# Polarizations and Hook Partitions

Ayah Almousa  Cornell University<EMAIL_ADDRESS>http://math.cornell.edu/~aalmousa and  Keller VandeBogert  University of South
Carolina<EMAIL_ADDRESS>http://people.math.sc.edu/kellerlv

###### Abstract.

In this paper, we relate combinatorial conditions for polarizations of powers
of the graded maximal ideal with rank conditions on submodules generated by
collections of Young tableaux. We apply discrete Morse theory to the
hypersimplex resolution introduced by Batzies–Welker to show that the
$L$-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal
is supported on a CW-complex. We then translate the “spanning tree condition”
of Almousa–Fløystad–Lohne characterizing polarizations of powers of the graded
maximal ideal into a condition about which sets of hook tableaux span a
certain Schur module. As an application, we give a complete combinatorial
characterization of polarizations of so-called “restricted powers” of the
graded maximal ideal.

###### 2020 Mathematics Subject Classification:

Primary: 13F20,13F55; Secondary: 55U10,05E40

AA was partially supported by the NSF GRFP under Grant No. DGE-1650441.

##  1\. Introduction

Let $S=k[x_{1},\dots,x_{n}]$ be a standard graded polynomial ring over a field
$k$. The Taylor resolution (introduced in [15]) is well-known to be a free
resolution of any ideal generated by monomials in $S$; this complex has many
convenient properties, not least of which is the fact that it is locally
isomorphic to an exterior algebra, implying that the Taylor resolution behaves
in a manner that is very similar to the Koszul complex. In [3], Bayer, Peeva,
and Sturmfels observed that this implies that every monomial ideal has a free
resolution supported on a simplicial complex. It is clear that not every
monomial ideal has a _minimal_ free resolution supported on a simplicial
complex, and indeed, work of Velasco [16] has shown that even the much weaker
notion of CW-complexes is not general enough to support all resolutions of
monomial ideals.

One natural question that arises from the above considerations is the
following: given that the minimal free resolution of every monomial ideal $I$
is a direct summand of the Taylor complex $T_{\bullet}$, how can one extract a
strictly smaller subcomplex $F_{\bullet}\subset T_{\bullet}$ satisfying

  1. (1)

$F_{\bullet}$ is still supported on a cell complex, and

  2. (2)

$F_{\bullet}$ is a free resolution of $I$?

Batzies and Welker give one possible answer to this question using discrete
Morse theory; the basic idea is as follows: suppose that $X$ is a cell complex
supporting the resolution of some ideal $I$. If the associated graph (see
Construction 2.9) admits an acyclic matching on some edge set $A$, then one
can construct an associated Morse complex $X_{A}$ that remains acyclic, is
closer to being minimal, and is also supported on a cell complex. Let
$\mathfrak{m}=(x_{1},\dots,x_{n})\subset S$. One application of these
techniques by Batzies is a proof that the Eliahou-Kervaire resolution for
resolving $\mathfrak{m}^{d}$ is supported on a cell complex (it was later
proved that the Eliahou-Kervaire in full generality is cellular independently
by Mermin [12] and Clark [5]).

As it turns out, there is another well-known complex providing a minimal free
resolution of powers of the maximal ideal, introduced by Buchsbaum and
Eisenbud in [4]. The free modules building these $L$-complexes are Schur
modules corresponding to appropriate hook partitions, whose basis elements are
represented by semistandard tableaux subject to so-called straightening
relations. We show that the $L$-complexes of Buchsbaum and Eisenbud are also
supported on a cell complex using techniques similar to those of Batzies; one
step in this proof is the observation that there is a simple bijection between
the basis elements of the hypersimplex resolution (see Definition 4.2) and
basis elements of an associated enveloping algebra.

Batzies’ hypersimplex resolution in particular keeps track of all possible
linear syzygies on the monomial minimal generating set of $\mathfrak{m}^{d}$,
which can be encoded as a graph. In [1], Almousa, Fløystad, and Lohne show
that this graph of linear syzygies can be used to characterize all possible
_polarizations_ (see Definition 5.1) of $\mathfrak{m}^{d}$. We use the
aforementioned bijection of basis elements to translate conditions on spanning
trees contained within the graph of linear syzygies to rank conditions on
submodules generated by elements of an appropriate Schur module. This yields a
dictionary between the notation and terminology introduced in [1] and well-
established notions arising in the context of Schur modules. Moreover, we
extend the results of [1] to polarizations of _restricted powers_ of the
maximal ideal, and give an explicit algorithm for checking whether a given
graph of linear syzygies induces a well-defined isotone map. This algorithm
opens the door to methods of computing all possible polarizations of
$\mathfrak{m}^{d}$ for any number of variables.

The paper is organized as follows. In Section 2, we recall the definition of a
cellular resolution and summarize the results of Batzies–Welker [2] on
discrete Morse theory for cellular resolutions. In Section 3, we recall the
construction for the $L$-complex of Buchsbaum and Eisenbud, which is a minimal
free resolution of powers of complete intersections. In Section 4, we begin
our study of the so-called _hypersimplicial complex_ $\mathcal{H}_{n}^{d}$
(Definition 4.2) of Batzies–Welker. We observe that the cells
$\mathcal{H}_{n}^{d}$ are correspond to hook tableaux, and that
$\mathcal{H}_{n}^{d}$ supports a free resolution of a power of the graded
maximal ideal coming from a certain double complex (see 4.3 and Figure 1).
With this perspective in mind, we apply discrete Morse theory in a novel way
to obtain a CW-complex supporting the $L$-complex of Buchsbaum and Eisenbud
for powers of the graded maximal ideal (Propositions 2.10 and 4.9).

In Section 5, we summarize the results from [1] giving a complete
characterization of all polarizations of powers of the graded maximal ideal.
In Section 6, we translate the machinery of Section 5 to the language of hook
tableaux (Propositon 6.2). We utilize this perspective to give a novel
characterization of polarizations of powers of the graded maximal ideal
$\mathbf{m}^{d}$ in terms of hook tableaux which span the Schur module
$L^{2}_{d}$ (see Theorem 6.5).

In Section 7, we extend the results from Sections 4 and 6 from powers of the
graded maximal ideal to a larger class of ideals called _restricted powers_ of
the graded maximal ideal. In particular, we give a complete characterization
of all polarizations of such ideals in Theorem 7.8, which is a direct
extension of the results on polarizations of $\mathfrak{m}^{d}$ in [1].

##  2\. Frames and Discrete Morse Theory for Cellular Resolutions

In this section, we recall some important notions on cellular resolutions and
Discrete Morse theory for cellular resolutions. For further exposition on
frames and cellular resolutions, we refer the reader to [13] and [14]. We will
use the terminology of frames to give a convenient framework (pun intended)
for defining cellular resolutions. Proposition 2.10 will be essential for
proving that the $L$-complexes of Buchsbaum and Eisenbud are cellular. We
begin by adopting the following setup:

######  Setup 2.1.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial ring over a field $k$. Let $M$ be
a monomial ideal in $S$ minimally generated by monomial $m_{1},\dots,m_{r}$.
Let $L_{M}$ denote the set of least common multiples of subsets of
$m_{1},\dots,m_{r}$. By convention, $1\in L_{M}$ is considered to be the lcm
of the empty set.

######  Definition 2.2.

Adopt notation and hypotheses of Setup 2.1. A _frame_ (or an $r$-frame)
$\mathbb{U}$ is a complex of finite $k$-vector spaces with differential
$\partial$ and a fixed basis that satisfies the following conditions:

  1. (1)

$U_{i}=0$ for $i<0$ and $i\gg 0$,

  2. (2)

$U_{0}=k$

  3. (3)

$U_{1}=k^{r}$

  4. (4)

$\partial(w_{j})=1$ for each basis vector $w_{j}$ in $U_{1}=k^{r}$.

Given a complex of free modules over some polynomial ring, it is easy to
obtain a frame by setting all variables equal to $1$. Conversely, given a
frame $\mathbb{U}$, one may construct a multigraded complex $\mathbb{G}$ of
finitely generated free multigraded $S$-modules with multidegrees in $L_{M}$
using the following construction due to Peeva and Velasco [14].

######  Construction 2.3.

Adopt notation and hypotheses of Setup 2.1. Let $\mathbb{U}$ be an $r$-frame.
Set

$G_{0}=S\text{ and }G_{1}=S(-m_{1})\oplus\dots\oplus S(-m_{r}).$

Let $\overline{v}_{1},\dots,\overline{v}_{p}$ and
$\overline{u}_{1},\dots,\overline{u}_{q}$ be the given bases of $U_{i}$ and
$U_{i-1}$, respectively. Let $u_{1},\dots,u_{q}$ be the basis of
$G_{i-1}=S^{q}$ chosen on the previous step of the induction. Introduce
$v_{1},\dots,v_{p}$ that will be a basis of $G_{i}=S^{p}$. If

$\partial(\overline{v}_{j})=\sum_{1\leq s\leq q}\alpha_{sj}\overline{u}_{s}$

with coefficients $\alpha_{sj}\in k$, then set

$\displaystyle\operatorname{mdeg}(v_{j})=\operatorname{lcm}(\operatorname{mdeg}(u_{s})\mid\alpha_{sj}\neq
0)$ $\displaystyle G_{i}=\bigoplus_{1\leq j\leq
p}S(-\operatorname{mdeg}(v_{j}))$ $\displaystyle d(v_{j})=\sum_{1\leq s\leq
q}\alpha_{sj}\frac{\operatorname{mdeg}(v_{j})}{\operatorname{mdeg}(u_{s})}\cdot
u_{s}.$

Clearly $\operatorname{coker}(d_{1})=S/M$ and the differential $d$ is
homogeneous by construction. Call $\mathbb{G}$ the _$M$ -homogenization_ of
$\mathbb{U}$.

The following simple criterion by Peeva and Velasco [14] determines when a
frame supports a graded free resolution of $S/M$. The abridged version of this
result states that exactness can be checked by only considering
multihomogeneous strands.

######  Proposition 2.4.

The sequence of modules and homomorphisms $\mathbb{G}$ as in Construction 2.3
is a complex. Moreover, if $\mathbb{G}(\leq m)$ is the subcomplex of
$\mathbb{G}$ generated by the multihomogeneous basis elements of multidegrees
dividing $m$, then $\mathbb{G}$ is a free multigraded resolution of $S/M$ if
and only if for all monomials $1\neq m\in L_{M}$, the frame of the complex
$G(\leq m)$ is exact.

A natural source of frames that can be used to support resolutions of monomial
ideals are provided by CW-complexes, since the conditions $(1)-(4)$ of
Definition 2.2 are trivially satisfied.

######  Notation 2.5.

Let $X$ be a regular CW-complex, and denote by $X^{(i)}$ the set of $i$-cells
of $X$ and by $X^{(\ast)}\coloneqq\bigcup_{i\geq 0}X^{(i)}$ the set of all
cells of $X$. Denote by $\mathcal{C}(X;k)$ the augmented oriented cellular
chain complex of $X$ over $k$ with

$\mathcal{C}(X;k)_{i}=\bigoplus_{c\in X^{(i)}}ke_{c}$

where $e_{c}$ denotes the basis element corresponding to the face $c\in
X^{(i)}$, and the differential $\partial$ acts as

$\partial(e_{c})=\sum_{c\geq c^{\prime}\in
X^{(i-1)}}[c,c^{\prime}]e_{c^{\prime}}$

where $[c,c^{\prime}]$ is the coefficient in the differential of the cellular
homology of $X$.

With the above notation in mind, we use the language of frames to define
cellular resolutions.

######  Definition 2.6.

Adopt notation of Notation 2.5. Assume that $\lvert X^{(0)}\rvert=r$ and
$M=(m_{1},\dots,m_{r})$ is a monomial ideal in a polynomial ring $S$. Label
each $0$-cell of $X$ by a minimal generator $m_{i}$ of $M$. After shifting
$\mathcal{C}(X;k)$ in homological degree, $\mathcal{C}(X;k)[-1]$ is a frame.
Denote by $\mathbb{F}_{X}$ the $M$-homogenization of $\mathcal{C}(X;k)$ as in
Construction 2.3. The complex $\mathbb{F}_{X}$ is _supported on_ $X$. The
complex $\mathbb{F}_{X}$ is a _cellular resolution_ if it is exact.

######  Definition 2.7.

Let $M=(m_{1},\dots,m_{r})$ be a monomial ideal in a polynomial ring $S$, and
let $X$ be a regular CW-complex with $0$-cells labeled by the generators of
$M$. The multidegree of each vertex of $X$ is given by its monomial label.
Define a face $c$ to have multidegree

$\operatorname{mdeg}(c)=\operatorname{lcm}(m_{i}\mid m_{i}\in c).$

By convention, $\operatorname{mdeg}(\emptyset)=1$. Define the following
subcomplexes of $X$:

$\displaystyle X_{\leq m}$ $\displaystyle\coloneqq\\{c\in
X\mid\operatorname{mdeg}(c)\text{ divides }m\\}$ $\displaystyle X_{<m}$
$\displaystyle\coloneqq\\{c\in X\mid\operatorname{mdeg}(c)\text{ strictly
divides }m\\}.$

The following Proposition is an immediate consequence of Proposition 2.4
combined with the notation and hypotheses introduced in Definition 2.7.

######  Proposition 2.8.

Let $M=(m_{1},\dots,m_{r})$ be a monomial ideal in a polynomial ring $S$, and
let $X$ be a regular $CW$-complex with $0$-cells labeled by the minimal
generators of $M$. The complex $\mathbb{F}_{X}$ from Definition 2.6 is a free
resolution of $S/M$ if and only if for all multidegrees $1\neq m\in L_{M}$,
the complex $X_{\leq m}$ is acyclic over $k$.

Next, we introduce some of the basic machinery of discrete Morse theory for
cellular resolutions. Discrete Morse theory was developed by Forman in [7] to
extend the ideas from Morse theory in differential geometry to CW complexes.
The interested reader is encouraged to consult Forman’s survey paper [8] for
further reading on discrete Morse theory. The application of discrete Morse
theory to the study of cellular resolutions was first explored by Batzies and
Welker in [2] as a method of “cutting down” a large cellular resolution in
such a manner that the resulting subcomplex is also a cellular resolution.

######  Construction 2.9.

Adopt Notation 2.5. Let $G_{X}$ be the directed graph on the set of cells of
$X$ whose set $E_{X}$ of edges is given by $c\rightarrow c^{\prime}$ for
$c^{\prime}\subseteq c$ and $\dim(c^{\prime})=\dim(c)-1$. A _discrete Morse
function_ arises from a set $A\subseteq E_{X}$ of edges in $G_{X}$ satisfying:

  1. (1)

each cell occurs in at most one edge of $A$, and

  2. (2)

the graph $G_{X}^{A}$ with edge set

$E_{X}^{A}\coloneqq(E_{X}\setminus A)\cup\\{c^{\prime}\rightarrow c\mid
c\rightarrow c^{\prime}\in A\\}$

is acyclic (i.e., it does not contain a directed cycle).

Such a set $A\subseteq E_{X}$ is called an _acyclic matrching_ of $G_{X}$. A
cell of $X$ is _$A$ -critical_ with respect to $A$ if it is not contained in
any edge of $A$. An acyclic matching is _homogeneous_ if $c\rightarrow
c^{\prime}\in A$ implies that
$\operatorname{mdeg}(c)=\operatorname{mdeg}(c^{\prime})$.

The proof of the following proposition can be found in the appendix of [2],
and shows that acyclic matchings can be used to induce acyclic subcomplexes
that are also supported on cell complexes.

######  Proposition 2.10.

Let $X$ be a regular CW-complex which supports a free resolution of a monomial
ideal $M$, and let $A$ be a homogeneous acyclic matching of $G_{X}$. Then
there is a (not necessarily regular) CW-complex $X_{A}$ whose $i$-cells are in
one-to-one correspondence with the $A$-critical $i$-cells of $X$ such that
$X_{A}$ is homotopy equivalent to $X$.

Moreover, $X_{A}$ inherits a multigrading from $X$, and for any multidegree
$\alpha$ and restriction $A_{\leq\alpha}$ of $A$ to $X_{\leq\alpha}$, one has

$X_{\leq\alpha}\simeq(X_{\leq\alpha})_{A_{\leq\alpha}}\cong(X_{A})_{\leq\alpha}.$

In particular, $X_{A}$ also supports a cellular resolution of the ideal $M$.

######  Definition 2.11.

The complex $X_{A}$ of Proposition 2.10 is called the _Morse complex_ of $X$
for the matching $A$.

######  Remark 2.12.

The explicit construction of the Morse complex $X_{A}$ from an acyclic
matching $A$ is quite technical, and will not be included in the current
paper. The interested reader is encouraged to consult the appendix of [2] for
more details.

##  3\. Background on L-complexes

The material up until Proposition 3.4, along with proofs, can be found in [4]
or Section $2$ of [6]. The goal of this section is to a give a brief jog
through the $L$-complexes of Buchsbaum and Eisenbud and to make clear our
conventions on Young tableaux. For further details on Schur modules and their
use in the construction of free resolutions, one may consult Weyman’s book
[17]. The following notation will be in play for the remainder of this
section.

######  Notation 3.1.

Let $R$ be a polynomial ring over a field $k$. Let $F$ be a free $R$-module of
rank $n$ with basis $f_{1},\dots,f_{n}$. Denote by $S_{d}(F)$ the $d$th
symmetric power of $F$, and by $\bigwedge^{n}F$ the $n$th exterior power of
$F$. Let $J=\\{j_{1}<j_{2}<\dots<j_{k}\\}\subset[n]$. Define

$f_{J}\coloneqq f_{j_{1}}\wedge\dots\wedge f_{j_{k}}\in\bigwedge^{k}F.$

If $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}$ such that
$\sum_{i}\alpha_{i}=d$, set

$f^{\alpha}\coloneqq f_{1}^{\alpha_{1}}f_{2}^{\alpha_{2}}\dots
f_{n}^{\alpha_{n}}\in S_{d}(F).$

######  Setup 3.2.

Let $F$ denote a free $R$-module of rank $n$, and $S=S(F)$ the symmetric
algebra on $F$ with the standard grading. Define a complex

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigwedge^{a+1}F\otimes_{R}S_{b-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{a+1,b-1}}$$\textstyle{\bigwedge^{a}F\otimes_{R}S_{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\kappa_{a,b}}$$\textstyle{\cdots}$

where the maps $\kappa_{a,b}$ are defined as the composition

$\begin{split}\bigwedge^{a}F\otimes_{R}S_{b}&\to\bigwedge^{a-1}F\otimes_{R}F\otimes_{R}S_{b}\\\
&\to\bigwedge^{a-1}F\otimes_{R}S_{b+1}\end{split}$

where the first map is comultiplication in the exterior algebra and the second
map is the standard module action (where we identify $F=S_{1}(F)$). Define

$L_{b}^{a}(F):=\ker\kappa_{a,b}.$

Let $\psi:F\to R$ be a morphism of $R$-modules with $\operatorname{im}(\psi)$
an ideal of grade $n$. Let
$\textrm{kos}^{\psi}:\bigwedge^{i}F\to\bigwedge^{i-1}F$ denote the standard
Koszul differential; that is, the composition

$\begin{split}\bigwedge^{i}F&\to
F\otimes_{R}\bigwedge^{i-1}F\quad\textrm{(comultiplication)}\\\
&\xrightarrow[]{\psi\otimes
1}R\otimes_{R}\bigwedge^{i-1}F\cong\bigwedge^{i-1}F\quad\textrm{(module
action).}\\\ \end{split}$

Explicitly, if $J=\\{j_{1}<\dots<j_{k}\\}$, then

$\textrm{kos}^{\psi}(f_{J})=\sum_{i\in[k]}(-1)^{i}\psi(f_{j_{i}})\cdot
f_{J\setminus j_{i}}.$

######  Definition 3.3.

Adopt notation and hypotheses of Setup 3.2. Define the complex

$\begin{split}&L(\psi,b):\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{L_{b}^{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
58.05333pt\raise 6.41666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.83333pt\hbox{$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 98.7089pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
68.7089pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 98.7089pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
125.98055pt\raise 6.41666pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.83333pt\hbox{$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 166.2089pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
136.2089pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 166.2089pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{L_{b}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
182.20946pt\raise 6.5pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{S_{b}(\psi)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 205.41779pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
205.41779pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
243.08792pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
243.08792pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\\\
\end{split}$

where $\textrm{kos}^{\psi}\otimes 1:L_{b}^{a}(F)\to L_{b}^{a-1}$ is induced by
making the following diagram commute:

$\textstyle{\bigwedge^{a}F\otimes
S_{b}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\textstyle{\bigwedge^{a-1}F\otimes
S_{b}(F)}$$\textstyle{L_{b}^{a}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\textstyle{L_{b}^{a-1}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

The following Proposition shows that the $L$-complexes constitute a minimal
free resolution of powers of complete intersections in general.

######  Proposition 3.4.

Let $\psi:F\to R$ be an $R$-module homomorphism from a free module $F$ of rank
$n$ such that the image $\operatorname{im}(\psi)$ is an ideal of grade $n$.
Then the complex $L(\psi,b)$ of Definition 3.3 is a minimal free resolution of
$R/\operatorname{im}(\psi)^{b}$

We also have (see Proposition $2.5(c)$ of [4], or just use Proposition 3.6)

$\begin{split}&\operatorname{rank}_{R}L_{b}^{a}(F)=\binom{n+b-1}{a+b}\binom{a+b-1}{a}.\\\
\end{split}$

Moreover, using the notation and language of Chapter $2$ of [17],
$L_{b}^{a}(F)$ is the Schur module $L_{(a+1,1^{b-1})}(F)$. This allows us to
identify a standard basis for such modules.

######  Notation 3.5.

We use the English convention for partition diagrams. That is, the partition
$(3,2,2)$ corresponds to the diagram

$\ytableau\ &\ \ \\\ \ \ \\\ \ \ \\\ .$

A Young tableau is standard if it is strictly increasing in both the columns
and rows. It is semistandard if it is strictly increasing in the columns and
nondecreasing in the rows.

######  Proposition 3.6.

Adopt notation and hypotheses as in Setup 3.2. Then a basis for $L_{b}^{a}(F)$
is represented by all Young tableaux of the form

$\ytableausetup{boxsize=2em}\ytableau i_{0}&j_{1}\cdots j_{b-1}\\\ i_{1}\\\
\vdots\\\ i_{a}\\\ $

with $i_{0}<\cdots<i_{a}$ and $i_{0}\leq j_{1}\leq\cdots\leq j_{b-1}$.

###### Proof.

See Proposition $2.1.4$ of [17] for a more general statement. ∎

######  Remark 3.7.

Adopt notation and hypotheses of Setup 3.2. Let $F$ have basis
$f_{1},\dotsc,f_{n}$. In the statement of Proposition 3.6, we think of the
tableau as representing the element

$\kappa_{a+1,b-1}(f_{i_{0}}\wedge\cdots\wedge f_{i_{a}}\otimes f_{j_{1}}\cdots
f_{j_{b-1}})\in\bigwedge^{a}F\otimes S_{b}(F).$

We will often write $f_{i_{0}}\wedge\cdots\wedge f_{i_{a}}\otimes
f_{j_{1}}\cdots f_{j_{b-1}}\in L_{b}^{a}(F)$, with the understanding that we
are identifying $L_{b}^{a}(F)$ with the cokernel of
$\kappa_{a+2,b-2}:\bigwedge^{a+2}F\otimes S_{b-2}(F)\to\bigwedge^{a+1}F\otimes
S_{b-1}(F)$.

The following Observation is sometimes referred to as the _shuffling_ or
_straightening_ relations satisfied by tableaux in the Schur module
$L_{b}^{a}(F)$.

######  Observation 3.8.

Any tableau of the form

$T=\ytableausetup{boxsize=2em}\ytableau i_{0}&j_{1}\cdots j_{b-1}\\\ i_{1}\\\
\vdots\\\ i_{a}\\\ $

with $j_{1}\leq\cdots\leq j_{b-1}$ viewed as an element in $L_{b}^{a}(F)$ with
$b\geq 2$ may be rewritten as a linear combination of other tableaux in the
following way:

$T=\sum_{k=0}^{a}(-1)^{k}\ytableausetup{boxsize=1.7em}\ytableau
j_{1}&i_{k}j_{2}\cdots j_{b-1}\\\ i_{0}\\\ \vdots\\\ \widehat{i_{k}}\\\
\vdots\\\ i_{a}\\\ .$

Notice that if $i_{0}>j_{1}$ and $i_{0}<\cdots<i_{a}$, then this rewrites $T$
as a linear combination of semistandard tableaux after reordering the row into
ascending order.

##  4\. The $L$-complex is cellular

In this section, we apply discrete Morse theory to the so-called _hypersimplex
resolution_ (see Definition 4.2) of $\mathfrak{m}^{d}$ in a novel way to
obtain a CW-complex which supports the $L$-complex with the exact basis
elements described in Proposition 3.6. In particular, Proposition 4.9 implies
that the L-complex of Buchsbaum and Eisenbud is CW-cellular. While Batzies and
Welker had previously obtained a minimal cellular resolution of
$\mathfrak{m}^{d}$ by finding an acyclic matching on the hypersimplex
resolution in [2], the minimal resolution they obtain is instead isomorphic to
the Eliahou-Kervaire resolution of $\mathfrak{m}^{d}$.

######  Notation 4.1.

The notation $\Delta(n,d)$ will denote the dilated $(n-1)$-simplex
$d\cdot\Delta^{n-1}$; that is,

$\Delta(n,d)=d\cdot\Delta^{n-1}:=\left\\{(y_{1},\dots,y_{n})\in\mathbb{R}^{n}\mid\sum_{i=1}^{n}y_{i}=d,y_{i}\geq
0\text{ for }i=1,\dots,n\right\\}.$

######  Definition 4.2.

Let $\mathcal{H}^{d}_{n}$ be the polytopal CW-complex with the underlying
space $\Delta(n,d)$, with CW-complex stucture induced by intersection with the
cubical CW-complex structure on $\mathbb{R}^{d}$ given by the integer lattice
$\mathbb{Z}^{d}$. That is, the closed cells of $\mathcal{H}^{d}_{n}$ are given
by all hypersimplices

$\displaystyle C_{\mathbf{a},J}$
$\displaystyle\coloneqq\Delta(n,d)\cap\left\\{(y_{1},\dots,y_{n})\in\mathbb{R}^{n}\mid
y_{i}=a_{i}\text{ for }i\in[d]\setminus J,\text{ and
}y_{j}\in[a_{j},a_{j}+1]\text{ for }j\in J\right\\}$
$\displaystyle=\operatorname{conv}\left(\mathbf{a}+\sum_{j\in
J}\ell_{j}\epsilon_{j}\mid\ell_{j}\in\\{0,1\\},\sum_{j\in
J}\ell_{j}=d-\lvert\mathbf{a}\rvert\right)$

with $\mathbf{a}\in\mathbb{N}^{n}$, $J\subset[n]$,
$\lvert\mathbf{a}\rvert\coloneqq\sum_{i\in[n]}a_{i}$, $\epsilon_{i}$ the $i$th
unit vector in $\mathbb{R}^{n}$, either subject to the conditions $\lvert
a\rvert=d$ and $J=\emptyset$ (these are the $0$-cells), or the condition
$1\leq d-\lvert\mathbf{a}\rvert\leq\lvert J\rvert-1$. The CW-complex
$\mathcal{H}^{d}_{n}$ is multigraded by setting
$\operatorname{lcm}(C_{\mathbf{a},J})\coloneqq\mathbf{a}+\sum_{j\in
J}\epsilon_{j}$. Call $\mathcal{H}^{d}_{n}$ the _hypersimplicial complex_.

Let $J=(j_{0}<\cdots<j_{r})$ and use the notation $J_{v}\coloneqq
J\setminus\\{j_{v}\\}$. Then the differential of $\mathcal{H}^{d}_{n}$ is
given by

$\displaystyle\partial(C_{\mathbf{a},J})=\begin{cases}\sum_{v=0}^{r}(-1)^{v}(C_{\mathbf{a},J_{v}}-C_{\mathbf{a}+\epsilon_{v},J_{v}})&\text{
if }2\leq d-\lvert\mathbf{a}\rvert\leq\lvert J\rvert-2\\\ \\\
\sum_{v=0}^{r}(-1)^{v}C_{\mathbf{a},J_{v}}&\text{ if
}1=d-\lvert\mathbf{a}\rvert\leq\lvert J\rvert-2\\\ \\\
\sum_{v=0}^{r}(-1)^{v}C_{\mathbf{a}+\epsilon_{j},J_{v}}&\text{ if }2\leq
d-\lvert\mathbf{a}\rvert=\lvert J\rvert-1\\\ \\\
C_{\mathbf{a}+e_{j_{0}},\emptyset}-C_{a+e_{j_{1}},\emptyset}&\text{ if
}d-\lvert\mathbf{a}\rvert=1\\\ \\\ 0&\text{ if
}\lvert\mathbf{a}\rvert=d.\end{cases}$ (1)

######  4.3.

Adopt notation and hypotheses of Setup 3.2. If $J=(j_{0}<j_{1}<\dots<j_{r})$
and $\mathbf{a}\in\mathbb{N}^{d}$, note that every $r$-dimensional cell
$C_{\mathbf{a},J}$ corresponds to the element $f_{J}\otimes f^{\mathbf{a}}$ in
$\bigwedge^{r+1}F\otimes S_{\lvert a\rvert}(F)$. These elements, in turn, can
be represented as hook tableaux with strictly increasing columns and weakly
increasing rows:

$C_{\mathbf{a},J}\quad\longleftrightarrow\quad f_{J}\otimes
f^{\mathbf{a}}\quad\longleftrightarrow\quad\ytableau j_{0}&1^{a_{1}}\dots
n^{a_{n}}\\\ j_{1}\\\ j_{2}\\\ \vdots\\\ j_{r}.$

We will implicitly use this correspondence to refer to cells
$C_{\mathbf{a},J}$ of $\mathcal{H}^{d}_{n}$ as tableaux or elements of
$\bigwedge^{r+1}F\otimes S_{\lvert a\rvert}(F)$. The
$\mathfrak{m}^{d}$-homogenization (see Construction 2.3) of
$\mathcal{H}^{d}_{n}[-1]$ therefore corresponds to the double complex in
Figure 1 where the maps $\kappa$ and $\textrm{kos}^{\psi}\otimes 1$ are as
defined in Setup 3.2.

${0}$${0}$${{\bigwedge^{n}F}}$${\iddots}$${\vdots}$${\vdots}$${0}$${\dots}$${{\bigwedge^{5}F\otimes
S_{d-3}}}$${{\bigwedge^{4}F\otimes S_{d-3}(F)}}$${0}$${{\bigwedge^{n}F\otimes
S_{d-2}(F)}}$${\dots}$${{\bigwedge^{4}F\otimes
S_{d-2}(F)}}$${{\bigwedge^{3}F\otimes
S_{d-2}(F)}}$${0}$${{\bigwedge^{n}F\otimes
S_{d-1}(F)}}$${\dots}$${{\bigwedge^{4}F\otimes
S_{d-1}(F)}}$${{\bigwedge^{3}F\otimes S_{d-1}(F)}}$${{\bigwedge^{2}F\otimes
S_{d-1}(F)}}$${{F\otimes
S_{d}(F)}}$${{S_{d}(F)}}$${0}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\kappa}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\kappa}$$\scriptstyle{\textrm{kos}^{\psi}\otimes
1}$

Figure 1. Double complex supported on the hypersimplex resolution of a power
of the graded maximal ideal.

$112$$113$$111$$222$$333$$133$$122$$223$$233$$123$ (a) The cell
$C_{(1,0,1),123}$ in $\mathcal{H}^{3}_{3}$.

$112$$113$$111$$222$$333$$133$$122$$223$$233$$123$ (b) The cell
$C_{(0,0,1),123}$ in $\mathcal{H}^{3}_{3}$.

######  Example 4.4.

The edge with vertices $(1,1,1)$ and $(1,0,2)$ in blue-green in Figure 2(a)
and 2(b) corresponds to the cell $C_{(1,0,1),23}$, which has the following
image in the double complex of Figure 1:

$\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\mapsto
x_{2}\cdot\ytableau 1&33\\\ -x_{3}\cdot\ytableau 1&23\\\ .$

The “up-simplex” corresponding to the cell $C_{(1,0,1),123}$ colored in Figure
2(a) has image

$\ytableau 1&13\\\ 2\\\ 3\mapsto
x_{1}\cdot\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}-x_{2}\cdot\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}+x_{3}\cdot\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2}.$

Finally, the “down-simplex” of Figure 2(b) corresponding to the cell
$C_{(0,0,1),123}$ has image

$\ytableau 1&3\\\ 2\\\
3\mapsto\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}-\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}+\ytableau{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}1}&{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}3}\\\
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}2}.$

The following simple observation turns out to be critical for applications in
Section 6.

######  Observation 4.5.

All elements of $\bigwedge^{2}F\otimes S_{d-1}$ contained in the one-skeleton
of a “down-triangle” are related by a straightening relation as in Observation
3.8.

Figure 3. The one-skeleton of $\mathcal{H}^{3}_{4}$.

######  Example 4.6.

Figure 3 depicts the one-skeleton of $\mathcal{H}^{3}_{4}$. In this case, the
hypersimplices which appear as maximal cells are not solely simplices: there
are four octahedra in the complex, which correspond to the four possible
elements of $\bigwedge^{4}F\otimes F$. The image of any one of these octahedra
in the double complex of Figure 1 is a linear combination of eight tableaux,
corresponding to the eight faces of the octahedron: four “up-triangles” coming
from its image under $\kappa$, and four “down-triangles” corresponding to its
image under $\textrm{kos}^{\psi}\otimes 1$.

######  Proposition 4.7 (see [2]).

Let $\mathfrak{m}^{d}=(x_{1},\dots,x_{n})^{d}\subset k[x_{1},\dots,x_{n}]$.
Then $\mathcal{H}^{d}_{n}$ defines a multigraded cellular free resolution of
$\mathfrak{m}^{d}$.

Batzies and Welker [2] use discrete Morse theory to show that the
Eliahou–Kervaire resolution for powers of the graded maximal ideal is
cellular. We apply their techniques to obtain a minimal cellular resolution
isomorphic to the $L$-complex as in Definition 3.3. To do this, we find an
acyclic matching on $\mathcal{H}^{d}_{n}$ distinct from the one in [2] which
has a corresponding Morse complex which supports the L-complex.

######  Proposition 4.8.

Let $\mathcal{H}_{n}^{d}$ be as in Definition 4.2. Consider the matching
$C_{\mathbf{a},J}\rightarrow C_{\mathbf{a}+\epsilon_{\min
J},J\setminus\min{J}}$, where

  1. (1)

$\mathbf{a}\in\mathbb{N}^{n}$,

  2. (2)

$J\subset[n]$ is such that $2\leq d-\lvert\mathbf{a}\rvert\leq\lvert
J\rvert-1$, and

  3. (3)

$\min J\leq\min\mathbf{a}$.

Then this is an acyclic homogeneous matching $A$ on $\mathcal{H}^{d}_{n}$ as
in Construction 2.9.

Moreover, if $\tilde{\mathcal{H}}^{d}_{n}=(\mathcal{H}^{d}_{n})_{A}$ denotes
the corresponding Morse complex, then the $A$-critical cells of
$\mathcal{H}^{d}_{n}$ are:

  1. (1)

the $0$-cells $C_{\mathbf{a},\emptyset}$, where
$\mathbf{a}\in\mathbb{N}^{n}\cap\Delta(n,d)$, and

  2. (2)

the cells $C_{\mathbf{a},J}$ such that $\min J\leq\min\mathbf{a}$ and
$\lvert\mathbf{a}\rvert=d-1$.

###### Proof.

$A$ is a matching because cells $C_{\mathbf{a},J}$ on the left hand side must
satisfy $\min J\leq\min\mathbf{a}$ while the cells
$C_{\mathbf{a}^{\prime},J^{\prime}}$ on the right hand side must satisfy $\min
J^{\prime}>\min\mathbf{a}^{\prime}$. Suppose, seeking contradiction, that $A$
contains a cycle. Observe that $\operatorname{mdeg}(C_{\mathbf{a},J})$ must be
weakly decreasing along every directed edge of the cycle, so in particular it
must be constant. Observe that every element $C_{\mathbf{a},J}$ at the head of
an edge directed upwards in $E^{A}$ satisfies $\min J\leq\min\mathbf{a}$, but
then every element at the head of an arrow pointing “down” from one of these
elements must have the same $\min J$. But this element
$C_{\mathbf{a}^{\prime},J^{\prime}}$ must also be at the tail of some other
element of $E^{A}$ pointing upwards, so it must also satisfy that $\min
J^{\prime}<\min\mathbf{a}$, which is a contradiction. ∎

We conclude this section with our main result, which states that the L-complex
from Definition 3.3 is supported on a CW-complex.

######  Proposition 4.9.

Let $\tilde{\partial}$ denote the differential of the Morse complex
$\tilde{H}_{n}^{d}=(H_{n}^{d})_{A}$ and $\tilde{C}_{\mathbf{a},J}$ the cell in
$\tilde{H}_{n}^{d}$ corresponding to the $A$-critical cell $C_{\mathbf{a},J}$
of $\mathcal{H}^{d}_{n}$. Then $\tilde{\mathcal{H}}_{n}^{d}$ supports a
minimal linear cellular resolution of a power of the graded maximal ideal
which is isomorphic to the $L$-complex from Definition 3.3.

###### Proof.

The $A$-critical cells of $\mathcal{H}^{d}_{n}$ are exactly the $0$-cells
$C_{\mathbf{a},\emptyset}$ for $\mathbf{a}\in\Delta_{n}(d)\cap\mathbb{N}^{n}$
and all cells $C_{\mathbf{a},J}$ such that $\min J\leq\min\mathbf{a}$ and
$\lvert\mathbf{a}\rvert=d-1$; in particular, the critical cells correspond to
exactly those standard hook tableaux which are basis elements of the modules
in the $L$-complex. Let $C_{\mathbf{a},J}$ be a critical cell with $p=\min J$
and $q=\min\\{i\mid i\in\operatorname{supp}(\mathbf{a})\\}$. Then the
differential $\partial$ of $\mathcal{H}_{n}^{d}$ applied to $C_{\mathbf{a},J}$
has at most one nonstandard tableau in its image, which would be
$C_{\mathbf{a},J_{p}}$. This element is matched with
$C_{\mathbf{a}-\epsilon_{q},J_{p}\cup q}$, which has an image under $\partial$
consisting of standard tableaux of the same shape and multidegree as
$C_{\mathbf{a},J_{p}}$, and potentially some tableaux corresponding to
elements which are matched with elements one dimension lower and therefore do
not appear in $\tilde{\partial}$. In particular, after homogenization,
$\tilde{\partial}$ corresponds exactly to the differential of the $L$-complex
in Definition 3.3. ∎

##  5\. Polarizations of Powers of Graded Maximal Ideals

The material in this section is a summary of the combinatorial
characterization of polarizations of powers of the graded maximal ideal
$\mathfrak{m}$ in a polynomial ring given by Almousa, Fløystad, and Lohne in
[1]. The idea is to put a set of partial orders $\geq_{i}$ on the generators
of $\mathfrak{m}^{d}$ and view a “potential polarization” as a set of isotone
maps. One may visualize these potential polarizations as a graph of linear
syzygies among the generators of the “potentially polarized” ideal, which is
in fact a subgraph of the one-skeleton of the hypersimplicial complex
$\mathcal{H}_{n}^{d}$ from the previous section. Theorem 5.16 gives a complete
characterization of which of these isotone maps give “honest” polarizations in
terms of their graphs of linear syzygies. The main result of this section is
given by Theorem 5.16, which will be retranslated in the context of Schur
modules in Section 6.

Much of the notation introduced in this section will be used in later sections
without reference. We begin this section with the definition of a
polarization. Intuitively, polarizations can be used to replace any monomial
ideal with a _squarefree_ monomial that is homologically indistinguishable.
One of the main insights of [1] is the fact that there are many ways to
polarize an ideal, and the family of all polarizations of a given monomial
ideal can be highly nontrivial. However, in the case of the ideal
$\mathfrak{m}^{d}$, a full characterization of all such polarizations is
possible; we recall this result in Proposition 5.16.

######  Definition 5.1 (Polarization).

Let $I$ be a monomial ideal in a polynomial ring $S=k[x_{1},\dots,x_{n}]$ over
a field $k$, and let $d_{i}$ be the highest power of $x_{i}$ that appears in a
minimal generator of $I$. Set $\check{X}_{i}=\\{x_{i1},\dots,x_{id_{i}}\\}$
for all $i$, and define the polynomial ring
$\tilde{S}=k[\check{X}_{1},\dots,\check{X}_{n}]$ in the union of all these
variables. Then $\tilde{I}\subset\tilde{S}$ is a _polarization_ of $I$ if

$\sigma=\bigcup_{i=1}^{n}\\{x_{i1}-x_{i2},x_{i1}-x_{i3},\dots,x_{i1}-x_{id_{i}}\\}$

is a regular $\tilde{S}/\tilde{I}$-sequence and
$\tilde{I}\otimes\tilde{S}/\langle\sigma\rangle\cong I$.

The first incarnation of polarizations appeared in Hartshorne’s thesis (see
[11]), where he used what he called “distractions” to prove the connectedness
of the Hilbert scheme.

######  Setup 5.2.

Fix integers $n$ and $d$, and let $S=k[x_{1},\dots,x_{n}]$ be a polynomial
over a field $k$. Let $\check{X}_{i}=\\{x_{i1},\dots,x_{id}\\}$ be a set of
variables, and let $\tilde{S}=k[\check{X}_{1},\dots,\check{X}_{n}]$ be a
polynomial ring in the union of all these variables. Denote by
$\mathfrak{m}=(x_{1},\dots,x_{n})$ the graded maximal ideal of $S$.

Let $\Delta(n,d)$ be the dilated $(n-1)$-simplex from Definition 4.1. Denote
by $\Delta^{\mathbb{Z}}(n,d)=\Delta(n,d)\cap\mathbb{Z}^{n}$ the set of lattice
points of the dilated simplex $d\cot\Delta_{n-1}$, i.e., the set of tuples
$\mathbf{a}=(a_{1},\dots,a_{n})$ of non-negative integers with
$\sum_{i}^{n}a_{i}=d$. Denote by $\operatorname{sk}_{1}(\mathcal{H}^{d}_{n})$
the one-skeleton of the hypersimplicial complex $\mathcal{H}^{d}_{n}$ from
Definition 4.2.

######  Remark 5.3.

Observe that the elements of $\Delta^{\mathbb{Z}}(n,d)$ are exactly the
exponent vectors of the minimal generating set of the ideal
$\mathfrak{m}^{d}$.

######  Notation 5.4.

Let $\epsilon_{i}\in\mathbb{N}^{n}$ be the $i$th unit vector in
$\mathbb{N}^{n}$. For a given $\mathbf{a}$, denote by
$\operatorname{Supp}(\mathbf{a})$ the _support_ of $\mathbf{a}$, that is, the
set of all $i$ such that $a_{i}>0$. If $B$ is a subset of $[n]$, denote by
$\mathbb{1}_{B}$ the $n$-tuple $\sum_{i\in B}\epsilon_{i}$. For example, if
$B=[n]$, then $\mathbb{1}_{B}=(1,\dots,1)$.

######  Definition 5.5.

Adopt notation and hypotheses of Setup 5.2. Fix an index $1\leq i\leq n$.
Define $(\Delta^{\mathbb{Z}}(n,d),\geq_{i})$ to be the poset with ground set
$\Delta^{\mathbb{Z}}(n,d)$ and partial order $\geq_{i}$ such that
$\mathbf{b}\geq_{i}\mathbf{a}$ if $b_{i}\geq a_{i}$ and $b_{j}\leq a_{j}$ for
$j\neq i$.

######  Observation 5.6.

The partial order $\geq_{i}$ as in Definition 5.5 is graded, where
$\mathbf{a}\in\Delta^{\mathbb{Z}}(n,d)$ has rank $a_{i}$.

In the following definitions, we introduce some key subgraphs of the one-
skeleton of $\mathcal{H}^{d}_{n}$ which will be critical for the combinatorial
characterization of polarizations of $\mathfrak{m}^{d}$. Complete down-graphs
will also be significant players in Section 6.

######  Definition 5.7 (Complete down-graph).

Given $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ and
$i,j\in\operatorname{Supp}(\mathbf{c})$, there is an edge between
$\mathbf{c}-\epsilon_{i}$ and $\mathbf{c}-\epsilon_{j}$ in
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$ denoted $(\mathbf{c};i,j)$. Every
edge in $\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$ can be realized as an
edge $(\mathbf{c};i,j)$ for unique $\mathbf{c},i,$ and $j$. An $n$-tuple
$\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ induces a subgraph of
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$ called the _complete down-graph_
$D(\mathbf{c})$ on the points $\mathbf{c}-\epsilon_{i}$ for
$i\in\operatorname{Supp}(\mathbf{c})$. If $R\subseteq[n]$, denote by
$D_{R}(\mathbf{c})$ the complete graph with edges $(\mathbf{c};r,s)$ for
$r,s\in R$.

######  Definition 5.8 (Complete up-graph).

Any $\mathbf{a}\in\Delta^{\mathbb{Z}}(n,d-1)$ also determines a subgraph of
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$: the _complete up-graph_
$U(\mathbf{a})$ consisting of points $\mathbf{a}+\epsilon_{i}$ for
$i=1,\dots,n$ with edges $(\mathbf{a}+\epsilon_{i}+\epsilon_{j};i,j)$ for
$i\neq j$.

######  Remark 5.9.

The complete down-graph $D(\mathbf{c})$ induces a simplex of full dimension
$d-1$ if and only if $c_{i}\geq 1$ for all $i$, i.e., $\mathbf{c}$ has full
support. For each $\mathbf{a}$ in $\Delta^{\mathbb{Z}}(n,d-1)$, the induced
simplex of the up-graph $U(\mathbf{a})$ always has full dimension $d-1$.

(3,0,0)(0,3,0)(0,0,3) Figure 4. The hypersimplicial complex
$\mathcal{H}^{3}_{3}$.

######  Example 5.10.

The one skeleton of $\mathcal{H}_{3}^{3}$ pictured in Figure 4 has three
“complete down-triangles” with full support corresponding to the vectors
$(2,1,1),(1,2,1)$, and $(1,1,2)$ in $\Delta^{\mathbb{Z}}(n,d+1)$. It also has
six “complete up-triangles”.

The maps in the following construction will be play an important role in the
combinatorial characterization of $\mathfrak{m}^{d}$.

######  Construction 5.11.

Adopt notation and hypotheses of Setup 5.2. Let $\mathcal{B}_{d}$ be the
Boolean poset on $[d]$ and $\\{X_{i}\\}_{1\leq i\leq n}$ be a set of rank-
preserving isotone maps

$\displaystyle
X_{i}:(\Delta^{\mathbb{Z}}(n,d),\leq_{i})\rightarrow\mathcal{B}_{d}.$

For any $\mathbf{a}\in\Delta^{\mathbb{Z}}(n,d)$, let
$m_{i}(\mathbf{a})=\prod_{j\in X_{i}(\mathbf{a})}x_{ij}$ and
$m(\mathbf{a})=\prod_{i=1}^{n}m_{i}(\mathbf{a})$. Let $J$ be the ideal in
$k[\check{X}_{1},\dots,\check{X}_{n}]$ generated by the $m(\mathbf{a})$.

######  Definition 5.12 (Linear syzygy edge).

Let $(\mathbf{c};i,j)$ be an edge of
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$, where
$\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$. Then $(\mathbf{c};i,j)$ is a
_linear syzygy edge_ (or _LS-edge_) if there is a monomial $\mathbf{m}$ of
degree $d-1$ such that

$m(\mathbf{c}-\epsilon_{i})=x_{jr}\cdot\mathbf{m}\quad\textrm{and}\quad
m(\mathbf{c}-\epsilon_{j})=x_{is}\cdot\mathbf{m},$

for suitable variables $x_{jr}\in\check{X}_{j}$ and $x_{is}\in\check{X}_{i}$.
This edge gives a linear syzygy between the monomials
$m(\mathbf{c}-\epsilon_{i})$ and $m(\mathbf{c}-\epsilon_{j})$. Equivalently,
in terms of the isotone maps,

$X_{p}(\mathbf{c}-\epsilon_{i})=X_{p}(\mathbf{c}-\epsilon_{j})$

for every $p\neq i,j$. Observe that both $m_{i}(\mathbf{c}-\epsilon_{i})$ and
$m_{j}(\mathbf{c}-\epsilon_{j})$ are common factors of
$m(\mathbf{c}-\epsilon_{i})$ and $m(\mathbf{c}-\epsilon_{j})$.

######  Notation 5.13.

For any $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$, let
$\mathrm{LS}(\mathbf{c})$ be the set of linear syzygy edges in the complete
down-graph $D(\mathbf{c})$. For any $\mathbf{b}\in\Delta^{\mathbb{Z}}(n,d-1)$,
denote by $\mathrm{LS}_{i}(\mathbf{b})$ the set of linear syzygy edges in the
induced subgraph of $U(\mathbf{b})$ on the vertex set
$\\{\mathbf{b}+\epsilon_{j}\mid j\neq i\\}$.

Sometimes, one may wish to consider whether two elements of
$\Delta^{\mathbb{Z}}(n,d)$ would share a linear syzygy edge with respect to a
subset of $[n]$.

######  Definition 5.14 ($R$-linear syzygy edge).

Let $R\subseteq[n]$ and $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ with $R$
contained in the support of $\mathbf{c}$. Let $r,s\in R$. Define
$(\mathbf{c};r,s)$ to be an _$R$ -linear syzygy edge_ if

$X_{p}(\mathbf{c}-\epsilon_{r})=X_{p}(\mathbf{c}-\epsilon_{s})\text{ for }p\in
R\setminus\\{r,s\\}.$

By the isotonicity of the $X_{p}$, for $p=r,s$,

$X_{r}(\mathbf{c}-\epsilon_{r})\subseteq X_{r}(\mathbf{c}-\epsilon_{s}),\quad
X_{s}(\mathbf{c}-\epsilon_{s})\subseteq X_{s}(\mathbf{c}-\epsilon_{r}).$

Let $D_{R}(\mathbf{c})$ be the complete graph with edges $(\mathbf{c};r,s)$
for $r,s\in R$.

The following lemma will be particularly useful in Section 6.

######  Lemma 5.15.

Let $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$. If the set of linear syzygy
edges in $\mathrm{LS}(\mathbf{c})$ contains a spanning tree for
$D(\mathbf{c})$, then for each $R\subseteq\operatorname{supp}(\mathbf{c})$,
the set of $R$-linear syzygy edges contains a spanning tree for
$D_{R}(\mathbf{c})$.

We conclude this section by presenting the main theorem of [1]: a complete
combinatorial characterization of all polarizations of $\mathfrak{m}^{d}$ in
terms of their graphs of linear syzygies.

######  Theorem 5.16 ([1]).

Adopt notation and hypotheses of Setup 5.2 and Construction 5.11. A set of
isotone maps $X_{1},\dots,X_{n}$ determines a polarization of the ideal
$(x_{1},\dots,x_{n})^{d}$ if and only if for every
$\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$, the linear syzygy edges
$\mathrm{LS}(\mathbf{c})$ contain a spanning tree for the down-graph
$D(\mathbf{c})$.

$x_{1}x_{2}y_{2}$$x_{1}x_{2}z_{2}$$x_{1}x_{2}x_{3}$$y_{1}y_{2}y_{3}$$z_{1}z_{2}z_{3}$$x_{1}z_{1}z_{2}$$x_{2}y_{1}y_{2}$$y_{1}y_{2}z_{2}$$y_{1}z_{2}z_{3}$$x_{1}y_{1}z_{2}$
Figure 5. An example of a polarization of $(x,y,z)^{3}$.

######  Example 5.17.

Figure 5 depicts the graph of linear syzygies for a polarization of
$(x,y,z)^{3}$. Notice that at most one edge is removed from each down-
triangle, so it satisfies the spanning tree condition of Theorem 5.16.

##  6\. Hook Tableaux and Polarizations

The goal of this section is to provide a dictionary between the notation and
terminology introduced in Section 5 and the Schur modules appearing in the
$L$-complexes of Section 3. More precisely, we give a new combinatorial
characterization of all polarizations of powers of the graded maximal ideal in
terms of generating sets of the Schur module $L^{1}_{d}(F)$. The main result
of this section is Theorem 6.5, which shows that the spanning tree condition
of Theorem 5.16 is equivalently asking that the Young tableaux canonically
associated to the linear syzygy edges form a basis for the associated Schur
module.

The actual dictionary for translating between the different aforementioned
frameworks is given by Proposition 6.2; these results will be employed in
Section 7 to extend the results of 5.1 to the case of restricted powers.

######  Setup 6.1.

Fix integers $n$ and $d$, and let $S=k[x_{1},\dots,x_{n}]$ be a polynomial
over a field $k$. Let $\check{X}_{i}=\\{x_{i1},\dots,x_{id}\\}$ be a set of
variables, and let $\tilde{S}=k[\check{X}_{1},\dots,\check{X}_{n}]$ be a
polynomial ring in the union of all these variables. Denote by
$\mathfrak{m}=(x_{1},\dots,x_{n})$ the graded maximal ideal of $S$.

Let $\Delta(n,d)$ be the dilated $(n-1)$-simplex from Definition 4.1. Denote
by $\Delta^{\mathbb{Z}}(n,d)=\Delta(n,d)\cap\mathbb{Z}^{n}$ the set of lattice
points of the dilated simplex $d\Delta_{n-1}$, i.e., the set of tuples
$\mathbf{a}=(a_{1},\dots,a_{n})$ of non-negative integers with
$\sum_{i}^{n}a_{i}=d$. Denote by $\operatorname{sk}_{1}(\mathcal{H}^{d}_{n})$
the one-skeleton of the hypersimplicial complex $\mathcal{H}^{d}_{n}$ from
Definition 4.2.

Let $L^{a}_{b}(F)$ be the Schur module defined in Setup 3.2.

######  Proposition 6.2.

Adopt notation and hypotheses of Setup 6.1. Then:

  1. (a)

There exists a bijection $\psi_{n,d}$ from $\Delta^{\mathbb{Z}}(n,d)$ to
$S_{d}(F)$.

  2. (b)

For any pair $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ and
$R\subseteq\operatorname{Supp}(\mathbf{c})$ such that $\lvert R\rvert=t$, the
complete subgraph $D_{R}(\mathbf{c})$ (see Definition 5.14) corresponds to a
unique element of $\bigwedge^{t}F\otimes S_{d-t+1}(F)$.

  3. (c)

There exists a bijection $\theta_{n,d}$ from the set of edges of
$\Delta^{\mathbb{Z}}(n,d)$ to $\bigwedge^{2}F\otimes S_{d-1}(F)$.

###### Proof.

For (a), the map

$\psi_{n,d}:\Delta^{\mathbb{Z}}(n,d)\rightarrow S_{d}(F)$ (2)

such that $\psi_{n,d}(\mathbf{a})=f^{\mathbf{a}}$ gives the desired bijection.

For (b), Let $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$. If
$R\subseteq\operatorname{Supp}(\mathbf{c})$, define

$\omega_{n,d}(\mathbf{c},R)\coloneqq f_{R}\otimes\psi_{n,d-\lvert
R\rvert+1}(\mathbf{c}_{R})$ (3)

where $\mathbf{c}_{R}=\mathbf{c}-\sum_{i\in R}\epsilon_{i}$. In particular, if
$R=\operatorname{Supp}(\mathbf{c})$, then the map

$\omega_{n,d}:\Delta^{\mathbb{Z}}(n,d+1)\rightarrow\bigwedge^{t}F\otimes
S_{d-t-1}(F)\\\ $ (4)

such that
$\omega_{n,d}(\mathbf{c})=f_{\operatorname{Supp}(\mathbf{c})}\otimes\psi_{n,d-t+1}(\mathbf{c}^{\prime})$
is a bijection between the down-triangles of
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$ and $\bigwedge^{t}F\otimes
S_{d-t-1}(F)$, where
$\mathbf{c}^{\prime}=\mathbf{c}-\sum_{i\in\operatorname{Supp}(\mathbf{c})}\epsilon_{i}$
and $t=\lvert\operatorname{Supp}(\mathbf{c})\rvert$.

For (c), If $(\mathbf{c};i,j)$ is an edge in
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$, then the map

$\theta_{n,d}(\mathbf{c};i,j)\coloneqq f_{i}\wedge
f_{j}\otimes\psi_{n,d-1}(\mathbf{c}-\epsilon_{i}-\epsilon_{j})$ (5)

gives a bijection between edges of
$\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$ and elements of
$\bigwedge^{2}F\otimes S_{d-1}(F)$. ∎

######  6.3.

Let $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ be a down-triangle and let
$R\subseteq\operatorname{Supp}(\mathbf{c})$. Set
$T=\omega(\mathbf{c},R)\in\bigwedge^{\lvert R\rvert}F\otimes S_{d-\lvert
R\rvert-1}(F)$. The tableaux appearing in the image $\kappa_{\lvert
R\rvert,n-\lvert R\rvert+1}(T)$ (where $\kappa_{a,b}$ is defined in Setup 3.2)
correspond exactly to the down-triangles $D_{P}(\mathbf{c})$ such that $\lvert
P\rvert=\lvert R\rvert-1$. In particular, as noted in Observation 4.5, if
$\lvert R\rvert=3$, $\kappa_{3,n-2}(T)$ corresponds to a linear combination of
the labels appearing on the three edges of $D_{R}(\mathbf{c})$.

######  Definition 6.4.

Let $\chi=\\{X_{i}\\}$ denote a set of isotone maps

$X_{i}:(\Delta^{\mathbb{Z}}(n,d),\geq_{i})\rightarrow\mathcal{B}_{d}$

as in Construction 5.11. Let $\mathrm{LS}_{\chi}$ be the set of linear syzygy
edges after applying $\chi$ to the generators of $\mathfrak{m}^{d}$ as in
Definition 5.12. Denote by $\mathrm{tab}(\chi)$ the set of tableaux in
$\bigwedge^{2}F\otimes S_{d-1}(F)$ associated to the edges
$\mathrm{LS}_{\chi}$ via the correspondence in 4.3.

######  Theorem 6.5.

Adopt notation and hypotheses of Setup 6.1. Let $\chi=\\{X_{i}\\}$ denote a
set of isotone maps as in Construction 5.11. Then the following are
equivalent:

  1. (1)

The elements of $\mathrm{tab}(\chi)$ span the module $L_{d}^{1}(F)$.

  2. (2)

For every $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$, $\mathrm{LS}(\mathbf{c})$
contains a spanning tree of the complete down-graph $D(\mathbf{c})$.

  3. (3)

The set of isotone maps $X_{1},\dots,X_{n}$ determine a polarization of
$(x_{1},\dots,x_{n})^{d}$.

###### Proof.

Note that (2)$\iff$(3) is Theorem 5.16.

(2)$\implies$(1): Take $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ and let
$\mathbf{a}=\mathbf{c}-\epsilon_{i}$ and $\mathbf{b}=\mathbf{c}-\epsilon_{j}$
for some $i,j\in\operatorname{Supp}(\mathbf{c})$. If $LS(\mathbf{c})$ contains
a spanning tree, then for any two vertices $\mathbf{a}$ and $\mathbf{b}$ in
$\mathbf{c}$, there exists a path in $LS(\mathbf{c})$ connecting them. It
suffices to show that for any $\mathbf{a}$ and $\mathbf{b}$ in
$D(\mathbf{c})$, $\theta_{n,d}(\mathbf{c};i,j)$ is in the span of
$\mathrm{tab}(\chi)$.

Proceed by induction on $k$, the number of edges in the shortest path from
$\mathbf{a}$ to $\mathbf{b}$. If $k=1$, then the tableau corresponding to the
edge between $\mathbf{a}$ and $\mathbf{b}$ is in $\mathrm{tab}(\chi)$. Now
assume that for any two vertices $\mathbf{c}-\epsilon_{i}$ and
$\mathbf{c}-\epsilon_{j}$ such that the shortest path in $LS(\mathbf{c})$
between them is length $k$, the tableau $\theta_{n,d}(\mathbf{c};i,j)$ is a
linear combination of elements in $\mathrm{tab}(\chi)$. Let $\mathbf{a}$ and
$\mathbf{b}$ be two vertices of $D(\mathbf{c})$ such that the shortest path
between them is length $k+1$, i.e., there is a set of vertices

$\mathbf{a}=\mathbf{d}^{1},\dots,\mathbf{d}^{k+2}=\mathbf{b}$

such that each $\mathbf{d}^{j}$ is a vertex in $D(\mathbf{c})$ and each pair
$(\mathbf{d}^{j},\mathbf{d}^{j+1})$ is connected by an edge in
$\mathrm{LS}(\mathbf{c})$. The length of the shortest path between
$\mathbf{a}$ and $\mathbf{d}^{k+1}$ is $k$, so by the induction hypothesis,
the tableau labeling the edge between them is spanned by the elements of
$\mathrm{tab}(\chi)$. If
$\mathbf{a}=\mathbf{c}-\epsilon_{i_{1}},\mathbf{b}=\mathbf{c}-\epsilon_{i_{2}},$
and $\mathbf{d}^{k+1}=\mathbf{c}-\epsilon_{i_{3}}$, set
$R=\\{i_{1},i_{2},i_{3}\\}$ and consider the “smaller” down-triangle
$D_{R}(\mathbf{c})$ (see Definition 5.14). By Observation
LABEL:obs:_triangleRels, the edge between $\mathbf{a}$ and $\mathbf{b}$ is a
linear combination of the other two edges of $D_{R}(\mathbf{c})$, which in
turn have been shown to be linear combinations of elements of
$\mathrm{tab}(\chi)$, hence proving the claim.

(1)$\implies$(2): Let $\mathbf{c}\in\Delta^{\mathbb{Z}}(n,d+1)$ be a complete
down-graph in $\operatorname{sk}_{1}(\mathcal{H}_{n}^{d})$. It suffices to
show that for any two vertices $\mathbf{a}=\mathbf{c}-\epsilon_{i}$ and
$\mathbf{b}=\mathbf{c}-\epsilon_{j}$, there exists a path from $\mathbf{a}$ to
$\mathbf{b}$ by edges labeled by tableaux in $\mathrm{tab}(\chi)$. Since
$\mathrm{tab}(\chi)$ must contain a basis of the module $L^{1}_{d}(F)$, one
may assume that $\mathrm{tab}(\chi)$ is a basis itself. Suppose
$\theta_{n,d}(\mathbf{c};i,j)$ is not in $\mathrm{tab}(\chi)$. The result
follows from the following claims:

  1. (i)

There exists some
$R=\\{i,j,\ell_{1}\\}\subseteq\operatorname{Supp}(\mathbf{c})$ such that
$D_{R}(\mathbf{c})$ has at least one edge labeled by an element of
$\mathrm{tab}(\chi)$.

  2. (ii)

Suppose $(\mathbf{c};i,\ell_{1})$ is the unique edge of $D_{R}(\mathbf{c})$
labeled by an element of $\mathrm{tab}(\chi)$. Then there exists some
$P=\\{\ell_{1},\ell_{2},j\\}\subseteq\operatorname{Supp}(\mathbf{c})$ such
that at least one edge of $D_{P}(\mathbf{c})$ has a label appearing in
$\mathrm{tab}(\chi)$ and $\ell_{2}\neq i$.

  3. (iii)

Let $\Gamma(\mathbf{c})$ be the subgraph of $D(\mathbf{c})$ with edges labeled
by elements of $\mathrm{tab}(\chi)$. If $\Gamma(\mathbf{c})$ contains a cycle,
then it corresponds to a linearly dependent subset of $\mathrm{tab}(\chi)$.

To see (i), suppose no tableaux corresponding to edges in any possible
$D_{R}(\mathbf{c})$ are in $\mathrm{tab}(\chi)$. Then no tableaux in any of
the possible straightening relations containing $\theta_{n,d}(\mathbf{c};i,j)$
coming from the image of any of the $\omega_{n,d}(\mathbf{c},R)$ under
$\kappa_{3,n-2}$ appear in $\mathrm{tab}(\chi)$. Hence,
$\theta_{n,d}(\mathbf{c};i,j)$ is not in the span of $\mathrm{tab}(\chi)$.

For (ii), observe that the image of $\omega_{n,d}(\mathbf{c},R)$ under
$\kappa_{3,n-2}$ gives that $\theta_{n,d}(\mathbf{c};i,j)$ is in the span of
$\theta_{n,d}(\mathbf{c};i,\ell_{1})$ and
$\theta_{n,d}(\mathbf{c};j,\ell_{1})$. By assumption,
$\theta_{n,d}(\mathbf{c};i,\ell_{1})\in\mathrm{tab}(\chi)$. Suppose every
other $D_{P}(\mathbf{c})$ with $\lvert P\rvert=3$ containing
$\theta_{n,d}(\mathbf{c};j,\ell_{1})$ has no edges labeled by tableaux in
$\mathrm{tab}(\chi)$. Then $\theta_{n,d}(\mathbf{c};j,\ell_{1})$ is not in the
span of $\mathrm{tab}(\chi)$, contradicting the assumption that
$\mathrm{tab}(\chi)$ spans $L^{1}_{d}(F)$.

To check (iii), proceed by induction on the length $k$ of the cycle. In the
base case where $k=3$, the cycle forms the edges of a down-triangle
$D_{R}(\mathbf{c})$ where $\lvert R\rvert=3$. The three tableaux labeling the
edges of $D_{R}(\mathbf{c})$ make up the straightening relation from the image
of $\omega_{n,d}(\mathbf{c},R)$ under $\kappa_{3,n-2}$, implying they are
linearly dependent. This contradicts the assumption that $\mathrm{tab}(\chi)$
forms a basis. Now assume that any cycle of length $k$ in $\Gamma(\mathbf{c})$
gives a linearly dependent subset of $\mathrm{tab}(\chi)$, and suppose there
is a cycle $\mathbf{a}^{1},\dots,\mathbf{a}^{k+1}$ such that the edges
$(\mathbf{a}^{j},\mathbf{a}^{j+1})$ and $(\mathbf{a}^{1},\mathbf{a}^{k})$ are
labeled by tableaux in $\mathrm{tab}(\chi)$ and each
$\mathbf{a}^{j}=\mathbf{c}-\epsilon_{i_{j}}$. Let
$R=\\{i_{1},i_{k},i_{k+1}\\}$. By the induction hypothesis,
$\theta_{n,d}(\mathbf{c};i_{1},i_{k})$ is equal to a linear combination of
tableaux $\theta_{n,d}(\mathbf{c};i_{k},i_{k+1})$ and
$\theta_{n,d}(\mathbf{c};i_{1},i_{k+1})$; but also by the induction
hypotheses, it is a linear combination of tableaux of the form
$\theta_{n,d}(\mathbf{c};i_{j},i_{j+1})$ where $1\leq j\leq k$. Therefore, the
cycle corresponds to a linearly dependent subset of $\mathrm{tab}(\chi)$.

With claims (i)-(iii) established, iterate the following process. Choose a
triangle $D_{R}(\mathbf{c})$ such that $R=\\{i,j,\ell\\}$ and some edge of
$D_{R}(\mathbf{c})$ is labeled by an element of $\mathrm{tab}(\chi)$. If both
$\theta_{n,d}(\mathbf{c};i,\ell_{1})$ and
$\theta_{n,d}(\mathbf{c};\ell_{1},j)$ are in $\mathrm{tab}(\chi)$, the claim
follows. Suppose that only one of these tableaux are in $\mathrm{tab}(\chi)$;
without loss of generality, assume that
$\theta_{n,d}(\mathbf{c};i,\ell)\in\mathrm{tab}(\chi)$. By (ii), there is some
triangle $D_{P}(\mathbf{c})$ such that $P=\\{\ell_{1},\ell_{2},j\\}$ and at
least one edge of $D_{P}(\mathbf{c})$ has a label appearing in
$\mathrm{tab}(\chi)$. If both $\theta_{n,d}(\mathbf{c};\ell_{1},\ell_{2})$ and
$\theta_{n,d}(\mathbf{c};\ell_{2},j)$ appear in $\mathrm{tab}(\chi)$, then the
claim follows. Otherwise, repeat this process. This process must terminate by
(iii), giving the desired path. ∎

##  7\. Cellular resolutions and Polarizations of Restricted Powers of the
Graded Maximal Ideal

In this section, we extend the results from Sections 5 and 6 to the case of
so-called _restricted powers_ of the graded maximal ideal. This class of
ideals comes from bounding the multidegrees appearing in the generators of
$\mathfrak{m}^{d}$. We show in Proposition 7.6 that these ideals also have a
minimal, linear, cellular resolution arising as a subcomplex of the L-complex.
In addition, we give two combinatorial characterizations of polarizations of
this class of ideals: one in terms of their graphs of linear syzygies, and one
in terms of spanning sets of a submodule of the associated Schur module. These
characterizations are given by Theorem 7.8.

######  Setup 7.1.

Let $I$ be a monomial ideal in a polynomial ring $S=k[x_{1},\dots,x_{n}]$ over
a field $k$ with generating set $\mathcal{G}(I)=(m_{1},\dots,m_{r})$. Let
$\mathbf{u}\in I$ be a monomial, and define the monomial ideal
$I_{\leq\mathbf{u}}$ to be the ideal generated by monomials $\\{m_{i}\mid
m_{i}\text{ divides }\mathbf{u}\\}$.

Let $d_{i}$ be the highest power of $x_{i}$ that appears in a minimal
generator of $I$. Set $\check{X}_{i}=\\{x_{i1},\dots,x_{id_{i}}\\}$ for all
$i$, and define the polynomial ring
$\widetilde{S}=k[\check{X}_{1},\dots,\check{X}_{n}]$ in the union of all these
variables. Observe that $\widetilde{S}$ has a $\mathbb{Z}^{n}$-grading induced
by the first indices of the variables in $\widetilde{S}$.

Let $\widetilde{I}$ be a polarization of $I$ as in Definition 5.1. Define
$\widetilde{I}_{\leq\mathbf{u}}$ to be generated by those elements of
$\widetilde{I}$ with $\mathbb{Z}^{n}$-multidegree bounded above by
$\mathbf{u}$.

The following useful proposition was observed in [10].

######  Proposition 7.2.

Let $I$ be a monomial ideal in a polynomial ring $S$. Fix a multihomogeneous
basis of a multigraded free resolution $\mathbb{F}_{I}$ of $S/I$. Denote by
$\mathbb{F}_{I}(\leq\mathbf{u})$ the subcomplex of $\mathbb{F}$ generated by
multihomogeneous basis elements of multidegrees dividing $\mathbf{u}$.

  1. (1)

The subcomplex $\mathbb{F}_{I}(\leq\mathbf{u})$ is a multigraded free
resolution of $S/(I_{\leq\mathbf{u}})$.

  2. (2)

If $\mathbb{F}_{I}$ is a minimal multigraded free resolution of $S/I$, then
$\mathbb{F}_{I}(\leq\mathbf{u})$ is independent of the choice of basis.

  3. (3)

If $\mathbb{F}_{I}$ is a minimal multigraded free resolution of $S/I$, then
the resolution $\mathbb{F}_{I}(\leq\mathbf{u})$ is also minimal.

######  Proposition 7.3.

Adopt Setup 7.1. Then $\widetilde{I}_{\leq\mathbf{u}}$ is a polarization of
$I_{\leq\mathbf{u}}$.

###### Proof.

Let $\widetilde{\mathbb{F}}$ be a free resolution of $\widetilde{I}$. Then
$\widetilde{\mathbb{F}}$ is a multigraded resolution with respect to the
$\mathbb{Z}^{n}$-multigrading where each variable $x_{i,j}$ in $\check{X}_{i}$
has multidegree $e_{i}$. By the definition of a polarization (Definition 5.1),
$\widetilde{\mathbb{F}}\otimes\widetilde{S}/\sigma\cong\mathbb{F}$, where
$\sigma$ is a regular $\widetilde{S}/\widetilde{I}$-sequence and $\mathbb{F}$
is a minimal free resolution of $I$. By Proposition 7.2, both
$\mathbb{F}(\leq\mathbf{u})$ and $\widetilde{\mathbb{F}}(\leq\mathbf{u})$ are
minimal multigraded free resolutions of $I_{\leq\mathbf{u}}$ and
$\widetilde{I}_{\leq\mathbf{u}}$, respectively; in particular, one has that

$\widetilde{\mathbb{F}}(\leq\mathbf{u})\otimes\widetilde{S}/\sigma\cong\mathbb{F}(\leq\mathbf{u}).$
(6)

It remains to check that $\sigma$ is indeed a regular sequence on
$\widetilde{S}/\widetilde{I}_{\leq\mathbf{u}}$. This follows immediately from
the string of isomorphisms:

$H_{\bullet}(\widetilde{\mathbb{F}}(\leq\mathbf{u})\otimes\widetilde{S}/\sigma)\cong\operatorname{Tor}_{\bullet}^{\widetilde{S}}(\widetilde{S}/\widetilde{I}_{\leq\mathbf{u}},\widetilde{S}/\sigma)\cong
H_{\bullet}(\widetilde{S}/\widetilde{I}_{\leq\mathbf{u}}\otimes
K(\sigma)_{\bullet}),$

where $K(\sigma)_{\bullet}$ denotes the Koszul complex on $\sigma$. Since
$\widetilde{F}(\leq\mathbf{u})\otimes\widetilde{S}/\sigma\cong\mathbb{F}(\leq\mathbf{u})$
is acyclic, it follows that $H_{>0}(\widetilde{S}/\widetilde{I}\otimes
K(\sigma)_{\bullet})=0$, so $\sigma$ is regular on
$\widetilde{S}/\widetilde{I}$ by, for instance, [13, Theorem 14.7]. ∎

We apply these results to extend results on cellular resolutions and
polarizations from powers of the maximal ideal to so-called _restricted
powers_ of the graded maximal ideal. This terminology conforms with that of
[9] and [10].

######  Definition 7.4.

Let $\mathfrak{m}=(x_{1},\dots,x_{n})$ be the graded maximal ideal in a
polynomial ring $S$ over a field $k$. For any vector
$\mathbf{u}\in\mathbb{N}^{n}$, define the _restricted power_ of $\mathfrak{m}$
to be $\mathfrak{m}^{\mathbf{d}}(\leq\mathbf{u})$ to be the ideal generated by
$(\mathbf{x}^{\mathbf{a}}\mid\sum_{i}^{n}a_{i}=d\text{ and }a_{i}\leq
u_{i}\text{ for all }i\in[n])$.

######  Setup 7.5.

Let $S=k[x_{1},\dots,x_{n}]$ be a polynomial over a field $k$. Denote by
$\mathfrak{m}=(x_{1},\dots,x_{n})$ the graded maximal ideal of $S$. For
$\mathbf{u}\in\mathbb{N}^{n}$, let $\mathfrak{m}^{d}(\leq\mathbf{u})$ be the
restricted power of the graded maximal ideal as in Definition 7.4. Let
$\check{X}_{i}=\\{x_{i1},\dots,x_{id}\\}$ be a set of variables, and let
$\widetilde{S}=k[\check{X}_{1},\dots,\check{X}_{n}]$ be a polynomial ring in
the union of all these variables. Let $\mathcal{B}_{d}(\leq u_{i})$ be the
_truncated Boolean poset_ on $[d]$ with elements of rank at most $u_{i}$ in
$\mathcal{B}_{d}$.

Let $\mathcal{H}_{n}^{d}$ be the hypersimplicial complex from Definition 4.2,
and let $\mathcal{H}_{n}^{d}(\leq\mathbf{u})$ be the induced subcomplex with
cells
$\\{C_{\mathbf{a},J}\mid\operatorname{mdeg}(C_{\mathbf{a},J})\leq\mathbf{u}\\}$.
Denote by $\operatorname{sk}_{0}(\mathcal{H}^{d}_{n}(\leq\mathbf{u}))$ and
$\operatorname{sk}_{1}(\mathcal{H}^{d}_{n}(\leq\mathbf{u}))$ the $0$-skeleton
and $1$-skeleton of $\mathcal{H}_{n}^{d}(\leq\mathbf{u})$, respectively. We
also use the notation $\Delta^{\mathbb{Z}}_{\leq\mathbf{u}}(n,d)$ for
$\operatorname{sk}_{0}(\mathcal{H}^{d}_{n}(\leq\mathbf{u}))$.

Let $L^{a}_{b,\leq\mathbf{u}}(F)$ be the Schur module defined in Setup 3.2
restricted to multidegrees $\leq\mathbf{u}$, where the multidegree of an
element $f_{J}\otimes f^{\alpha}\in\bigwedge F^{a}\otimes S_{b}(F)$ is defined
to be $\alpha+\sum_{j\in J}\epsilon_{j}$, where $\epsilon_{j}$ is the $j$’th
unit vector in $\mathbb{N}^{n}$.

######  Proposition 7.6.

Adopt notation and hypotheses of Setup 7.5. Then:

  1. (1)

The induced subcomplex $\mathcal{H}^{d}_{n}(\leq\mathbf{u})$ supports a
polyhedral cellular resolution of $\mathfrak{m}^{d}(\leq\mathbf{u})$.

  2. (2)

The induced subcomplex $\widetilde{\mathcal{H}}^{d}_{n}(\leq\mathbf{u})$
supports a minimal CW cellular resolution of
$\mathfrak{m}^{d}(\leq\mathbf{u})$ which is isomorphic to a subcomplex of the
L-complex.

In particular, $\mathfrak{m}^{d}(\leq\mathbf{u})$ has a linear minimal free
resolution.

###### Proof.

Apply Proposition 7.2. ∎

######  Corollary 7.7.

Adopt notation and hypotheses of Setup 7.1 and let
$\mathbb{1}=(1,1,\dots,1)\in\mathbb{N}^{n}$. The ideal generated by all
squarefree monomials of a given degree in $S$ has a non-minimal resolution
supported on the polyhedral cell complex
$\mathcal{H}_{n}^{d}(\leq\mathbb{1})$, and it has a minimal free resolution
supported on the CW-complex $\widetilde{\mathcal{H}}_{n}^{d}(\leq\mathbb{1})$,
the restriction of the Morse complex from Proposition 4.8.

Moreover, one can extend the characterizations of polarizations of powers of
the graded maximal ideal in Theorem 6.5 to restricted powers of the maximal
ideal. Observe that all the definitions in Section 5 work in this context,
exchanging $\Delta^{\mathbb{Z}}(n,d)$ with
$\operatorname{sk}_{0}(\mathcal{H}_{n}^{d}(\leq\mathbf{u}))$ and exchanging
$\mathcal{B}_{d}$ with $\mathcal{B}_{d}(\leq u_{i})$ as required.

######  Theorem 7.8.

Adopt notation and hypotheses of Setup 7.1. Let $\chi=\\{X_{i}\\}_{i\in[n]}$
denote a set of rank-preserving isotone maps

$X_{i}:(\Delta^{\mathbb{Z}}_{\leq\mathbf{u}}(n,d),\leq_{i})\rightarrow\mathcal{B}_{d}(\leq
u_{i})$

as in Construction 5.11. Denote by $\mathrm{tab}(\chi)$ be the set of tableaux
in $\bigwedge^{2}\mathbb{F}\otimes S_{d-1}(F)$ associated to the linear syzygy
edges in $\operatorname{sk}_{1}(\mathcal{H}_{n}^{d}(\leq\mathbf{u}))$ after
applying the isotone maps in $\chi$ to its vertices. Then the following are
equivalent:

  1. (1)

The elements of $\mathrm{tab}(\chi)$ span the module
$L_{d,\leq\mathbf{u}}^{1}(F)$.

  2. (2)

For every $\mathbf{c}\in\Delta^{\mathbb{Z}}_{\leq\mathbf{u}}(n,d+1)$,
$\mathrm{LS}(\mathbf{c})$ contains a spanning tree of the complete down-graph
$D(\mathbf{c})$.

  3. (3)

The set of isotone maps $X_{1},\dots,X_{n}$ determine a polarization of
$\mathfrak{m}^{d}(\leq\mathbf{u})$.

###### Proof.

(1) $\implies$ (2): The proof is identical to that of Theorem 6.5.

(2) $\implies$ (3): This follows from Proposition 7.3.

(3) $\implies$ (1): Suppose the set of isotone maps
$\chi=\\{X_{i}\\}_{i\in[n]}$ determine a polarization
$\widetilde{\mathfrak{m}^{d}(\leq\mathbf{u})}$ of
$\mathfrak{m}^{d}(\leq\mathbf{u})$. Let $\widetilde{\mathbb{F}}$ be a (not
necessarily minimal) free resolution of
$\widetilde{\mathfrak{m}^{d}(\leq\mathbf{u})}$ with linear syzygies
corresponding to the set of linear syzygy edges induced by $\chi$. Let
$\mathbb{F}=\widetilde{\mathbb{F}}\otimes\widetilde{S}/\sigma$, be the
depolarization of $\widetilde{\mathbb{F}}$, where $\sigma$ is a regular
sequence of variable differences. Then $\mathbb{F}$ is a free resolution of
$\mathfrak{m}^{d}(\leq\mathbf{u})$ with linear syzygies which are in bijection
with $\mathrm{tab}(\chi)$. Since $\mathbb{F}$ must be a free resolution of
$\mathfrak{m}^{d}(\leq\mathbf{u})$, $\mathrm{tab}(\chi)$ must span
$L^{1}_{d,\leq\mathbf{u}}(F)$.

∎

## Acknowledgments

We would like to thank Gunnar Fløystad for his helpful feedback on early
drafts of this paper.

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