}_\pi $
is a discrete polymatroid on
$[2p]=\{1,\ldots ,2p\}$
. In particular, the statement of Conjecture 1.2 holds.
Our approach to prove Theorem A can be summarized in the following quote by Miller and Sturmfels [Reference Miller and Sturmfels18, Introduction to Chapter 15]: ‘We consider the finest possible multigrading, which demands the refined toolkit of a new generation of combinatorialists’. More precisely, we utilize the result that double Schubert polynomials equal the multidegree polynomial of Schubert determinantal ideals with the aforementioned ‘finest possible multigrading’ (see [Reference Miller and Sturmfels18, Theorem 15.40]), and then we develop a method of standardization of ideals. This process of standardization allows us to study multidegrees in certain nonstandard multigradings by reducing the problem to a standard multigraded setting. Our main tool is Theorem 2.2 from [Reference Castillo, Cid-Ruiz, Li, Montaño and Zhang3] which shows that the support of the multidegree polynomial of any multihomogeneous prime ideal (with usual standard multigrading) is a discrete polymatroid. Here, we extend this theorem to the family of nonstandard multigradings that we study.
Much interest has been paid to the important conjectures proposed in [Reference Monical, Tokcan and Yong19] and a number of them have already been confirmed (see [Reference Fink, Mészáros and Dizier8]). Therefore, Theorem A settles a hitherto remaining conjecture from [Reference Monical, Tokcan and Yong19] and gives further evidence to the ubiquity of the SNP property in many ‘combinatorially defined polynomials’. Theorem A also gives more evidence for the presence of the Lorentzian property in double Schubert polynomials as conjectured in [Reference Huh, Matherne, Mészáros and St. Dizier12].
The structure of the paper is as follows. We review the notion of multidegrees in Section 2 and recall the connection between double Schubert polynomials and Schubert determinantal ideals in Section 3. Section 4 contains our main results, in particular the proof of Theorem A.
2 A short recap on multidegrees
In this short section, we briefly recall the notion of multidegrees and some of its basic properties; for more details the reader is referred to [Reference Miller and Sturmfels18, Reference Cid-Ruiz5].
Let
$\mathbb {k}$
be a field and
$R = \mathbb {k}[x_1,\ldots ,x_n]$
be a
$ \mathbb {Z}^p$
-graded polynomial ring (for now, we do not assume the grading to be positive). Let M be a finitely generated
$ \mathbb {Z}^p$
-graded module and
$F_\bullet $
be a
$ \mathbb {Z}^p$
-graded free R-resolution
$ F_\bullet : \; \cdots \rightarrow F_i \rightarrow F_{i-1} \rightarrow \cdots \rightarrow F_1 \rightarrow F_0 $
of M. Let
$t_1,\ldots ,t_p$
be variables over
$ \mathbb {Z}$
and consider the polynomial ring
$ \mathbb {Z}[{ \mathbf {t}}] = \mathbb {Z}[t_1,\ldots ,t_p]$
, where the variable
$t_i$
corresponds with the i-th elementary vector
${ \mathbf {e}}_i \in \mathbb {Z}^p$
. If we write
$F_i = \bigoplus _{j} R(-\mathbf {b}_{i,j})$
with
$\mathbf {b}_{i,j} = (\mathbf {b}_{i,j,1},\ldots ,\mathbf {b}_{i,j,p}) \in \mathbb {Z}^p$
, then we define the Laurent polynomial
$ \left [F_i\right ]_{ \mathbf {t}} \, := \, \sum _{j} { \mathbf {t}}^{\mathbf {b}_{i,j}} = \sum _{j} t_1^{\mathbf {b}_{i,j,1}} \cdots t_p^{\mathbf {b}_{i,j,p}}. $
Then, the K-polynomial of M is defined by
$$ \begin{align*}\mathcal{K}(M;{\mathbf{t}}) \, := \, \sum_{i} {(-1)}^i \left[ F_i \right]_{\mathbf{t}}. \end{align*} $$
It turns out that, even if the grading of R is nonpositive and we do not have a well-defined notion of Hilbert series, the above definition of K-polynomial is an invariant of the module M and it does not depend on the chosen free R-resolution
$F_\bullet $
(see [Reference Miller and Sturmfels18, Theorem 8.34]).
Definition 2.1. The multidegree polynomial of a finitely generated
$ \mathbb {Z}^p$
-graded R-module M is the homogeneous polynomial
$\mathcal {C}(M; { \mathbf {t}}) \in \mathbb {Z}[{ \mathbf {t}}]$
given as the sum of all terms in
$$ \begin{align*} \mathcal{K}(M; \mathbf{1} - {\mathbf{t}}) = \mathcal{K}(M; 1-t_1,\ldots,1-t_p) \end{align*} $$
having total degree
${ \mathrm {codim}}(M) = n - \dim (M)$
.
One case of particular interest is when R is a standard multigraded polynomial ring. We say that R is standard
$ \mathbb {Z}^p$
-graded if the total degree of each variable
$x_i$
is equal to one (i.e., for each
$1 \le i \le n$
, we have
$\deg (x_i) = { \mathbf {e}}_{k_i} \in \mathbb {Z}^p$
with
$1 \le k_i \le p$
). The study of standard multigraded algebras is of utmost importance as they correspond with closed subschemes of a product of projective spaces (see, e.g., [Reference Castillo, Cid-Ruiz, Li, Montaño and Zhang3] and the references therein). Since the coefficients of the multidegree polynomial are nonnegative in the standard multigraded case, it becomes natural to address the positivity of these coefficients. For each subset
$\mathfrak {J} = \{j_1,\ldots ,j_k\} \subseteq [p] = \{1, \ldots , p\}$
, denote by
$R_{({\mathfrak {J}})}$
the
$ \mathbb {Z}^k$
-graded
$\mathbb {k}$
-algebra given by
$$ \begin{align*}R_{({\mathfrak{J}})} := \bigoplus_{\substack{i_1\ge 0,\ldots, i_p\ge 0\\ i_{j} = 0 \mathrm{ if } j \not\in {\mathfrak{J}}}} {\left[R\right]}_{(i_1,\ldots,i_p)}, \end{align*} $$
and for any R-homogeneous ideal
$I \subset R$
we define
$I_{({\mathfrak {J}})}$
as the contraction
$I_{({\mathfrak {J}})} := I \cap R_{({\mathfrak {J}})}$
. The following theorem completely characterizes the positivity of multidegrees and is our main tool to prove Theorem A.
Theorem 2.2 [Reference Castillo, Cid-Ruiz, Li, Montaño and Zhang3].
Let
$R=\mathbb {k}[x_1,\ldots ,x_n]$
be a standard
$ \mathbb {Z}^p$
-graded polynomial ring. Let
$I \subset R$
be an R-homogeneous prime ideal. Write the multidegree polynomial of
$\mathcal {C}(R/I;{ \mathbf {t}})$
as
$$ \begin{align*} \mathcal{C}(R/I;{\mathbf{t}}) \,=\, \sum_{\substack{{\mathbf{n}} \in \mathbb{N}^p \\ |{\mathbf{n}}| = {\mathrm{codim}}(I)}} c_{{\mathbf{n}}} {\mathbf{t}}^{\mathbf{n}} \; \in \; \mathbb{N}[t_1,\ldots, t_p]. \end{align*} $$
Then, for all
${ \mathbf {n}} = (n_1,\ldots ,n_p) \in \mathbb {N}^p$
with
$|{ \mathbf {n}}| = { \mathrm {codim}}(I) = n - \dim (R/I)$
, we have that
$c_{ \mathbf {n}}> 0$
if and only if for each
${\mathfrak {J}} = \{j_1,\ldots ,j_k\} \subseteq [p]$
the inequality
$ n_{j_1} + \cdots + n_{j_k} \, \ge \, { \mathrm {codim}}\left (I_{({\mathfrak {J}})}\right ) $
holds. Furthermore, the support of
$\mathcal {C}(R/I; { \mathbf {t}})$
is a discrete polymatroid.
Proof. Consider the standard
$ \mathbb {Z}^{p}$
-graded polynomial ring
$R' = R[x_{n+1},\ldots ,x_{n+p}]$
with
$\deg (x_{n+i}) = { \mathbf {e}}_i \in \mathbb {Z}^p$
, and notice that
$\mathcal {C}(R/I;{ \mathbf {t}}) = \mathcal {C}(R'/IR';{ \mathbf {t}})$
. Thus, we assume that I is a relevant prime (i.e.,
$I \not \supset \bigoplus _{i_1\ge 1,\ldots , i_p\ge 1} {\left [R\right ]}_{(i_1,\ldots ,i_p)}$
), and so
$ \mathrm {MultiProj}(R/I) \neq \emptyset $
. We embed
$X = \mathrm {MultiProj}(R/I)$
as a closed subscheme of a multiprojective space
${ \mathbb {P}}:={ \mathbb {P}}_{\mathbb {k}}^{m_1} \times _{\mathbb {k}} \cdots \times _{\mathbb {k}} { \mathbb {P}}_{\mathbb {k}}^{m_p}$
. From [Reference Castillo, Cid-Ruiz, Li, Montaño and Zhang3, Remark 2.9], we have that
${ \mathbf {n}} \in { \mathrm {supp}}(\mathcal {C}(R/I;{ \mathbf {t}}))$
if and only if
$\deg _{ \mathbb {P}}^{{ \mathbf {m}}-{ \mathbf {n}}}(X)>0$
where
${ \mathbf {m}} - { \mathbf {n}}=(m_1-n_1,\ldots ,m_p-n_p)$
. Then, [Reference Castillo, Cid-Ruiz, Li, Montaño and Zhang3, Theorem A] implies that
${ \mathbf {n}} \in { \mathrm {supp}}(\mathcal {C}(R/I;{ \mathbf {t}}))$
if and only if
$|{ \mathbf {n}}| = { \mathrm {codim}}(I)$
and
$\sum _{j \in {\mathfrak {J}} }n_{j} \ge { \mathrm {codim}}\left (I_{({\mathfrak {J}})}\right )$
for each
${\mathfrak {J}} \subseteq [p]$
. Equivalently, we obtain that
${ \mathbf {n}} \in { \mathrm {supp}}(\mathcal {C}(R/I;{ \mathbf {t}}))$
if and only if
$|{ \mathbf {n}}| = { \mathrm {codim}}(I)$
and
$$ \begin{align*} \sum_{j \in {\mathfrak{J}} }n_{j} \,\le\, {\mathrm{codim}}(I) - {\mathrm{codim}}\left(I_{([p] \setminus {\mathfrak{J}})}\right) \,=\, \sum_{j \in {\mathfrak{J}}} m_j + r([p] \setminus {\mathfrak{J}}) - r([p]) \end{align*} $$
for each
${\mathfrak {J}} \subseteq [p]$
, where
$r : 2^{[p]} \rightarrow \mathbb {N}$
is the rank function
$r({\mathfrak {J}}) := \dim \big ( \mathrm {MultiProj}\big (R_{({\mathfrak {J}})}/I_{({\mathfrak {J}})}\big )\big )$
(see [Reference Castillo, Cid-Ruiz, Li, Montaño and Zhang3, Proposition 5.1]). Finally, we can check that
$s : 2^{[p]} \rightarrow \mathbb {N}$
with
$s({\mathfrak {J}}) := \sum _{j \in {\mathfrak {J}}} m_j + r([p] \setminus {\mathfrak {J}}) - r([p])$
is a rank function (see, e.g., [Reference Schrijver20, §44.6f]), and so it follows that
${ \mathrm {supp}}(\mathcal {C}(R/I;{ \mathbf {t}}))$
is a polymatroid.
3 Schubert determinantal ideals
Here, we recall the connection between double Schubert polynomials and Schubert determinantal ideals (for more details, the reader is referred to [Reference Miller and Sturmfels18, Chapters 15, 16]).
First, we define matrix Schubert varieties and Schubert determinantal ideals by following [Reference Miller and Sturmfels18, Chapter 15]. Let
$\mathbb {k}$
be an algebraically closed field and
$M_p(\mathbb {k})$
be the
$\mathbb {k}$
-vector space of
$p\times p$
matrices with entries in
$\mathbb {k}$
. As an affine variety we define its coordinate ring as
$\widetilde {R} =\mathbb {k}[x_{i,j} \mid (i,j)\in [p]\times [p]]$
. Furthermore, we consider a
$( \mathbb {Z}^p\oplus \mathbb {Z}^p)$
-grading on
$\widetilde {R}$
by setting
$\deg (x_{i,j}) = { \mathbf {e}}_i \oplus -{ \mathbf {e}}_j \in \mathbb {Z}^p \oplus \mathbb {Z}^p$
, where
${ \mathbf {e}}_i \in \mathbb {Z}^p$
denotes the i-th elementary vector.
Definition 3.1 (see [Reference Miller and Sturmfels18, Chapter 15]).
Let
$\pi $
be a permutation matrix. The matrix Schubert variety
$\overline {X_\pi }\subset M_p(\mathbb {k})$
is the subvariety given by
$ \overline {X_\pi }=\{Z\in M_p(\mathbb {k})\mid \mathrm {rank}(Z_{m\times n})\leq \mathrm {rank}(\pi _{m\times n}) \; \mathrm { for all }\; m,n\}, $
where
$Z_{m\times n}$
is the restriction to the first m rows and n columns. The Schubert determinantal ideal
$I_\pi \subset \widetilde {R}$
is the
$\widetilde {R}$
-homogeneous ideal generated by all minors in
$\mathbf {X}_{m \times n}$
of size
$1 + \mathrm {rank}(\pi _{m \times n})$
for all m and n, where
$\mathbf {X} = (x_{i,j})$
is the
$p\times p$
matrix with the variables of
$\widetilde {R}$
.
The following theorem collects several results of fundamental importance to our approach. In particular, it shows that double Schubert polynomials equal the multidegree polynomial of matrix Schubert varieties. To define multidegrees over
$\widetilde {R}$
with its
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-grading, we consider the polynomial ring
$ \mathbb {Z}[{ \mathbf {t}},{ \mathbf {s}}] = \mathbb {Z}[t_1,\ldots ,t_p,s_1,\ldots ,s_p]$
, where
$t_i$
has degree
${ \mathbf {e}}_i \oplus \mathbf {0} \in \mathbb {Z}^p \oplus \mathbb {Z}^p$
and
$s_i$
has degree
$\mathbf {0} \oplus { \mathbf {e}}_i \in \mathbb {Z}^p \oplus \mathbb {Z}^p$
.
Theorem 3.2. Let
$\pi \in \mathscr {S}_p$
be a permutation, and denote also by
$\pi $
the corresponding permutation matrix. Then, the following statements hold:
-
1.
$I_\pi $
is a prime ideal, and so it coincides with the ideal
$I(\overline {X_\pi })$
of polynomials vanishing on the matrix Schubert variety
$\overline {X_\pi }$
. ([Reference Fulton10], [Reference Miller and Sturmfels18, Corollary 16.29]) -
2.
$\widetilde {R}/I_\pi $
is a Cohen–Macaulay ring. ([Reference Fulton10], [Reference Miller and Sturmfels18, Corollary 16.44]) -
3.
${\mathfrak {S}}_\pi ({ \mathbf {t}}, \mathbf {s}) = \mathcal {C}(\widetilde {R}/I_\pi; { \mathbf {t}}, { \mathbf {s}})$
. ([Reference Fehérand and Rimányi7],[Reference Knutson and Miller14], [Reference Miller and Sturmfels18, Theorem 15.40])
The next technical lemma will allow us to substitute the grading of
$\widetilde {R}$
which has negative components for the degrees of the variables. Let
$R =\mathbb {k}[x_{i,j} \mid (i,j) \in [p]\times [p]]$
with induced
$( \mathbb {Z}^p\oplus \mathbb {Z}^p)$
-grading by setting
$\deg (x_{i,j}) = { \mathbf {e}}_i \oplus { \mathbf {e}}_j \in \mathbb {Z}^p \oplus \mathbb {Z}^p$
. As for
$\widetilde {R}$
, define multidegrees over R in the polynomial ring
$ \mathbb {Z}[{ \mathbf {t}}, { \mathbf {s}}]$
.
Lemma 3.3. Let
$I \subset \widetilde {R}$
be an
$\widetilde {R}$
-homogeneous ideal, and denote also by I the corresponding R-homogeneous ideal in R. Then we have
$ \mathcal {C}(\widetilde {R}/I; t_1,\ldots ,t_p, s_1,\ldots ,s_p) = \mathcal {C}(R/I; t_1,\ldots , t_p, -s_1,\ldots ,-s_p). $
Proof. Notice that, if
$\widetilde {F}_\bullet $
is a
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-graded free
$\widetilde {R}$
-resolution of
$\widetilde {R}/I$
with
$\widetilde {F}_i = \bigoplus _{j} \widetilde {R}(-\mathbf {a}_{i,j},-\mathbf {b}_{i,j})$
, then there is a corresponding
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-graded free R-resolution
$F_\bullet $
of
$R/I$
with
$F_i = \bigoplus _{j} R(-\mathbf {a}_{i,j},\mathbf {b}_{i,j})$
. By definition, this yields the equality of K-polynomials
$$ \begin{align*} \mathcal{K}(\widetilde{R}/I;{\mathbf{t}},{\mathbf{s}}) = \mathcal{K}(\widetilde{R}/I;t_1,\ldots,t_p,s_1,\ldots,s_p) = \mathcal{K}(R/I;t_1,\ldots,t_p,s_1^{-1},\ldots,s_p^{-1}) = \mathcal{K}(R/I;{\mathbf{t}},{\mathbf{s}}^{\mathbf{-1}}). \end{align*} $$
From [Reference Miller and Sturmfels18, Claim 8.54], we have
$\mathcal {K}(R/I;\mathbf {1-t},\mathbf {1-s}) = \mathcal {C}(R/I;{ \mathbf {t}},{ \mathbf {s}}) + Q({ \mathbf {t}},{ \mathbf {s}})$
, where
$Q({ \mathbf {t}},{ \mathbf {s}})$
is a polynomial with terms of degree at least
${ \mathrm {codim}}(I) + 1$
. Equivalently, we get
$\mathcal {K}(R/I;{ \mathbf {t}},{ \mathbf {s}}) = \mathcal {C}(R/I;\mathbf {1-t},\mathbf {1-s}) + Q(\mathbf {1-t},\mathbf {1-s})$
. It then follows that
$$ \begin{align*} \mathcal{K}(\widetilde{R}/I;\mathbf{1-t},\mathbf{1-s}) = \mathcal{C}(R/I; t_1,\ldots,t_p,1-\tfrac{1}{1-s_1},\ldots,1-\tfrac{1}{1-s_p}) + Q(t_1,\ldots,t_p,1-\tfrac{1}{1-s_1},\ldots,1-\tfrac{1}{1-s_p}). \end{align*} $$
By expanding the right-hand side of the above equality, the result of the lemma is obtained.
4 Standardization of ideals
In this section, we develop a process of standardization of ideals in a certain nonstandard multigrading. This process will allow us to show that the support of the multidegree polynomial of any Cohen–Macaulay prime ideal is a discrete polymatroid in the nonstandard multigradings that we consider. The following setup is used throughout this section.
Setup 4.1. Let
$p \ge 1$
be a positive integer and
$\mathbb {k}$
be a field. Let R and S be the polynomial rings
$R=\mathbb {k}[{ \mathbf {x}}]$
and
$S=\mathbb {k}[\mathbf {w}, \mathbf {z}]$
over the set of variables
${ \mathbf {x}} = \{x_{i,j}\}_{1\le i,j\le p}$
,
$\mathbf {w} = \{w_{i,j}\}_{1\le i,j\le p}$
and
$\mathbf {z} = \{z_{i,j}\}_{1\le i,j\le p}$
. We consider R and S as
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-graded rings by setting that
$$ \begin{align*} \deg(x_{i,j}) = {\mathbf{e}}_i \oplus {\mathbf{e}}_j, \quad \deg(w_{i,j}) = {\mathbf{e}}_i \oplus \mathbf{0} \quad \mathrm{and} \quad \deg(z_{i,j}) = \mathbf{0} \oplus {\mathbf{e}}_j, \end{align*} $$
where
${ \mathbf {e}}_i \in \mathbb {Z}^p$
denotes the i-th elementary vector and
$\mathbf {0} \in \mathbb {Z}^p$
denotes the zero vector. We define the
$\mathbb {k}$
-algebra homomorphism
$$ \begin{align*} \phi: R=\mathbb{k}[{\mathbf{x}}] \longrightarrow S = \mathbb{k}[\mathbf{w}, \mathbf{z}], \quad \phi(x_{i,j}) = w_{i,j}z_{i,j}. \end{align*} $$
For an R-homogeneous ideal
$I \subset R$
, we say that the extension
$\phi (I) S$
is the standardization of I, as
$\phi (I) S$
is an S-homogeneous ideal in the standard multigraded polynomial ring S. Let
${ \mathbf {t}} = \{t_1,\ldots ,t_p\}$
and
${ \mathbf {s}} = \{s_1,\ldots ,s_p\}$
be variables indexing the
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-grading, where
$t_i$
corresponds with
${ \mathbf {e}}_i \oplus \mathbf {0} \in \mathbb {Z}^p \oplus \mathbb {Z}^p$
and
$s_i$
corresponds with
$\mathbf {0} \oplus { \mathbf {e}}_i \in \mathbb {Z}^p \oplus \mathbb {Z}^p$
. Given a finitely generated graded R-module M and a finitely generated graded S-module N, by a slight abuse of notation, we consider both multidegrees
$\mathcal {C}(M;{ \mathbf {t}},{ \mathbf {s}})$
and
$\mathcal {C}(N;{ \mathbf {t}},{ \mathbf {s}})$
as elements of the same polynomial ring
$ \mathbb {Z}[{ \mathbf {t}}, { \mathbf {s}}]= \mathbb {Z}[t_1,\ldots ,t_p,s_1,\ldots ,s_p]$
.
First, we show some basic properties of the process of standardization.
Proposition 4.2. Assume Setup 4.1. Let
$I \subset R$
be an R-homogeneous ideal and
$J = \phi (I)S$
be its standardization. Then, the following statements hold:
-
(i)
${ \mathrm {codim}}(I) = { \mathrm {codim}}(J)$
. -
(ii)
$\mathcal {C}(R/I;{ \mathbf {t}},{ \mathbf {s}}) = \mathcal {C}(S/J;{ \mathbf {t}},{ \mathbf {s}})$
. -
(iii) If
$R/I$
is a Cohen–Macaulay ring, then
$S/J$
also is. -
(iv) Let
$>$
be a monomial order on R and
$>'$
be a monomial order on S which is compatible with
$\phi $
(i.e., if
$f,g \in R$
with
$f> g$
, then
$\phi (f)>' \phi (g)$
). Then
${ \mathrm {in}}_{>'}(J) = \phi ({ \mathrm {in}}_{>}(I))S$
.
Proof. Let T be the polynomial ring
$T=\mathbb {k}[{ \mathbf {x}},\mathbf {w},\mathbf {z}] \cong R \otimes _{\mathbb {k}} S$
with its natural
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-grading induced from the ones of R and S. We now think of R and S as subrings of T. Consider the quotient ring
$T/IT$
, and notice that
$\{x_{i,j} - w_{i,j}z_{i,j}\}_{1\le i,j \le p}$
is a regular sequence of homogeneous elements over
$T/IT$
. We also have the following natural isomorphism
$$ \begin{align*} \frac{T}{IT + \left(\{x_{i,j} - w_{i,j}z_{i,j}\}_{1\le i,j \le p}\right)} \;\cong\; S/J. \end{align*} $$
As the natural inclusion
$R \hookrightarrow T$
is a polynomial extension, we have that
$\dim (T/IT) = \dim (R/I) + \dim (S) = \dim (R) + \dim (S) - { \mathrm {codim}}(I)$
and that
$T/IT$
is Cohen–Macaulay when
$R/I$
is. So, by cutting out with the regular sequence described above, we obtain that
$ \dim (S/J) = \dim (T/IT) - \dim (R) = \dim (S) - { \mathrm {codim}}(I) $
and that
$S/J$
is Cohen–Macaulay when
$T/IT$
is. This completes the proofs of parts (i) and (iii).
Let
$F_\bullet : \cdots \xrightarrow {f_2} F_1 \xrightarrow {f_1} F_0$
be a graded free R-resolution of
$R/I$
. Since
$\{x_{i,j} - w_{i,j}z_{i,j}\}_{1\le i,j \le p}$
is a regular sequence on both T and
$T/IT$
, it follows that
${ \mathrm {Tor}}_k^T\left (T/IT, T/(\{x_{i,j} - w_{i,j}z_{i,j}\}_{1\le i,j \le p})\right ) = 0$
for all
$k> 0$
, and so
$G_\bullet = F_\bullet \otimes _R T/(\{x_{i,j} - w_{i,j}z_{i,j}\}_{1\le i,j \le p})$
provides (up to isomorphism) a graded free S-resolution of
$S/J$
. The identification of
$G_\bullet $
as a resolution of S-modules is the same as
$\phi (F_\bullet )$
(more precisely,
$G_\bullet $
has the same shiftings as
$F_\bullet $
in the
$( \mathbb {Z}^p \oplus \mathbb {Z}^p)$
-grading and the i-th differential matrix of
$G_\bullet $
is given by the substitution
$\phi (f_i)$
of
$f_i$
). Therefore, by definition, we obtain the equality
$\mathcal {C}(R/I;{ \mathbf {t}},{ \mathbf {s}}) = \mathcal {C}(S/J;{ \mathbf {t}},{ \mathbf {s}})$
that shows part (ii).
To show part (iv) we can use Buchberger’s algorithm (see, e.g., [Reference Eisenbud6, Chapter 15]). Indeed, we can perform essentially the same steps of the algorithm in a set of generators of I and the corresponding set of generators for J.
The following theorem provides the main result of this section. It shows that the support of the multidegree polynomial is a discrete polymatroid for Cohen–Macaulay prime ideals in R. The proof is carried out by performing a standardization process that allows us to invoke Theorem 2.2.
Theorem 4.3. Assume Setup 4.1. Let
$I \subset R$
be an R-homogeneous Cohen–Macaulay prime ideal. Then, the support of the multidegree polynomial
$\mathcal {C}(R/I;{ \mathbf {t}}, { \mathbf {s}})$
is a discrete polymatroid.
Proof. Let
$\mathcal {L} = \{ (i,j) \mid x_{i,j} \in I \}$
be the set of indices such that the corresponding variable belongs to I. We consider the polynomial rings
$ R' = \mathbb {k}[x_{i,j} \mid (i,j) \not \in \mathcal {L}] \subset R $
and
$ S' = \mathbb {k}[w_{i,j}, z_{i,j} \mid (i,j) \not \in \mathcal {L}] \subset S. $
Let
$I' \subset R'$
be the (unique) ideal that satisfies the condition
$I = I'R + \left (x_{i,j} \mid (i,j) \in \mathcal {L}\right )$
. By construction, we have that
$x_{i,j} \not \in I'$
for all
$x_{i,j} \in R'$
. Since
$R/I \cong R'/I'$
, it follows that
$I'$
is also a Cohen–Macaulay prime ideal.
Let
$J' = \phi (I'R)S \cap S'$
. For any
$w_{i,j}z_{i,j} \in S'$
, Proposition 4.2(i) and the fact that the corresponding
$x_{i,j}$
does not belong to the prime
$I'$
imply that
$$ \begin{align*} {\mathrm{codim}}(J'S+w_{i,j}z_{i,j}S) = {\mathrm{codim}}(I'R + x_{i,j}R) = {\mathrm{codim}}(I'R) + 1 = {\mathrm{codim}}(J'S) + 1. \end{align*} $$
By Proposition 4.2(iii),
$S/J'S$
is Cohen–Macaulay, and so it necessarily follows that
$w_{i,j}z_{i,j}$
is a nonzerodivisor over
$S/J'S$
for all
$w_{i,j}z_{i,j} \in S'$
. Consequently, we obtain that
$S'/J'$
is a domain if and only if
$\left (S'/J'\right )_{\prod w_{i,j}z_{i,j}}$
is a domain. Let
$B = \mathbb {k}[w_{i,j},z_{i,j},z_{i,j}^{-1} \mid (i,j) \not \in \mathcal {L}]$
, and consider the automorphism given by
$$ \begin{align*} \psi : B \rightarrow B, \quad w_{i,j} \mapsto \frac{w_{i,j}}{z_{i,j}}, \quad z_{i,j} \mapsto {z_{i,j}}. \end{align*} $$
The ideal
$\psi (J'B)$
coincides with the extension of
$I'$
in B under the ring homomorphism
$R' \rightarrow B, x_{i,j} \mapsto w_{i,j}$
, and so it follows that
$\psi (J'B)$
and, consequently,
$J'B$
are prime ideals. We then conclude that
$J'$
is a prime ideal.
Since the variables
$x_{i,j}$
with indices in
$\mathcal {L}$
form a regular sequence over
$R/I'R$
, we obtain the equation
$$ \begin{align} \mathcal{C}(R/I;{\mathbf{t}}, {\mathbf{s}}) = \prod_{(i,j) \in \mathcal{L}} (t_i+s_j) \cdot \mathcal{C}(R/I'R;{\mathbf{t}}, {\mathbf{s}}) \end{align} $$
(see, e.g., [Reference Miller and Sturmfels18, Exercise 8.12]). To conclude the proof, it is now sufficient to show that the support of
$\mathcal {C}(R/I'R;{ \mathbf {t}}, { \mathbf {s}})$
is a discrete polymatroid; indeed, we would obtain that the support of
$\mathcal {C}(R/I;{ \mathbf {t}}, { \mathbf {s}})$
is a Minkowski sum of a finite number of discrete polymatroids which in turn is also a discrete polymatroid by [Reference Schrijver20, Corollary 46.2c]. Finally, this condition follows by applying Theorem 2.2 to the prime ideal
$J'S$
and exploiting the equality
$\mathcal {C}(R/I'R;{ \mathbf {t}}, { \mathbf {s}}) = \mathcal {C}(S/J'S;{ \mathbf {t}}, { \mathbf {s}})$
from Proposition 4.2(ii).
We are now ready to prove the main result of this paper.
Proof of Theorem A.
As we already have all the necessary ingredients, the proof follows straightforwardly by combining Theorem 3.2, Lemma 3.3 and Theorem 4.3.
Furthermore, we determine the defining inequalities of the discrete polymatroids in Theorem 4.3 and, accordingly, in Theorem A. Similarly to Section 2, for any two subsets
${\mathfrak {J}}_1, {\mathfrak {J}}_2 \subseteq [p]$
, we denote by
$R_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)} \subseteq R$
and
$S_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)} \subseteq S$
the
$( \mathbb {Z}^{|{\mathfrak {J}}_1|} \oplus \mathbb {Z}^{|{\mathfrak {J}}_2|})$
-graded
$\mathbb {k}$
-algebras obtained by restricting to the positions in
${\mathfrak {J}}_1$
for the first part
$ \mathbb {Z}^p \oplus \mathbf {0}$
of the grading, and to the ones in
${\mathfrak {J}}_2$
for the second part of the grading
$\mathbf {0} \oplus \mathbb {Z}^p$
.
Theorem 4.4. Assume Setup 4.1. Let
$I\subset R$
be an R-homogeneous Cohen–Macaulay prime ideal. Then, we have that the coefficient of
$\mathbf {t}^{\mathbf {r}}\mathbf {s}^{\mathbf {c}} = t_1^{r_1}\cdots t_p^{r_p} s_1^{c_1}\cdots s_p^{c_p}$
is nonzero in
$\mathcal {C}(R/I;{ \mathbf {t}}, { \mathbf {s}})$
if and only if
-
(i)
$\sum _{j \in [p]} r_j+ \sum _{j \in [p]} c_j = { \mathrm {codim}}(I)$
-
(ii) For every
$\mathfrak {J}_1,\mathfrak {J}_2\subseteq [p]$
, we have that
$\sum _{j\in \mathfrak {J}_1}r_j+\sum _{j\in \mathfrak {J}_2}c_j\geq { \mathrm {codim}} \big (I_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}\big )$
, where
$I_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}$
is the contracted ideal
$I_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)} = I \cap R_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}$
.
Proof. By Theorem 4.3, we know that the Newton polytope of
$\mathcal {C}(R/I;{ \mathbf {t}}, { \mathbf {s}})$
is a base polymatroid polytope, and so, under the condition
$\sum _{j \in [p]} r_j+ \sum _{j \in [p]} c_j = { \mathrm {codim}}(I)$
, all its defining inequalities are of the form
$$ \begin{align} \sum_{j\in\mathfrak{J}_1}r_j+\sum_{j\in\mathfrak{J}_2}c_j \geq C({\mathfrak{J}}_1,{\mathfrak{J}}_2), \end{align} $$
for some constant
$C({\mathfrak {J}}_1,{\mathfrak {J}}_2)$
that depends on the subsets
${\mathfrak {J}}_1,{\mathfrak {J}}_2 \subseteq [p]$
. We now determine
$C({\mathfrak {J}}_1,{\mathfrak {J}}_2)$
.
We keep the same notation of the proof of Theorem 4.3, in particular,
$I = I'R + \left (x_{i,j} \mid (i,j) \in \mathcal {L}\right )$
. Equation (1) decomposes the Newton polytope of
$\mathcal {C}(R/I;{ \mathbf {t}}, { \mathbf {s}})$
as the Minkowski sum of the Newton polytopes of
$\prod _{(i,j) \in \mathcal {L}} (t_i+s_j)$
and
$\mathcal {C}(S/J'S;{ \mathbf {t}}, { \mathbf {s}})$
, both of which are also base polymatroid polytopes. So, we analyze the minimum of the sum in Equation (2) with the two contributions.
-
1.
$\textrm {Newton}(\prod _{(i,j) \in \mathcal {L}} (t_i+s_j))$
is determined by the equality
$\sum _{j \in [p]} r_j+ \sum _{j \in [p]} c_j = |\mathcal {L}|$
and the inequalities
$\sum _{j\in \mathfrak {J}_1}r_j+\sum _{j\in \mathfrak {J}_2}c_j \ge \big |\{(i,j)\in \mathcal {L} \mid i\in {\mathfrak {J}}_1 \mathrm { and } j\in {\mathfrak {J}}_2\}\big |$
. -
2. Due to Theorem 2.2,
$\textrm {Newton}(\mathcal {C}(S/J'S;{ \mathbf {t}}, { \mathbf {s}}))$
is determined by the equality
$\sum _{j \in [p]} r_j+ \sum _{j \in [p]} c_j = { \mathrm {codim}}(J'S)$
and the inequalities
$\sum _{j\in \mathfrak {J}_1}r_j+\sum _{j\in \mathfrak {J}_2}c_j \ge { \mathrm {codim}} \big ({(J'S)}_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}\big )$
, where
${(J'S)}_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}$
is the contracted ideal
${(J'S)}_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)} = J'S \cap S_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}$
.
Notice that Proposition 4.2(i) yields the equality
$$\begin{align*}{\mathrm{codim}} \big(I_{({\mathfrak{J}}_1,{\mathfrak{J}}_2)}\big) = {\mathrm{codim}} \big(J_{({\mathfrak{J}}_1,{\mathfrak{J}}_2)}\big) = {\mathrm{codim}} \big({(J'S)}_{({\mathfrak{J}}_1,{\mathfrak{J}}_2)}\big) + \big|\{(i,j)\in\mathcal{L} \mid i\in{\mathfrak{J}}_1 \textrm{ and } j\in{\mathfrak{J}}_2\}\big|. \end{align*}$$
Therefore, since we can split the value of
$C({\mathfrak {J}}_1,{\mathfrak {J}}_2)$
in terms of the sum of the defining inequalities of the Newton polytopes of
$\prod _{(i,j) \in \mathcal {L}} (t_i+s_j)$
and
$\mathcal {C}(S/J'S;{ \mathbf {t}}, { \mathbf {s}})$
, it follows that
$C({\mathfrak {J}}_1,{\mathfrak {J}}_2) = { \mathrm {codim}} \big (I_{({\mathfrak {J}}_1,{\mathfrak {J}}_2)}\big )$
. This concludes the proof of the theorem.
Finally, we perform a simple computation out of the six possible permutations in
$\mathscr {S}_3$
(see [Reference Miller and Sturmfels18, Examples 15.4, 15.42]).
Example 4.5 (
$p = 3$
and
$\pi = (1, 3, 2)$
).
The Schubert determinantal ideal and the double Schubert polynomials are given by
$I_{132} = \left (x_{1,1}x_{2,2}-x_{1,2}x_{2,1}\right )$
and
${\mathfrak {S}}_{132} = t_1 + t_2 - s_1 - s_2$
. The standardization of
$I_{132}$
is the ideal
$J = \left (w_{1,1}w_{2,2}z_{1,1}z_{2,2}-w_{1,2}w_{2,1}z_{1,2}z_{2,1}\right ) \in S$
. The ideal J is prime and S has a standard
$( \mathbb {Z}^3 \oplus \mathbb {Z}^3)$
-grading. One can compute that
$$ \begin{align*} \mathcal{C}(S/J; {\mathbf{t}}, {\mathbf{s}}) = t_1+t_2+s_1+s_2 \end{align*} $$
(see [Reference Miller and Sturmfels18, Exercise 8.12], or just utilize the built-in command multidegree on the computer algebra system Macaulay2 [Reference Grayson and Stillman11]). Coinciding with the claim of Theorem 2.2, the support of
$\mathcal {C}(S/J; { \mathbf {t}}, { \mathbf {s}})$
is a discrete polymatroid. Notice that
${\mathfrak {S}}_{132} = t_1 + t_2 - s_1 - s_2 = \mathcal {C}(S/J; { \mathbf {t}}, -{ \mathbf {s}})$
, as shown by Lemma 3.3.
Remark 4.6. From the conjectures stated by Monical, Tokcan and Yong [Reference Monical, Tokcan and Yong19], a remaining open one is to show that Grothendieck polynomials also satisfy the SNP property (see [Reference Monical, Tokcan and Yong19, Conjecture 5.5]). In [Reference Castillo, Cid-Ruiz, Mohammadi and Montaño4], we settled a particular case of this conjecture. More precisely, we showed that the support of a Grothendieck polynomial is a generalized polymatroid when the Schubert polynomial is zero-one (see [Reference Castillo, Cid-Ruiz, Mohammadi and Montaño4, Theorem B]).
Acknowledgements
F.C. thanks Ghent University for their hospitality. We thank the reviewer for helpful comments and suggestions.
Competing interest
The authors have no competing interest to declare.
Funding statement
F.C. was partially supported by FONDECYT Grant 1221133. Y.C.R. was partially supported by an FWO Postdoctoral Fellowship (1220122N). F.M. was partially supported by FWO grants (G023721N, G0F5921N), the KU Leuven iBOF/23/064 grant and the UiT Aurora MASCOT project. J.M. was partially supported by NSF Grant DMS #2001645/2303605.
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