# Gröbner geometry for
regular nilpotent Hessenberg Schubert cells

Mike Cummings  ,  Sergio Da Silva  ,  Megumi Harada  and  Jenna Rajchgot
Dept. of Mathematics and Statistics, McMaster University, 1280 Main Street
West, Hamilton, Ontario L8S 4K1, Canada<EMAIL_ADDRESS>Dept. of
Mathematics and Economics, Virginia State University, 1 Hayden Drive,
Petersburg, Virginia 23806, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>Dept. of
Mathematics and Statistics, McMaster University, 1280 Main Street West,
Hamilton, Ontario L8S 4K1, Canada<EMAIL_ADDRESS>Dept. of Mathematics
and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario
L8S 4K1, Canada<EMAIL_ADDRESS>
###### Abstract.

A regular nilpotent Hessenberg Schubert cell is the intersection of a regular
nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe
a set of minimal generators of the defining ideal of a regular nilpotent
Hessenberg Schubert cell in the type $A$ setting. We show that these minimal
generators are a Gröbner basis for an appropriate lexicographic monomial
order. As a consequence, we obtain a new computational-algebraic proof, in
type $A$, of Tymoczko’s result that regular nilpotent Hessenberg varieties are
paved by affine spaces. In addition, we prove that these defining ideals are
complete intersections, are geometrically vertex decomposable, and compute
their Hilbert series. We also produce a Frobenius splitting of each Schubert
cell that compatibly splits all of the regular nilpotent Hessenberg Schubert
cells contained in it. This work builds on, and extends, work of the second
and third author on defining ideals of intersections of regular nilpotent
Hessenberg varieties with the (open) Schubert cell associated to the Bruhat-
longest permutation.

##  1\. Introduction

Hessenberg varieties are subvarieties of the full flag variety
$\mathrm{Flags}(\mathbb{C}^{n})$.111In this manuscript, we restrict to the
case of Lie type $A$, i.e., when the flag variety corresponds to the group
$GL_{n}(\mathbb{C})$ (or $SL_{n}(\mathbb{C})$). Much can also be said about
other Lie types. They were first introduced to algebraic geometers by De Mari,
Procesi, and Shayman [7], and their study lies in the intersection of
algebraic geometry, representation theory, and combinatorics, among other
research areas. For example: these varieties arise in the study of quantum
cohomology of flag varieties, they are generalizations of the Springer fibers
which arise in geometric representation theory, total spaces of families of
suitable Hessenberg varieties support interesting integrable systems [1], and
the study of some of their cohomology rings suggests that there is a rich
Hessenberg analogue to the theory of Schubert calculus on
$\mathrm{Flags}(\mathbb{C}^{n})$.

The motivation of the present paper stems mainly from Schubert geometry.
Moreover, this manuscript can and should be viewed as a natural companion
paper to the recent work [6] of the second and third authors. Let us briefly
recall the setting. The main objects of discussion are the local patches of
Hessenberg varieties — i.e., intersections of Hessenberg varieties with
certain choices of affine Zariski-open subsets of
$\mathrm{Flags}(\mathbb{C}^{n})$. (Throughout this manuscript, we restrict to
the so-called regular nilpotent case – to be defined precisely below in
Section 2.2.) In particular, we may study the local Hessenberg patch ideal,
denoted $I_{w_{0},h}$, for the special case of a regular nilpotent Hessenberg
variety (to be defined in Section 2.2), intersected with the affine coordinate
chart $w_{0}B^{-}B$ of $\mathrm{Flags}(\mathbb{C}^{n})\cong
GL_{n}(\mathbb{C})/B$ centered at the permutation flag corresponding to the
maximal element $w_{0}$ in $S_{n}$; this is the case studied in [6]. (Here $B$
is the usual Borel subgroup of upper-triangular invertible matrices in
$GL_{n}(\mathbb{C})$ and $B^{-}$ is the opposite Borel of lower-triangular
invertible matrices.) This affine coordinate chart is also a Schubert cell,
$B(\text{Id})B/B$ of the identity permutation $\text{Id}\in S_{n}$.

The main contribution of this manuscript is that we prove analogues of the
results of [6] for _all_ possible regular nilpotent Hessenberg Schubert cells
– that is, for all possible intersections of regular nilpotent Hessenberg
varieties, denoted $\text{Hess}(\mathsf{N},h)$, with Schubert cells $BwB/B$,
for any $w\in S_{n}$. It turns out that the perspectives used in [6] can be
treated in a similar manner to the $w_{0}$-chart, as long as we restrict our
attention to the Hessenberg Schubert cells (instead of looking at the entire
local Hessenberg patch). A rough statement of our main results, which is
stated more precisely as Corollary 4.5 and Theorem 4.6, is as follows:

######  Theorem.

Let $n$ be a positive integer and let $h:[n]\to[n]$ be an indecomposable
Hessenberg function. Let $w\in S_{n}$ and suppose that the Hessenberg Schubert
cell $\mathrm{Hess}(\mathsf{N},h)\cap BwB/B$ is non-empty. Let
$J_{w,h}\subseteq\mathbb{C}[BwB/B]$ denote the (radical) defining ideal of
$\mathrm{Hess}(\mathsf{N},h)\cap BwB/B$. Then there exists a lexicographic
monomial order $<$ such that $\mathrm{in}_{<}(J_{w,h})$ is generated by
$|\lambda_{h}|$-many indeterminates, where $\lambda_{h}$ is a partition
associated to the Hessenberg function $h$. Moreover, there is a natural list
of generators of $J_{w,h}$ which form a Gröbner basis for $J_{w,h}$ with
respect to $<$.

The above results easily imply that each non-empty Hessenberg Schubert cell is
isomorphic to an affine space $\mathbb{A}^{r_{w}-|\lambda_{h}|}$ where $r_{w}$
is the dimension of the Schubert cell $BwB/B$, $w\in S_{n}$. Using this, in
Corollary 5.5, we recover the known result that each regular nilpotent
Hessenberg variety is paved by affine spaces. This was initially proved by
Tymoczko in [19] and generalized to the regular (not necessarily nilpotent)
case by Precup in [17]. Tymoczko and Precup’s proofs use the language of Lie
theory (and apply beyond type $A$) and are delicate combinatorial arguments.
Thus, one contribution of this paper is a new, quick proof of this paving-by-
affines result in the type $A$ regular nilpotent case, using purely
computational-algebraic methods.

As in [6], our main theorem has a number of immediate consequences, including
geometric vertex decomposability, Frobenius splitting, and Hilbert series.
These applications are discussed in detail, and more precisely, in Section 5.

######  Corollary.

Let $n$ be a positive integer and let $h:[n]\to[n]$ be an indecomposable
Hessenberg function. Let $w\in S_{n}$ and suppose that the Hessenberg Schubert
cell $\mathrm{Hess}(\mathsf{N},h)\cap BwB/B$ is non-empty. Let
$J_{w,h}\subseteq\mathbb{C}[BwB/B]$ denote the (radical) defining ideal of
$\mathrm{Hess}(\mathsf{N},h)\cap BwB/B$. Then,

  1. (1)

each $J_{w,h}\subseteq\mathbb{C}[BwB/B]$ is a complete intersection ideal;

  2. (2)

the Hilbert series of $\mathbb{C}[BwB/B]/J_{w,h}$, with respect to
$\mathbb{Z}$-grading induced by the natural circle action on the Hessenberg
Schubert cell, is given by

$H_{R/J_{w,h}}(t)=\frac{\displaystyle\prod_{\begin{subarray}{c}k>h(\ell)\\\
v_{w}(k)>v_{w}(\ell)+1\end{subarray}}(1-t^{v_{w}(k)-v_{w}(\ell)-1})}{\displaystyle\prod_{\begin{subarray}{c}i<w(j)\\\
j<w^{-1}(i)\end{subarray}}\left(1-t^{w(j)-i}\right)}.$

  3. (3)

each ideal $J_{w,h}$ is geometrically vertex decomposable;

  4. (4)

upon replacing $\mathbb{C}$ by a algebraically closed field $\mathbb{K}$ of
positive characteristic, there is a natural Frobenius splitting of
$\mathbb{K}[BwB/B]$ which compatibly splits all ideals
$J_{w,h}\subseteq\mathbb{K}[BwB/B]$.

We briefly describe the contents of the manuscript. In Section 2 we recall the
key definitions and background. In Section 3 we introduce coordinates on the
intersections of Schubert cells with regular nilpotent Hessenberg varieties,
and define their local defining ideals via explicit relations. Then in Section
4 we prove our main results, and in Section 5 we give a detailed discussion of
the many immediate applications of our main results, as itemized above.

### Remark on the field.

For simplicity, as well as consistency with some of the literature that we
cite, we work over $\mathbb{C}$. We note, however, that all of our
computational-algebraic arguments work more generally. For example, our
Gröbner basis arguments work over arbitrary base fields and our Frobenius
splitting results hold over perfect fields of positive characteristic.

### Acknowledgements

We thank the anonymous referee for a careful reading of this manuscript and
their helpful comments. Cummings was supported in part by an NSERC CGS-M and
The Milos Novotny Fellowship. Da Silva was supported in part by an NSERC
Postdoctoral Fellowship of Canada. Harada was supported in part by NSERC
Discovery Grant 2019-06567 and a Canada Research Chair Tier 2 award. Rajchgot
was supported in part by NSERC Discovery Grant 2017-05732.

##  2\. Background

In this section we recall some background on flag varieties, regular nilpotent
Hessenberg varieties, Hessenberg patch ideals, and a certain circle action on
regular nilpotent Hessenberg varieties. We follow [6] as our main reference
for notational conventions. For the purposes of this manuscript we restrict to
Lie groups of type $A$, although Hessenberg varieties can be defined in
arbitrary Lie type.

###  2.1. Flag varieties

The (full) flag variety ${\rm Flags}(\mathbb{C}^{n})$ is the set of sequences
of nested subspaces of $\mathbb{C}^{n}$

${\rm Flags}(\mathbb{C}^{n})\coloneqq\\{V_{\bullet}=(0=V_{0}\subsetneq
V_{1}\subsetneq V_{2}\subsetneq\cdots\subsetneq
V_{n}=\mathbb{C}^{n})\mid\dim_{\mathbb{C}}(V_{i})=i\\}.$

Let $B$ denote the Borel subgroup of $GL_{n}(\mathbb{C})$ of upper-triangular
invertible matrices and let $U^{-}$ denote the subgroup of
$GL_{n}(\mathbb{C})$ of lower-triangular matrices with $1$’s along the
diagonal. By representing $V_{\bullet}\in{\rm Flags}(\mathbb{C}^{n})$ as an
element of $GL_{n}(\mathbb{C})$ by the matrix whose first $i$ columns span
$V_{i}$, we can identify ${\rm Flags}(\mathbb{C}^{n})$ with
$GL_{n}(\mathbb{C})/B$. Then $U^{-}B$, the set of cosets $uB$ for $u\in
U^{-}$, can be viewed as a coordinate chart on $GL_{n}(\mathbb{C})/B\cong{\rm
Flags}(\mathbb{C}^{n})$, which we show precisely below. In fact, $U^{-}B$ is
an open dense subset of ${\rm Flags}(\mathbb{C}^{n})$.

Let $w\in S_{n}$ be a permutation of the symmetric group on $n$ letters. By
slight abuse of notation, we will often take $w$ to also mean its
corresponding permutation matrix. Here we take the convention that we record
the permutation “along columns” to obtain the corresponding permutation
matrix; that is, we write

(2.1) $w=\begin{bmatrix}|&|&&|\\\ e_{w(1)}&e_{w(2)}&\cdots&e_{w(n)}\\\
|&|&&|\end{bmatrix},$

where $e_{j}$ is the $j^{\rm th}$ standard basis vector, written as a column
vector, with a 1 in the $j^{\rm th}$ entry and 0’s elsewhere.

Given the above notation, we define an open cell (a coordinate chart) in
$GL_{n}(\mathbb{C})/B$ containing the permutation $w\in S_{n}$ by the
translation

$\mathcal{N}_{w}\coloneqq wU^{-}B\subseteq GL_{n}(\mathbb{C})/B.$

To describe $\mathcal{N}_{w}$ more concretely, let $M$ denote a generic
element of $U^{-}$,

(2.2) $M\coloneqq\begin{bmatrix}1\\\ \star&1\\\ \vdots&\vdots&\ddots\\\
\star&\star&\cdots&1\\\ \star&\star&\cdots&\star&1\end{bmatrix},$

where the $\star$’s are complex numbers. Since the entries below the diagonal
are free of constraints, it is clear that $U^{-}$, the set of all such $M$’s,
is isomorphic to affine space $\mathbb{A}^{n(n-1)/2}$. For any $w\in S_{n}$,
the map $M\mapsto wMB\in GL_{n}(\mathbb{C})/B$ is an embedding
$U^{-}\cong\mathbb{A}^{n(n-1)/2}\xrightarrow{\cong}\mathcal{N}_{w}$, showing
that $\mathcal{N}_{w}$ is isomorphic to affine space as well. Henceforth, we
can identify a point in $\mathcal{N}_{w}$ with a translation by $w$ of a
matrix of the form (2.2). In particular, a point in $\mathcal{N}_{w}$ is
uniquely identified by a matrix $wM$ whose $(i,j)$-th entries are given by

(2.3) $[wM]_{i,j}=\begin{cases}1&\text{if }i=w(j),\\\ 0&\text{if
}j>w^{-1}(i),\\\ x_{i,j}&\text{otherwise, for some
}x_{i,j}\in\mathbb{C}.\end{cases}$

######  Example 2.4.

Denote by $w_{0}\in S_{n}$ the longest permutation in Bruhat order. If $n=4$,
then $w_{0}=4321$ in one-line notation and

$w_{0}M=\begin{bmatrix}x_{1,1}&x_{1,2}&x_{1,3}&1\\\ x_{2,1}&x_{2,2}&1&0\\\
x_{3,1}&1&0&0\\\ 1&0&0&0\end{bmatrix}.$

∎

It follows from (2.3) that there is an isomorphism between the coordinate ring
of $\mathcal{N}_{w}$ and the polynomial ring in $n(n-1)/2$ indeterminates, the
$x_{i,j}$ that satisfy $j<w^{-1}(i)$. We denote this polynomial ring by
$\mathbb{C}[\mathbf{x}_{w}]$ where $\mathbf{x}_{w}$ denotes the finite set of
indeterminates labelled $x_{i,j}$ for $j<w^{-1}(i)$.

###  2.2. Hessenberg varieties and Hessenberg patch ideals

We next review the definition of a Hessenberg variety. Let
$[n]\coloneqq\\{1,2,\ldots,n\\}$. A function $h:[n]\to[n]$ is a Hessenberg
function if $h(i)\geq i$ for all $i\in[n]$ and $h(i+1)\geq h(i)$ for all
$i\in[n-1]$. An indecomposable Hessenberg function additionally satisfies
$h(i)\geq i+1$ for all $i\in[n-1]$. To a Hessenberg function $h$, we may
associate the Hessenberg subspace of $\mathfrak{gl}_{n}(\mathbb{C})$, defined
by

$H(h):=\\{(a_{i,j})_{i,j\in[n]}\,\mid\,a_{i,j}=0\,\textup{if}\,i>h(j)\\},$

where $\mathfrak{gl}_{n}(\mathbb{C})$ is the Lie algebra of
$GL_{n}(\mathbb{C})$ consisting of $n\times n$ matrices with entries in
$\mathbb{C}$. Given an indecomposable Hessenberg function $h$ and a linear
operator $A:\mathbb{C}^{n}\to\mathbb{C}^{n}$, we define the Hessenberg variety
associated to $A$ and $h$ to be the following subvariety of ${\rm
Flags}(\mathbb{C}^{n})$,

${\rm Hess}(A,h)\coloneqq\\{(V_{0}\subsetneq\cdots\subsetneq V_{n})\in{\rm
Flags}(\mathbb{C}^{n})\mid AV_{i}\subseteq V_{h(i)}\text{ for all
}i\in[n]\\}.$

Equivalently, viewing a flag as a coset $MB$ in $GL_{n}(\mathbb{C})/B$ and
representing it by a matrix $M$ in the coset $MB$, and viewing $A$ as an
element in $\mathfrak{gl}_{n}(\mathbb{C})$, we may also describe
$\mathrm{Hess}(A,h)$ as

(2.5) $\mathrm{Hess}(A,h)\coloneqq\\{MB\in
GL_{n}(\mathbb{C})/B\,\mid\,M^{-1}AM\in H(h)\\}.$

For more on this equivalence for all regular Hessenberg varieties, see [14,
Section 4].

In this manuscript we focus on the case that $A$ is a regular nilpotent
operator. Specifically, we restrict attention to the Hessenberg varieties
${\rm Hess}(A,h)$ where $A=\mathsf{N}$ is defined by

(2.6) $\mathsf{N}\coloneqq\begin{bmatrix}0&1&0&\cdots&0\\\ 0&0&1&\cdots&0\\\
\vdots&&&\ddots&\vdots\\\ 0&0&0&\cdots&1\\\ 0&0&0&\cdots&0\end{bmatrix}.$

The remainder of this section will be devoted to reviewing the defining
equations for the affine variety $\mathcal{N}_{w}\cap{\rm Hess}(\mathsf{N},h)$
from [6, Section 2.2]. In the next section, we will describe equations which
define the intersection of ${\rm Hess}(\mathsf{N},h)$ with a _Schubert cell_.

In the definition below, as well as in the discussion that follows, we view
$M$ as a matrix with entries $0,1,$ or an indeterminate in the set
$\mathbf{x}_{w}=\\{x_{i,j}\\}$ as in Section 2.1, and view each matrix entry
of $(wM)^{-1}\mathsf{N}(wM)$ as an element in $\mathbb{C}[\mathbf{x}_{w}]$.
Observe that since $\det(wM)=\pm 1$, the entries of $(wM)^{-1}$ are
polynomials in the entries of $wM$, and in particular, are not rational
functions.

######  Definition 2.7 ([2, Definition 3.3]).

Let $h$ be a Hessenberg function. Let $w\in S_{n}$ and $k,\ell\in[n]$.
Following the notation of (2.3) and (2.6), define polynomials in
$\mathbb{C}[\mathbf{x}_{w}]$ by

$f^{w}_{k,\ell}\coloneqq[(wM)^{-1}\mathsf{N}(wM)]_{k,\ell}.$

For an indecomposable Hessenberg function $h$, we define the Hessenberg patch
ideal corresponding to $w$ and $h$ to be the ideal

$I_{w,h}\coloneqq\langle f^{w}_{k,\ell}\mid
k>h(\ell)\rangle\subseteq\mathbb{C}[\mathbf{x}_{w}].$

From (2.5) it follows that the $f^{w}_{k,\ell}$ with $k>h(\ell)$ must vanish
on $\mathrm{Hess}(\mathsf{N},h)$ and they (set-theoretically) cut out
$\mathrm{Hess}(\mathsf{N},h)$ in $\mathcal{N}_{w}$. In fact, the following is
known.

######  Proposition 2.8 ([2, Proposition 3.7]).

The Hessenberg patch ideals $I_{w,h}$ defined above are radical and are the
defining ideals of the affine variety $\mathcal{N}_{w}\cap{\rm
Hess}(\mathsf{N},h)$.

######  Example 2.9.

Consider $w_{0}\in S_{4}$ as in Example 2.4. Then,

$(w_{0}M)^{-1}\mathsf{N}(w_{0}M)=\begin{bmatrix}0&0&0&0\\\ 1&0&0&0\\\
-x_{2,2}+x_{3,1}&1&0&0\\\
-x_{1,2}+x_{1,3}(x_{2,2}-x_{3,1})+x_{2,1}&-x_{1,3}+x_{2,2}&1&0\end{bmatrix}.$

For example, for the indecomposable Hessenberg function $h=(2,3,4,4)$, the
corresponding Hessenberg patch ideal corresponding to $w_{0}$ and $h$ is given
by

$I_{w_{0},h}=\langle-
x_{2,2}+x_{3,1},\,-x_{1,2}+x_{1,3}(x_{2,2}-x_{3,1})+x_{2,1},\,-x_{1,3}+x_{2,2}\rangle.$

∎

In work with Abe, DeDieu, and Galetto, the third author showed that there
exists an explicit inductive formula for the polynomials $f^{w_{0}}_{k,\ell}$
[2, Equation (3.6)]. Building on this work, the second and third authors found
an explicit formula for the initial term for the polynomials
$f^{w_{0}}_{k,\ell}$ and recursive equations relating the $f^{w_{0}}_{k,\ell}$
for varying $k$ and $\ell$ [6]. To proceed further, we require a monomial
order, which in [6] is defined as follows.

######  Definition 2.10 ([6, Definition 4.11]).

We denote by $<_{n}$ the lexicographical monomial order on
$\mathbb{C}[\mathbf{x}_{w_{0}}]$ defined by
$x_{i,j}>_{n}x_{i^{\prime},j^{\prime}}$ if $i<i^{\prime}$, or, $i=i^{\prime}$
and $j<j^{\prime}$.

Moreover, there exists an ordering of the polynomials $f^{w_{0}}_{k,\ell}$
such that the initial term of one polynomial does not appear in any of the
polynomials later in the order. This sequence can be obtained by reading from
the matrix $(w_{0}M)^{-1}\mathsf{N}(w_{0}M)$ left-to-right along the bottom
row, then left-to-right along the penultimate row, and so on. Explicitly, this
is the sequence

(2.11)
$f^{w_{0}}_{n,1},\,f^{w_{0}}_{n,2},\,\ldots,\,f^{w_{0}}_{n,n-1},\,f^{w_{0}}_{n-1,1},\,f^{w_{0}}_{n-1,2},\,\ldots.$

It is shown in [6] that the initial terms of the polynomials
$f^{w_{0}}_{k,\ell}$ with respect to $<_{n}$ form a list of distinct
indeterminates. We have the following.

######  Lemma 2.12 ([6, Lemma 4.13]).

Let $k,\ell\in[n]$ satisfy $k>\ell+1$. Then the following hold.

  1. (1)

With respect to the monomial order $<_{n}$ of Definition 2.10,
$\mathrm{in}_{<_{n}}(f^{w_{0}}_{k,\ell})=-x_{n+1-k,\ell+1}$. In particular,
the initial term is an indeterminate (up to sign).

  2. (2)

The initial term $-x_{n+1-k,\ell+1}$ of $f^{w_{0}}_{k,\ell}$ does not appear
in any $f^{w_{0}}_{k^{\prime},\ell^{\prime}}$ which are listed after
$f^{w_{0}}_{k,\ell}$ in the sequence (2.11).

  3. (3)

The indeterminate $x_{n+1-k,\ell+1}$ appears exactly once in
$f^{w_{0}}_{k,\ell}$ and all other indeterminates $x_{i,j}$ appearing in
$f^{w_{0}}_{k,\ell}$ either satisfy $i>n+1-k$ or $i=n+1-k$ and $j>\ell+1$.

Let $h$ be an indecomposable Hessenberg function. By Definition 2.7, the ideal
$I_{w_{0},h}$ is generated by the polynomials $f^{w_{0}}_{k,\ell}$ for all
$k,\ell\in[n]$ with $k>h(\ell)$. By definition of indecomposable Hessenberg
functions, if $k>h(\ell)$ then $k\geq\ell+1$, so the above lemma applies. We
additionally have the following useful result from [6].

######  Proposition 2.13 ([6, Corollary 4.15]).

For any indecomposable Hessenberg function $h$, the generators
$f^{w_{0}}_{k,\ell}$ form a Gröbner basis for the Hessenberg patch ideal
$I_{w_{0},h}$ with respect to $<_{n}$.

In this paper, we generalize Proposition 2.13 to all regular nilpotent
Hessenberg Schubert cells.

######  Remark 2.14.

Throughout this manuscript, we restrict to indecomposable Hessenberg
functions. This restriction is quite natural since, in type $A$, any regular
nilpotent Hessenberg variety is the product of indecomposable regular
nilpotent Hessenberg varieties [8, Theorem 4.5].

###  2.3. $\mathsf{S}$-actions on regular nilpotent Hessenberg varieties

In this manuscript we will use a certain circle action on
$\mathrm{Hess}(\mathsf{N},h)$ which allows us to restrict attention to an
appropriate subset of Schubert cells for our analysis. It is the same circle
action as is used in [6], so we keep the exposition brief and refer the reader
to [6, Section 2] for details.

Let $\mathsf{S}\cong\mathbb{C}^{*}$ denote the circle subgroup of the maximal
torus of $GL_{n}(\mathbb{C})$ given by

$\mathsf{S}\coloneqq\\{\underline{t}\coloneqq\mathrm{diag}(t,t^{2},\ldots,t^{n})\,\mid\,t\in\mathbb{C}^{*}\\}.$

It is straightforward to check that $\mathsf{S}$ preserves
$\mathrm{Hess}(\mathsf{N},h)$. Moreover, since $\mathsf{S}$ is a subgroup of
the usual maximal torus of $GL_{n}(\mathbb{C})$ and the maximal torus
preserves the coordinate patches $\mathcal{N}_{w}$, it follows that
$\mathsf{S}$ also preserves the cell $\mathcal{N}_{w}$, hence also the
intersections $\mathrm{Hess}(\mathsf{N},h)\cap\mathcal{N}_{w}$.

Additionally, we can also concretely compute the action of $\mathsf{S}$ on any
local coordinate patch $\mathcal{N}_{w}$. Recall that the standard maximal
torus action on $GL_{n}(\mathbb{C})/B$ is given by left multiplication on left
cosets; since $\mathsf{S}$ is a subgroup, it acts in the same way. More
precisely, given a matrix $wM$ representing a flag $[wM]\in
GL_{n}(\mathbb{C})/B$, we have

$\underline{t}\cdot[wM]=[\underline{t}(wM)]$

for any $\underline{t}\in\mathsf{S}$. To express $[\underline{t}(wM)]$ in
terms of the coordinate chart $\mathcal{N}_{w}$, we must now find a matrix
$M^{\prime}$ with entries described by (2.3) such that
$\underline{t}wM=wM^{\prime}$. The simple observation here is that left
multiplication by the diagonal matrix $\underline{t}$ changes the entries in
the $(w(j),j)$-th spots to be $t^{w(j)}$ instead of a $1$, so in order to
return the matrix to the required form, it is necessary to multiply on the
right by the diagonal element
$\mathrm{diag}(t^{-w(1)},t^{-w(2)},\ldots,t^{-w(n)})$, i.e. the diagonal
matrix whose $j$-th diagonal entry is $t^{-w(j)}$. From this simple
observation it is not hard to see that the matrix $wM^{\prime}$ satisfies

$[wM^{\prime}]_{i,j}=\begin{cases}1&\text{if }i=w(j),\\\ 0&\text{if
}j>w^{-1}(i),\\\ t^{i-w(j)}x_{i,j}&\text{otherwise}\end{cases}$

where the $x_{i,j}$ are the entries of $wM$ as in (2.3).

The above-described torus action on the patch $\mathcal{N}_{w}$ induces an
$\mathsf{S}$-grading on the coordinate ring $\mathbb{C}[\mathbf{x}_{w}]$,
where the indeterminate $x_{i,j}$ has degree $w(j)-i$. In Section 3, we will
restrict this action further to the Schubert cell at $w$, but the principle
remains the same. In fact, in this setting, we will see that this
$\mathbb{Z}$-grading is positive.

##  3\. Coordinates on regular nilpotent Hessenberg Schubert cells

In this section, we will relate coordinates on the regular nilpotent
Hessenberg Schubert cells to certain coordinates on the $w_{0}$-chart of the
same Hessenberg variety. This will allow us, in the next section, to adapt
results about the $w_{0}$-chart from [6] to yield results for the regular
nilpotent Hessenberg Schubert cells, allowing us to draw conclusions about,
e.g., the initial terms, recurrence relations, and Gröbner bases. Indeed,
finding this explicit relation with the $w_{0}$-chart allows us to avoid
reproving the technical details of [6].

The Schubert cell of a permutation $w$ is defined by

$X^{w}_{\circ}\coloneqq BwB/B\subseteq
GL_{n}(\mathbb{C})/B\cong\mathrm{Flags}(\mathbb{C}^{n}).$

By the well-known theory of Bruhat decomposition, the Schubert cells are
disjoint and their union is $GL_{n}(\mathbb{C})/B$ (see [10, §23.4] for
further background on the Bruhat decomposition). Furthermore, each
$X^{w}_{\circ}$ is isomorphic to a complex affine space of dimension
$\ell(w)$, where $\ell(w)$ denotes the length of the permutation $w$. In fact,
in a manner similar to how we described elements in $GL_{n}(\mathbb{C})/B$ in
the previous section, we can parametrize the Schubert cell by viewing an
element of $X^{w}_{\circ}$ as represented by a matrix of the form
$\Omega_{w}$, where the $(i,j)$-th entry is given by:

(3.1) $[\Omega_{w}]_{i,j}=\begin{cases}1&\text{if }i=w(j),\\\ 0&\text{if
}i>w(j)\text{ or }j>w^{-1}(i),\\\ z_{i,j}&\text{otherwise (i.e., }i<w(j)\text{
and }j<w^{-1}(i)).\end{cases}$

Note that the condition $i>w(j)$ corresponds to the matrix entries occurring
below the $1$’s and the condition $j>w^{-1}(i)$ corresponds to the entries
occurring to the right of the $1$’s. This parametrization illustrates that
$X^{w}_{\circ}$ is isomorphic to an affine space, and we can view
$\mathbf{z}_{w}:=\\{z_{i,j}\,\mid\,i<w(j)\,\textup{ and }\,j<w^{-1}(i)\\}$ as
indeterminates corresponding the affine coordinates on $X^{w}_{\circ}$. In
particular, the affine coordinate ring of $X^{w}_{\circ}$ can be identified
with the polynomial ring $\mathbb{C}[\mathbf{z}_{w}]$. Note that, in the
discussion below, we will reserve the coordinates $x_{i,j}$ for entries of
$wM$ as in (2.3), and the coordinates $z_{i,j}$ for entries of $\Omega_{w}$ as
in (3.1).

We now wish to relate the coordinates $\mathbf{z}_{w}$ with the coordinates
$\mathbf{x}_{w_{0}}$. To do this, our first observation is that we can write
the $z_{i,j}$ coordinates of the Schubert cell as a specialization of the
$x_{i,j}$ coordinates of the $w_{0}$-chart (by setting certain variables to
zero after an appropriate relabelling). We start with a motivational example
to demonstrate this behaviour, and describe this process more precisely
afterwards.

######  Example 3.2.

Let $n=4$ and $w=3421$. As we see below, for an element
$w_{0}M\in\mathcal{N}_{w_{0}}$, there exists a permutation matrix $v_{w}$ such
that if we right-multiply $w_{0}M$ by $v_{w}$ and then set some variables
equal to $0$, then we obtain an element of the cell $X^{w}_{\circ}$. Indeed,
in our example, $v_{w}$ turns out to be the permutation matrix

$v_{w}=\begin{bmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{bmatrix}$

which corresponds to the permutation $v_{w}=2134$. We can compute the result
of right-multiplication by $v_{w}$ and setting $x_{3,1}$ to $0$, and obtain

(3.3) $\left.\left(\begin{bmatrix}x_{1,1}&x_{1,2}&x_{1,3}&1\\\
x_{2,1}&x_{2,2}&1&0\\\ x_{3,1}&1&0&0\\\
1&0&0&0\end{bmatrix}\begin{bmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix}\right)\right|_{x_{3,1}=0}=\begin{bmatrix}x_{1,2}&x_{1,1}&x_{1,3}&1\\\
x_{2,2}&x_{2,1}&1&0\\\ 1&0&0&0\\\ 0&1&0&0\end{bmatrix}\in X^{w}_{\circ}.$

For an element of the cell $X^{w}_{\circ}$, we may represent it as a matrix
with entries as in (3.1) by

(3.4) $\Omega_{w}=\begin{bmatrix}z_{1,1}&z_{1,2}&z_{1,3}&1\\\
z_{2,1}&z_{2,2}&1&0\\\ 1&0&0&0\\\ 0&1&0&0\end{bmatrix}.$

Using the correspondence (3.3), we can identify the coordinates of
$\Omega_{w}$ with those of $w_{0}M$ by a ring homomorphism $\psi_{w}$ from
$\mathbb{C}[\mathbf{x}_{w_{0}}]\rightarrow\mathbb{C}[\mathbf{z}_{w}]$ defined
by comparing the matrix of (3.3) entry-by-entry to $\Omega_{w}$ in (3.4). For
instance, in this example, $z_{1,1}$ is identified with $x_{1,2}$, $z_{1,2}$
with $x_{1,1}$, and so on. The reader may verify that this is exactly the
correspondence given in Definition 3.8 below, which formalizes and generalizes
the phenomenon seen in this example. ∎

Following the example, given a permutation $w\in S_{n}$, define
$v_{w}\coloneqq w_{0}w$. Denote the $(i,j)$-th entry of the matrix $w_{0}M$ by
$x_{i,j}$ for $i,j\in[n]$ with $i+j\leq n+1$. (Here we set
$x_{i,n-i+1}\coloneqq 1$ for all $1\leq i\leq n$, i.e., the anti-diagonal
entries are set to be equal to $1$.) Under right-multiplication by $v_{w}$
(and by our convention (2.1) for permutation matrices), the columns of
$w_{0}M$ are permuted, and in particular, the $(i,j)$-th entry of the matrix
$(w_{0}M)v_{w}$ is the $(i,v_{w}(j))$-th entry of the matrix $w_{0}M$.
Equivalently, we have

$[(w_{0}M)v_{w}]_{i,j}=[w_{0}M]_{i,v_{w}(j)}.$

Moreover, since it is precisely the anti-diagonal entries of $w_{0}M$ which
are equal to $1$, the above equality implies that the $(i,j)$-th entry of
$(w_{0}M)v_{w}$ is equal to $1$ precisely when
$i=n+1-v_{w}(j)=w_{0}(v_{w}(j))=w(j)$, where the last equality is by our
choice of $v_{w}$. This means that the $1$’s in the matrix $(w_{0}M)v_{w}$ are
located precisely at the same locations as the $1$’s in the permutation matrix
corresponding to $w$. Therefore, if the variables appearing in $(w_{0}M)v_{w}$
below the 1’s and to the right of the $1$’s are all equal to $0$, then the
matrix represents an element of the cell $X^{w}_{\circ}$.

Indeed, from (3.1), it then follows that in order for the matrix
$(w_{0}M)v_{w}$ to be in $X^{w}_{\circ}$, we must have that
$[(w_{0}M)v_{w}]_{i,j}=0$ if either

$w(j)<i$

or

$j>w^{-1}(i).$

We can then ask which indeterminates appearing as entries in $w_{0}M$,
obtained by right-multiplication by $v_{w}$, have to be equal to $0$ in order
for the corresponding matrix $(w_{0}M)v_{w}$ to be contained in
$X^{w}_{\circ}$. Indeed, let $x_{a,b}$ denote the $(a,b)$-th entry of $w_{0}M$
for $a+b<n+1$. (As we already saw, if $a+b=n+1$ then $x_{a,b}=1$ so we do not
consider this case.) Based on the above paragraphs, it is straightforward now
to see that if $(w_{0}M)v_{w}$ is in $X^{w}_{\circ}$ it is necessary to have
$x_{a,b}=0$ if either

(3.5) $w(v_{w}^{-1}(b))<a$

or

(3.6) $v_{w}^{-1}(b)>w^{-1}(a).$

Since $v_{w}=w_{0}w$ by definition, the inequality (3.5) simplifies to
$w_{0}(b)<a$ or equivalently $n+1-b<a$, which can never happen since we
started with $a+b<n+1$. Therefore, the only non-vacuous condition is (3.6).

The preceding discussion motivates the following definitions. (In the previous
paragraph, we used indices $a$ and $b$ to distinguish the indices used for
$w_{0}M$ versus $(w_{0}M)v_{w}$. We now revert to using $i,j$ to denote
indices for the matrix $w_{0}M$.) Let $D_{w}$ denote the collection of
indeterminates $x_{i,j}$ with $(i,j)$ satisfying (3.6); more precisely, let

(3.7) $D_{w}\coloneqq\\{x_{i,j}\in\mathbf{x}_{w_{0}}\mid i+j\leq n\text{ and
}v_{w}^{-1}(j)>w^{-1}(i)\\}.$

######  Definition 3.8.

Define a homomorphism of rings
$\psi_{w}:\mathbb{C}[\mathbf{x}_{w_{0}}]\to\mathbb{C}[\mathbf{z}_{w}]$ by

$\psi_{w}(x_{i,j})\coloneqq\begin{cases}0&\text{if }x_{i,j}\in D_{w},\\\
z_{i,v_{w}^{-1}(j)}&\text{if }x_{i,j}\notin D_{w}.\\\ \end{cases}$

######  Remark 3.9.

Geometrically, the ring homomorphism $\psi_{w}$ corresponds to the embedding
of the Schubert cell $X^{w}_{\circ}$ into $\mathcal{N}_{w_{0}}$, given by
right multiplication of an element $\Omega_{w}\in X^{w}_{\circ}$ by
$v_{w}^{-1}$. Then $\Omega_{w}v_{w}^{-1}\in\mathcal{N}_{w_{0}}$, and certain
coordinates are equal to $0$ by the discussion above, so the corresponding map
on coordinate rings sets those coordinates to $0$.

In what follows, if $A$ is a matrix whose entries are in
$\mathbb{C}[\mathbf{x}_{w_{0}}]$, we denote by $\psi_{w}(A)$ the matrix
obtained by applying the ring homomorphism $\psi_{w}$ to each matrix entry of
$A$, i.e., if $a_{i,j}$ is the $(i,j)$-th entry of $A$, then
$\psi_{w}(a_{i,j})$ is the $(i,j)$-th entry of $\psi_{w}(A)$. By construction,
this results in a matrix with entries in $\mathbb{C}[\mathbf{z}_{w}]$. With
this notation in place, we have the following observations.

######  Remark 3.10.

For any $w\in S_{n}$, we have that $\Omega_{w}=\psi_{w}((w_{0}M)v_{w})$ by
construction of $\psi_{w}$.

We also have the following.

######  Lemma 3.11.

For any $w\in S_{n}$,
$\left(\psi_{w}((w_{0}M)v_{w})\right)^{-1}=\psi_{w}\left(((w_{0}M)v_{w})^{-1}\right)$.

###### Proof.

This follows from the facts that $\psi_{w}$ is a ring homomorphism applied
entry-by-entry and that any entry of the inverse of a matrix can be expressed
as a polynomial in terms of the entries of the given matrix. Indeed, the
determinant of $M$ is equal to $1$ and determinants of permutation matrices
are $\pm 1$. Hence the determinant of $(w_{0}M)v_{w}$ is $\pm 1$ and so the
determinant in the denominator of the usual formula for a matrix inverse does
not appear in this computation, and all entries of the matrix inverse are
polynomials and not rational functions. ∎

######  Definition 3.12.

Let $w\in S_{n}$. As in Definition 2.7, define polynomials $g^{w}_{k,\ell}$ in
$\mathbb{C}[\mathbf{z}_{w}]$ for $k,\ell\in[n]$ by

$g^{w}_{k,\ell}\coloneqq\left[\Omega_{w}^{-1}\mathsf{N}\Omega_{w}\right]_{k,\ell}.$

For an indecomposable Hessenberg function $h$, we define the regular nilpotent
Hessenberg Schubert cell ideal corresponding to $w$ and $h$ to be the ideal

(3.13) $J_{w,h}\coloneqq\langle g^{w}_{k,\ell}\mid
k>h(\ell)\rangle\subseteq\mathbb{C}[\mathbf{z}_{w}].$

As in the case of the generators $f^{w}_{k,\ell}$ in Definition 2.7, it is
straightforward to see from (2.5) that the $g^{w}_{k,\ell}$ set-theoretically
cut out the intersection $\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$ from
$X^{w}_{\circ}$, justifying the terminology in Definition 3.12 above. It will
also follow from our algebraic arguments below that $J_{w,h}$ is radical (see
Corollary 4.7), and hence $J_{w,h}$ also scheme-theoretically defines
$\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$.

######  Notation 3.14.

It will also be useful in what follows to have notation for the number of
generators $g^{w}_{k,\ell}$ of $J_{w,h}$ as listed in (3.13). Let
$\lambda_{h}$ denote the partition $(n-h(1),n-h(2),\ldots,n-h(n-1),n-h(n)=0)$.
Then $\lvert\lambda_{h}\rvert$, the sum of the parts of $\lambda_{h}$, is
$\lvert\lambda_{h}\rvert=n^{2}-\sum_{i=1}^{n}h(i)$. This is equal to the
number of generators $g^{w}_{k,\ell}$ as listed in (3.13).

######  Example 3.15.

Continuing Example 3.2, we now note that we can express the generators for the
regular nilpotent Hessenberg Schubert cell ideal in terms of the Hessenberg
$w_{0}$-patch ideal via the map $\psi_{w}$. First, notice that

(3.16) $\Omega_{w}^{-1}\mathsf{N}\Omega_{w}=\begin{bmatrix}0&1&0&0\\\
0&0&0&0\\\ 1&-z_{2,1}&0&0\\\
-z_{1,3}+z_{2,1}&-z_{1,1}+z_{1,3}z_{2,1}+z_{2,2}&1&0\end{bmatrix}.$

Meanwhile,

$F\coloneqq(w_{0}M)^{-1}\mathsf{N}(w_{0}M)=\begin{bmatrix}0&0&0&0\\\
1&0&0&0\\\ -x_{2,2}+x_{3,1}&1&0&0\\\
-x_{1,2}+x_{1,3}(x_{2,2}-x_{3,1})+x_{2,1}&-x_{1,3}+x_{2,2}&1&0\end{bmatrix}$

and

(3.17) $v_{w}^{-1}Fv_{w}=\begin{bmatrix}0&1&0&0\\\ 0&0&0&0\\\
1&-x_{2,2}+x_{3,1}&0&0\\\
-x_{1,3}+x_{2,2}&-x_{1,2}+x_{1,3}(x_{2,2}-x_{3,1})+x_{2,1}&1&0\end{bmatrix}.$

Since $\Omega_{w}=\psi_{w}((w_{0}M)v_{w})$ as observed in Remark 3.10, it
follows by Lemma 3.11 that
$\Omega_{w}^{-1}=\psi_{w}(v_{w}^{-1}(w_{0}M)^{-1})$. Applying $\psi_{w}$
entry-by-entry, it follows that
$\psi_{w}(v_{w}^{-1}Fv_{w})=\Omega_{w}^{-1}\mathsf{N}\Omega_{w}$. In
particular, the generators satisfy
$g^{w}_{k,\ell}=\psi_{w}([v_{w}^{-1}Fv_{w}]_{k,\ell})$. For example, comparing
the $(3,2)$-th entries of the RHS of (3.16) and (3.17), we can see that

$\begin{split}\psi_{w}(-x_{2,2}+x_{3,1})&=\psi_{w}(-x_{2,2})+\psi_{w}(x_{3,1})\\\
&=-z_{2,1}+0=-z_{2,1}\end{split}$

as claimed. We will formalize this computation in Lemma 3.20 below. ∎

We have just seen in the above example that the generators $g^{w}_{k,\ell}$ in
this special case have a simple algebraic relation to certain of the
$f^{w_{0}}_{i,j}$ generators from Definition 2.7, since the matrix entries of
$F$ from the above example are precisely these generators. In the discussion
that follows, the overarching theme is that we can use this correspondence to
effectively translate results on the $w_{0}$-patch as obtained in [6] to the
regular nilpotent Hessenberg Schubert cells $\mathrm{Hess}(\mathsf{N},h)\cap
X^{w}_{\circ}$. Before proceeding further into the details of the translation,
however, we need first to make precise the permutations $w$ for which this
translation makes sense. As we will show below, it turns out that the only
Schubert cells $X^{w}_{\circ}$ for which the intersection
$\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$ is non-empty (and hence the
questions about local defining ideals is non-vacuous) are the permutations $w$
lying in the $\mathsf{S}$-fixed point set of $\mathrm{Hess}(\mathsf{N},h)$.
The next lemma makes this precise.

######  Lemma 3.18.

Let $w\in S_{n}$. Let $\mathsf{S}$ denote the copy of $\mathbb{C}^{*}$ acting
on $\mathrm{Hess}(\mathsf{N},h)$ as described in Section 2.3. Then
$\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}\neq\emptyset$ if and only if
$w\in\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$.

###### Proof.

Since $w\in X^{w}_{\circ}$ for any $w\in S_{n}$, one direction is clear.
Indeed, since $w\in X^{w}_{\circ}$, if $w\in\mathrm{Hess}(\mathsf{N},h)$ then
$\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}\neq\emptyset$.

It remains to show the other direction. Suppose there exists a point
$gB\in\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$, so it is non-empty.
Following our notational convention in (3.1) for points in the Schubert cell
$X^{w}_{\circ}$, we may represent this point $gB$ as a matrix $\Omega_{w}$
whose $(i,j)$-th entry is $1$ if $i=w(j)$, $0$ if $i>w(j)$ or $j>w^{-1}(i)$,
and denoted $z_{i,j}$ otherwise. We already observed in Section 2.3 that
$\mathrm{Hess}(\mathsf{N},h)$ is preserved under the $\mathsf{S}$-action for
our choice of $\mathsf{S}$. Schubert cells are also preserved, since they are
preserved by the maximal torus action, and $\mathsf{S}$ is a subgroup of the
maximal torus. Thus $\mathsf{S}$ preserves
$\mathrm{Hess}(\mathsf{N},h)\cap\Omega_{w}$, and we conclude that the
$\mathsf{S}$-orbit $\mathsf{S}\cdot gB$ is also contained in
$\mathrm{Hess}(\mathsf{N},h)\cap\Omega_{w}$. Using the matrix representation
of the point $gB$ and the definition of $\mathsf{S}$, it is straightforward to
compute, following the outline given in Section 2.3, that for an element
$\underline{t}\coloneqq\mathrm{diag}(t,t^{2},\ldots,t^{n})$ for
$t\in\mathbb{C}^{*}$, the element $\underline{t}\cdot gB$ can be represented
by a matrix in $GL_{n}(\mathbb{C})$ whose $(i,j)$-th entry is $0$ if $i>w(j)$
or $j>w^{-1}(i)$, is $t^{i}$ if $i=w(j)$, and $t^{i}z_{i,j}$ otherwise. Such a
matrix is not in the form given in (3.1) so we adjust by multiplication on the
right by a torus element in $B$ given by
$\mathrm{diag}(t^{-w(1)},t^{-w(2)},\ldots,t^{-w(n)})$, which then yields a
matrix specifying the same flag, the entries of which are $0$ if $i>w(j)$ or
$j>w^{-1}(i)$, $1$ if $i=w(j)$, and $t^{i-w(j)}z_{i,j}$ otherwise. Now we
observe that by assumption, if the $(i,j)$-th entry is $t^{i-w(j)}z_{i,j}$,
then $i<w(j)$ and therefore $i-w(j)<0$. This implies that as we take the limit
as $t\to\infty$, the limit point of this orbit is precisely the permutation
flag $w$. Since $\mathrm{Hess}(\mathsf{N},h)$ is closed, the limit point $w$
is also contained in $\mathrm{Hess}(\mathsf{N},h)$, and it is evident that $w$
is $\mathsf{S}$-fixed by the above computation. Hence
$w\in\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$, as desired. ∎

The above lemma shows that we can restrict attention to permutations $w$
contained in $\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$. The third author and
Abe, Horiguchi, and Masuda gave an explicit characterization of this set, as
follows.

######  Lemma 3.19 ([3, Lemma 2.3]).

Let $h$ be any Hessenberg function. Then,

${\rm Hess}(\mathsf{N},h)^{\mathsf{S}}=\\{w\in S_{n}\mid w^{-1}(w(j)-1)\leq
h(j)\text{ for all }j\in[n]\\},$

where we take the convention that $w(0)=0$ for all $w\in S_{n}$.

We can now relate the $f^{w_{0}}_{i,j}$ polynomials on the $w_{0}$-chart to
the $g^{w}_{k,\ell}$ polynomials on the regular nilpotent Hessenberg Schubert
cell. The following is a straightforward computation.

######  Lemma 3.20.

For any $w\in S_{n}$ and any $k,\ell\in[n]$, we have
$g^{w}_{k,\ell}=\psi_{w}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)})$.

###### Proof.

By Remark 3.10, Lemma 3.11, and Definition 3.12 of the $g^{w}_{k,\ell}$, we
can compute that

$\displaystyle g_{k,\ell}^{w}$
$\displaystyle=[\Omega_{w}^{-1}\mathsf{N}\Omega_{w}]_{k,\ell}=[(\psi_{w}((w_{0}M)v_{w}))^{-1}\mathsf{N}(\psi_{w}((w_{0}M)v_{w}))]_{k,\ell}$
$\displaystyle=[\psi_{w}(((w_{0}M)v_{w})^{-1})\mathsf{N}(\psi_{w}((w_{0}M)v_{w}))]_{k,\ell}$
and since $\psi_{w}$ is a ring homomorphism,
$\displaystyle=[\psi_{w}((v_{w}^{-1}(w_{0}M)^{-1})\mathsf{N}(w_{0}M)v_{w})]_{k,\ell}$
$\displaystyle=\psi_{w}([v_{w}^{-1}(w_{0}M)^{-1}\mathsf{N}(w_{0}M)v_{w}]_{k,\ell})$

where the last equality holds since $\psi_{w}$ is applied to the matrix entry-
by-entry. The claim of the lemma then follows from the facts that
multiplication of $(w_{0}M)^{-1}\mathsf{N}(w_{0}M)$ on the left by
$v_{w}^{-1}$ permutes rows and multiplication on the right by $v_{w}$ permutes
columns. ∎

Although the above result holds for any $k,\ell\in[n]$, in the next sections
we will always impose further restrictions on $(k,\ell)$ so that they
correspond to the generators $g^{w}_{k,\ell}$ for the ideal $J_{w,h}$.

##  4\. A Gröbner basis for defining ideals of regular nilpotent Hessenberg
Schubert cells

In the last section, we provided an explicit relationship between the
generators $g^{w}_{k,\ell}$ of the local defining ideal of regular nilpotent
Hessenberg Schubert cells and their corresponding entries
$f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}$ in the $w_{0}$-chart. The main purpose of
the present section is to show that these generators in fact form a Gröbner
basis for the ideal $J_{w,h}$. The main strategy is to concretely describe the
relationship between the initial terms of the $g^{w}_{k,\ell}$ and those of
the corresponding $f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}$ through the ring
homomorphism $\psi_{w}$.

We first note that if a polynomial $f^{w_{0}}_{a,b}$ is equal to either $0$ or
$1$, which occurs when $a\leq b+1$, then $\psi_{w}$ will map $f^{w_{0}}_{a,b}$
to itself and the initial terms are vacuously preserved. From Lemma 2.12(1) it
follows that the case in which $f^{w_{0}}_{a,b}$ has a non-constant initial
term is when $a>b+1$. Therefore, in comparing initial terms of
$g^{w}_{k,\ell}$ with the corresponding $f^{w_{0}}_{v_{w}(k),v_{k}(\ell)}$ via
Lemma 3.20, we need only consider the case when $v_{w}(k)>v_{w}(\ell)+1$.

In order to discuss Gröbner bases, we first need to fix a monomial order. Our
first step is therefore to adapt the monomial order $<_{n}$ from Definition
2.10 to a monomial order on $\mathbb{C}[\mathbf{z}_{w}]$. We do this using the
permutation $v_{w}$.

######  Definition 4.1.

Define a lexicographic monomial order $<_{n}^{w}$ on
$\mathbb{C}[\mathbf{z}_{w}]$ by $z_{i,j}>_{n}^{w}z_{i^{\prime},j^{\prime}}$ if
$i<i^{\prime}$ or $i=i^{\prime}$ and $v_{w}(j)<v_{w}(j^{\prime})$.

######  Remark 4.2.

In the case that $w=w_{0}$, the permutation $v_{w}$ is the identity
permutation, so we recover from $<_{n}^{w}$ exactly the monomial order $<_{n}$
from Definition 2.10.

######  Lemma 4.3.

Let $w\in\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$. Assume $k,\ell\in[n]$
satisfy $k\geq h(\ell)$ and $v_{w}(k)>v_{w}(\ell)+1$. Then,

  1. (1)

The initial term of $f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}$ does not get mapped to
zero under $\psi_{w}$.

  2. (2)

Furthermore, if $(k^{\prime},\ell^{\prime})$ also satisfies the hypotheses and
$(k^{\prime},\ell^{\prime})\neq(k,\ell)$, then
$\psi_{w}(\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k^{\prime}),v_{w}(\ell^{\prime})}))\neq\psi_{w}(\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)})).$
That is, $\psi_{w}$ is injective when we restrict its domain to
$\mathbb{C}[\mathbf{x}_{w_{0}}\setminus D_{w}]$.

###### Proof.

Suppose $k,\ell$ satisfy $k\geq h(\ell)$ and $v_{w}(\ell)+1>v_{w}(k)$. By
Lemma 2.12(2), the initial term is the indeterminate
$\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)})=-x_{n+1-v_{w}(k),v_{w}(\ell)+1}$.
To show that this does not map to $0$ under $\psi_{w}$, we will show that
$x_{n+1-v_{w}(k),v_{w}(\ell)+1}\not\in D_{w}$. By (3.7), this is equivalent to
showing that

$n+1-v_{w}(k)+v_{w}(\ell)+1\leq n\quad\textup{and}\quad
v_{w}^{-1}(v_{w}(\ell)+1)\leq w^{-1}(n+1-v_{w}(k)).$

The first inequality is equivalent to $v_{w}(\ell)+2\leq v_{w}(k)$, and this
follows from the assumption on $k$ and $\ell$. For the second inequality,
observe that

$n+1-v_{w}(k)=w_{0}(v_{w}(k))$

and hence

$w^{-1}(n+1-v_{w}(k))=w^{-1}(w_{0}v_{w}(k))=(w_{0}v_{w})^{-1}(w_{0}v_{w}(k))=k.$

Hence, to prove the second inequality it suffices to show that

$v_{w}^{-1}(v_{w}(\ell)+1)\leq k.$

Since $v_{w}=w_{0}w$, we may also compute the left-hand side as

$\begin{split}v_{w}^{-1}(v_{w}(\ell)+1)&=w^{-1}w_{0}(w_{0}w(\ell)+1)\\\
&=w^{-1}w_{0}(n+1-w(\ell)+1)\\\ &=w^{-1}(w(\ell)-1)\end{split}$

so we need to prove

$w^{-1}(w(\ell)-1)\leq k.$

Recall that $w\in\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$. By Lemma 3.19 we
know therefore that

$w^{-1}(w(\ell)-1)\leq h(\ell)$

and by our assumption $h(\ell)\leq k$, and hence we obtain the inequality

$w^{-1}(w(\ell)-1)\leq k$

as desired. This proves the first claim.

Now if $\psi_{w}(x_{i,j})=\psi_{w}(x_{i^{\prime},j^{\prime}})$ and are
nonzero, by Definition 3.8 we must have $i=i^{\prime}$ and
$v_{w}^{-1}(j)=v_{w}^{-1}(j^{\prime})$, which forces $j=j^{\prime}$, so
$x_{i,j}=x_{i^{\prime},j^{\prime}}$. That is,
$\psi_{w}|_{\mathbb{C}[\mathbf{x}_{w_{0}}\setminus D_{w}]}$ is injective. This
proves the second claim. ∎

######  Lemma 4.4.

Under the hypotheses of Lemma 3.20, $\psi_{w}$ respects the monomial orders
$<_{n}^{w}$ and $<_{n}$. That is,
$\mathrm{in}_{<_{n}^{w}}(\psi_{w}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}))=\psi_{w}(\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}))$.

###### Proof.

By the result of Lemma 2.12(3), the initial term of
$f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}$ is an indeterminate that appears exactly
once in $f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}$ and by Lemma 4.3, this initial term
does not get mapped to zero. As a result, it suffices to show that if
$x_{i,j},x_{i^{\prime},j^{\prime}}\notin D_{w}$ with
$x_{i,j}>_{n}x_{i^{\prime},j^{\prime}}$, then
$\psi_{w}(x_{i,j})>_{n}^{w}\psi_{w}(x_{i^{\prime},j^{\prime}})$.

So suppose that $x_{i,j},x_{i^{\prime},j^{\prime}}\notin D_{w}$ with
$x_{i,j}>_{n}x_{i^{\prime},j^{\prime}}$. By Definition 2.10, we have that
either $i<i^{\prime}$, or, $i=i^{\prime}$ and $j<j^{\prime}$. Then by
Definition 3.8, we have that
$\psi_{w}(x_{i,j})>_{n}^{w}\psi_{w}(x_{i^{\prime},j^{\prime}})$ if and only if
$z_{i,v_{w}^{-1}(j)}>_{n}^{w}z_{i^{\prime},v_{w}^{-1}(j^{\prime})}$. The case
$i<i^{\prime}$ is immediate, so assume that $i=i^{\prime}$. Then, the
inequality holds if and only if
$v_{w}(v_{w}^{-1}(j))<v_{w}(v_{w}^{-1}(j^{\prime}))$, i.e., $j<j^{\prime}$. ∎

######  Corollary 4.5.

Let $J_{w,h}$ be the regular nilpotent Hessenberg Schubert cell ideal defining
$\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$ as a subset of
$X^{w}_{\circ}$. Then, with respect to the monomial order $<^{w}_{n}$ defined
above, the initial terms of the generators $g^{w}_{k,\ell}$ of $J_{w,h}$ are
distinct indeterminates.

###### Proof.

Let $\\{g^{w}_{k_{1},\ell_{1}},\ldots,g^{w}_{k_{r},\ell_{r}}\\}$ be the set of
generators of $J_{w,h}$ from Definition 3.12. By Lemma 3.20, this is equal to
the set
$\\{\psi_{w}(f^{w_{0}}_{v_{w}(k_{1}),v_{w}(\ell_{1})}),\ldots,\psi_{w}(f^{w_{0}}_{v_{w}(k_{r}),v_{w}(\ell_{r})})\\}$.
Then, by Lemma 4.4,
$\mathrm{in}_{<_{n}^{w}}(g^{w}_{k_{i},\ell_{i}})=\psi_{w}(\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k_{i}),v_{w}(\ell_{i})}))$
and by Lemma 4.3, the initial terms of $g^{w}_{k_{i},\ell_{i}}$ with respect
to $<_{n}^{w}$ are distinct indeterminates, as claimed. ∎

It is now straightforward to see that the generators $\\{g_{k,\ell}^{w}\\}$ of
$J_{w,h}$ form a Gröbner basis with respect to $<_{n}^{w}$.

######  Theorem 4.6.

For any indecomposable Hessenberg function $h$ and any $w\in{\rm
Hess}(\mathsf{N},h)^{\mathsf{S}}$, the generators $g_{k,\ell}^{w}$ for the
regular nilpotent Hessenberg Schubert cell ideal $J_{w,h}$ form a Gröbner
basis for $J_{w,h}$ with respect to $<_{n}^{w}$. Moreover, the initial ideal
$\mathrm{in}_{<^{w}_{n}}(J_{w,h})$ is an ideal generated by distinct
indeterminates.

###### Proof.

We saw in Corollary 4.5 that the initial terms of the generators are distinct
indeterminates. This implies that the $S$-polynomial for any pair of
generators $g^{w}_{k,\ell}$ and $g^{w}_{k^{\prime},\ell^{\prime}}$ is given by
$S(g^{w}_{k,\ell},g^{w}_{k^{\prime},\ell^{\prime}})=\mathrm{in}_{<_{n}^{w}}(g^{w}_{k^{\prime},\ell^{\prime}})g^{w}_{k,\ell}-\mathrm{in}_{<_{n}^{w}}(g^{w}_{k,\ell})g^{w}_{k^{\prime},\ell^{\prime}}$.
Buchberger’s Criterion [5, Chapter 2.6, Theorem 6] immediately implies that
the generators $g^{w}_{k,\ell}$ form a Gröbner basis with respect to
$<_{n}^{w}$. Since we saw in Corollary 4.5 that the initial terms of the
$g^{w}_{k,\ell}$ are distinct indeterminates, and since these form a Gröbner
basis, the initial ideal is an ideal generated by distinct indeterminates as
claimed. ∎

We now provide a sketch of an alternate proof using the fact that the image of
a Gröbner basis is again a Gröbner basis under a specialization map if the
initial terms appearing do not map to zero.

###### Alternate proof sketch of Theorem 4.6.

Suppose that $\\{g^{w}_{k_{1},\ell_{1}},\ldots,g^{w}_{k_{r},\ell_{r}}\\}$ are
the generators for $J_{w,h}$ from Definition 3.12. By Lemma 3.20, each
$g^{w}_{k,\ell}$ is the image of a certain $f^{w_{0}}_{a,b}$ under the ring
homomorphism $\psi_{w}$. Furthermore by Lemma 4.3, the initial terms of these
$f^{w_{0}}_{a,b}$ with respect to $<_{n}$ are not mapped to $0$ by $\psi_{w}$.
Let us define a new lexicographic monomial order $\prec^{w_{0},w}_{n}$ on
$\mathbb{C}[\mathbf{x}_{w_{0}}]$ which respects the ordering $<_{n}$ except
for all $x_{i,j}\in D_{w}$ which receive the least weight. That is, if
$x_{i,j}\in D_{w}$ and $x_{i^{\prime},j^{\prime}}\notin D_{w}$, then
$x_{i,j}\prec_{n}^{w_{0},w}x_{i^{\prime},j^{\prime}}$, and if
$x_{i,j},x_{i^{\prime},j^{\prime}}\in D_{w}$ or
$x_{i,j},x_{i^{\prime},j^{\prime}}\notin D_{w}$, then
$x_{i,j}\prec_{n}^{w_{0},w}x_{i^{\prime},j^{\prime}}$ if
$x_{i,j}<_{n}x_{i^{\prime},j^{\prime}}$.

Then
$\\{f^{w_{0}}_{v_{w}(k_{1}),v_{w}(\ell_{1})},\ldots,f^{w_{0}}_{v_{w}(k_{r}),v_{w}(\ell_{r})}\\}$
is a Gröbner basis for the ideal it generates with respect to
$\prec^{w_{0},w}_{n}$, since the initial terms of the $f^{w_{0}}_{a,b}$ with
respect to $<_{n}$ are distinct indeterminates not in $D_{w}$. Then per [11,
Definition 4], $\prec^{w_{0},w}_{n}$ is a $\psi_{w}$-admissible order, so the
result of [11, Theorem 1], together with Lemmas 4.3 and 4.4, completes the
proof. (Technically, $\psi_{w}$ is a composition of a specialization map with
a change of variable names, but this is a small adjustment to the statement of
[11, Theorem 1].) ∎

From the above results, it follows immediately that the regular nilpotent
Hessenberg Schubert cell ideal $J_{w,h}$ is radical and hence is the defining
ideal of ${\rm Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$.

######  Corollary 4.7.

Let $w\in\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$. Then the ideal $J_{w,h}$
is radical.

###### Proof.

Since $\mathrm{in}_{<_{n}^{w}}(J_{w,h})$ is an ideal of indeterminates, it is
a radical ideal. Thus, so too is $J_{w,h}$ (e.g., see [5, Exercise 4.2.16]). ∎

##  5\. Applications of the Main Results

In this section, we discuss some consequences of our main results. The results
of the previous sections imply that the regular nilpotent Hessenberg Schubert
cells are affine spaces, as we show in Proposition 5.5. This allows us to
obtain several immediate applications of our main results which connect to
existing literature and illustrate the usefulness of our computational-
algebraic perspective.

The content of this section is as follows. First, in Section 5.1 we see that
the local defining ideals of regular nilpotent Hessenberg Schubert cells are
complete intersections. Second, in Section 5.2 we show – as advertised above –
that the regular nilpotent Hessenberg varieties $\mathrm{Hess}(\mathsf{N},h)$
are paved by affine spaces. Third, in Section 5.3 we compute the Hilbert
series of the ideals of $\mathrm{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$. The
content of Section 5.4 is to show that these cell ideals $J_{w,h}$ are
geometrically vertex decomposable. Finally, in Section 5.5 we state and prove
some results on Frobenius splitting related to regular nilpotent Hessenberg
Schubert cells.

###  5.1. Complete intersections

As an immediate corollary of the results in Section 3, we see that, in analogy
with the regular nilpotent Hessenberg patch ideal at the $w_{0}$-chart (as
seen in [6]), regular nilpotent Hessenberg Schubert cell ideals are also
complete intersection ideals. We begin with the following.

######  Lemma 5.1.

Let $\\{g^{w}_{k_{1},\ell_{1}},\ldots,g^{w}_{k_{r},\ell_{r}}\\}$ be the set of
nonzero generators of $J_{w,h}$ from Definition 3.12, ordered so that
$\mathrm{in}_{<_{n}^{w}}(g^{w}_{k_{1},\ell_{1}})>_{n}^{w}\ldots>_{n}^{w}\mathrm{in}_{<_{n}^{w}}(g^{w}_{k_{r},\ell_{r}})$.
Then $\mathrm{in}_{<_{n}^{w}}(g^{w}_{k_{i},\ell_{i}})$ does not divide any
term of $g^{w}_{k_{j},\ell_{j}}$ for any $j>i$.

###### Proof.

By Lemma 3.20, the ordered set of generators
$\\{g^{w}_{k_{1},\ell_{1}},\ldots,g^{w}_{k_{r},\ell_{r}}\\}$ is equal to the
ordered set
$\\{\psi_{w}(f^{w_{0}}_{v_{w}(k_{1}),v_{w}(\ell_{1})}),\ldots,\psi_{w}(f^{w_{0}}_{v_{w}(k_{r}),v_{w}(\ell_{r})})\\}$.
Furthermore, we have that

$\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k_{1}),v_{w}(\ell_{1})})>_{n}\ldots>_{n}\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k_{r}),v_{w}(\ell_{r})})$

and so $\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k_{i}),v_{w}(\ell_{i})})$ does
not divide any term of $f^{w_{0}}_{v_{w}(k_{j}),v_{w}(\ell_{j})}$ for $j>i$ by
[6, Lemma 4.13]. Applying $\psi_{w}$, we see that
$\mathrm{in}_{<_{n}^{w}}(g^{w}_{k_{i},\ell_{i}})$ does not divide any term of
$g^{w}_{k_{j},\ell_{j}}$ for any $j>i$, completing the proof. ∎

Before stating the main result of this subsection we include some notation for
the height of each Hessenberg Schubert cell ideal. As discussed at the
beginning of Section 4, the nonzero generators for the Hessenberg Schubert
cell ideal $J_{w,h}$ are the $g^{w}_{k,\ell}$ that satisfy both $k>h(\ell)$
and $v_{w}(k)>v_{w}(\ell)+1$. We denote by $\Lambda_{w,h}$ the number of
nonzero generators $g^{w}_{k,\ell}$ for the ideal $J_{w,h}$.

We now recall from [6] the definition of a triangular complete intersection
and then state the main result of this subsection.

######  Definition 5.2 ([6, Definition 3.3]).

A complete intersection ideal $I\subseteq\mathbb{C}[x_{1},\ldots,x_{n}]$ of
height $r$ is called a triangular complete intersection of height $r$ with
respect to a monomial order $<$ if there exists an ordered set of generators
$f_{1},\ldots,f_{r}$ of $I$ satisfying:

  1. (1)

the initial terms $\mathrm{in}_{<}(f_{i})$ are unique indeterminates up to
multiplication by units;

  2. (2)

$\mathrm{in}_{<}(f_{i})$ does not divide any term of $f_{j}$ for any $j>i$.

######  Theorem 5.3.

For any indecomposable Hessenberg function $h$ and any $w\in{\rm
Hess}(\mathsf{N},h)^{\mathsf{S}}$, the regular nilpotent Hessenberg Schubert
cell ideal $J_{w,h}$ is a complete intersection of height $\Lambda_{w,h}$ and
the nonzero $g_{k,\ell}^{w}$ defining $J_{w,h}$ form a minimal generating set.
Furthermore, $J_{w,h}$ is a triangular complete intersection.

###### Proof.

The result of Theorem 4.6 says that $\mathrm{in}_{<_{n}^{w}}(J_{w,h})$ is an
ideal of indeterminates and hence is a complete intersection. Thus, by [12,
Corollary 19.3.8], $J_{w,h}$ is also a complete intersection. That the nonzero
generators $g^{w}_{k,\ell}$ form a minimal generating set follows from Lemma
5.1.

To see that $J_{w,h}$ has height $\Lambda_{w,h}$, it suffices to show that
$J_{w,h}$ is prime [13, Theorem 1.8A(b)]. Relabel the variables in the ring as
$z_{1},\ldots,z_{n}$. By Lemma 5.1 we may reduce the generators
$g^{w}_{k,\ell}$ to a reduced Gröbner basis for $J_{w,h}$ with respect to
$<_{n}^{w}$ as $g_{1},\ldots,g_{\Lambda_{w,h}}$ where
$\mathrm{in}_{<_{n}^{w}}(g_{i})$ is an indeterminate that does not divide any
term of $g_{j}$ for any distinct $i$ and $j$. By a relabeling of the
variables, we can write $g_{i}=z_{i}-q_{i}$ for each $1\leq
i\leq\Lambda_{w,h}$ where $q_{i}$ is a polynomial in the variables
$z_{\Lambda_{w,h}+1},\ldots,z_{n}$. It follows that

$\mathbb{C}[\mathbf{z}_{w}]/J_{w,h}\cong\mathbb{C}[q_{1},\ldots,q_{\Lambda_{w,h}},z_{\Lambda_{w,h}+1},\ldots,z_{n}]\cong\mathbb{C}[z_{\Lambda_{w,h}+1},\ldots,z_{n}].$

Hence $J_{w,h}$ is prime and [13, Theorem 1.8A(b)] implies that $J_{w,h}$ has
height $\Lambda_{w,h}$, as desired. That $J_{w,h}$ is a triangular complete
intersection now follows from the present argument, Corollary 4.5, and Lemma
5.1. ∎

In the above proof, we have also shown the following.

######  Corollary 5.4.

If $I\subseteq\mathbb{C}[x_{1},\ldots,x_{n}]$ is a triangular complete
intersection of height $r$, then $I$ is a prime ideal and
$\mathbb{V}(I)\cong\mathbb{A}^{n-r}$.

###  5.2. Affine pavings

An affine paving of an algebraic variety $X$ is an ordered partition of $X$
into disjoint $X_{0},X_{1},X_{2},\dots$, such that

  * •

each $X_{i}$ is homeomorphic to an affine space; and

  * •

each of the finite unions $\cup_{i=0}^{r}X_{i}$ is a Zariski closed subset of
$X$.

Tymoczko proved in [19] that regular nilpotent Hessenberg varieties in the
classical Lie type are paved by affine spaces, using delicate combinatorial
and Lie-theoretic arguments. In this subsection, we provide a quick alternate
proof of this result in Lie type $A$.

Combining Corollary 5.4 with Theorem 5.3 immediately yields the following
result. Recall from the previous subsection that $\Lambda_{w,h}$ denotes the
height of the regular nilpotent Hessenberg Schubert cell ideal $J_{w,h}$.

######  Proposition 5.5.

Let $h$ be an indecomposable Hessenberg function and
$w\in\mathrm{Hess}(\mathsf{N},h)^{\mathsf{S}}$. Let $r_{w}$ denote the
dimension of the Schubert cell $X^{w}_{\circ}$. Then, the Hessenberg Schubert
cell $\mathrm{Hess}(\mathsf{N},h)\cap X_{\circ}^{w}$ is isomorphic to the
affine space $\mathbb{A}^{r_{w}-\Lambda_{w,h}}$.

Next recall that the flag variety $GL_{n}(\mathbb{C})/B$ has an affine paving.
The disjoint sets $X_{0},X_{1},X_{2},\dots$ in the paving are the Schubert
cells and they are ordered by any total order which refines Bruhat order. We
will now see that this affine paving induces an affine paving on each regular
nilpotent Hessenberg variety.

######  Theorem 5.6.

Let $h$ be an indecomposable Hessenberg function. Then ${\rm
Hess}(\mathsf{N},h)\subseteq GL_{n}(\mathbb{C})/B$ has an affine paving by the
set of all ${\rm Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$, $w\in{\rm
Hess}(\mathsf{N},h)^{\mathsf{S}}$, totally ordered by any order which refines
Bruhat order.

###### Proof.

By Lemma 3.18, $\text{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$ is non-empty if
and only if $w\in\text{Hess}(\mathsf{N},h)^{\mathsf{S}}$. By Proposition 5.5,
each non-empty $\text{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$ is isomorphic to
affine space. The result now follows by noting that the set of all Schubert
cells form an affine paving of $GL_{n}(\mathbb{C})/B$ and the Hessenberg
variety $\text{Hess}(\mathsf{N},h)$ is a closed subvariety of
$GL_{n}(\mathbb{C})/B$. ∎

###  5.3. Hilbert series

In this section, we compute the Hilbert series of regular nilpotent Hessenberg
Schubert cell ideals. The computation is a straightforward application of
Theorem 5.3, together with well-known techniques regarding Hilbert series of
complete intersections and initial ideals. We start by quickly reviewing this
background on Hilbert series.

Recall that the Hilbert series of a positively $\mathbb{Z}$-graded ring
$R=\mathbb{C}\oplus R_{1}\oplus R_{2}\oplus\ldots$ is the series

$H_{R}(t)=\sum_{k=0}^{\infty}\dim_{\mathbb{C}}(R_{k})t^{k}.$

The following results are well-known but are stated here for completeness.

######  Lemma 5.7.

[9, Theorem 15.26 and Exercise 21.17] Suppose that $I$ is a homogeneous ideal
of $R=\mathbb{C}[x_{1},\ldots,x_{n}]$ with respect to a positive
$\mathbb{Z}$-grading on $R$. Let $<$ be a monomial order on $R$. Then

  1. (1)

$H_{R/I}(t)=H_{R/\mathrm{in}_{<}(I)}(t)$.

  2. (2)

Let $I=\langle f_{1},\ldots,f_{k}\rangle$ be a complete intersection of height
$k$ where each $f_{i}$ is homogeneous and $\mathrm{deg}(f_{i})=d_{i}$. Then

$H_{R/I}(t)=H_{R}(t)\prod_{i=1}^{k}(1-t^{d_{i}}).$

Although the generators $g^{w}_{k,\ell}$ are not homogeneous with respect to
the standard grading, we again use the $\psi_{w}$ map to define a _non-
standard_ grading on $\mathbb{C}[\mathbf{z}_{w}]$. It will turn out, as we see
below, that with respect to this non-standard grading, the $g^{w}_{k,\ell}$
are homogeneous.

In Section 2.3 of this manuscript we defined, following [6], a
$\mathbb{Z}$-grading on $\mathbb{C}[\mathbf{x}_{w}]$ by
$\deg(x_{i,j}):=w(j)-i$. In the special case $w=w_{0}$, this becomes a
positive $\mathbb{Z}$-grading on $\mathbb{C}[\mathbf{x}_{w_{0}}]$ given by
$\deg(x_{i,j}):=w_{0}(j)-i$ (see Section 2.3 and [6, Lemma 2.18]). We now
define a $\mathbb{Z}$-grading on $\mathbb{C}[\mathbf{z}_{w}]$ by using this
grading on $\mathbb{C}[\mathbf{x}_{w_{0}}]$ together with the algebra
homomorphism $\psi_{w}$, namely, we define
$\deg(z_{i,j})\coloneqq\deg(\psi_{w}^{-1}(z_{i,j}))=w_{0}(v_{w}(j))-i$, for
all $z_{i,j}\in\mathbf{z}_{w}$. That this is well-defined follows from Lemma
4.3(2).

######  Remark 5.8.

The grading on $\mathbb{C}[\mathbf{z}_{w}]$ given above agrees with the
grading that arises from the $\mathsf{S}$-action on ${\rm
Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$, defined in Section 2.3. Indeed, for
any $z_{i,j}\in\mathbf{z}_{w}$, the grading from the $\mathsf{S}$-action
yields $\deg(z_{i,j})=w(j)-i$. By the construction of $v_{w}$, we know that
$w=w_{0}v_{w}$. From this it follows that the two definitions agree.

######  Example 5.9.

We continue Example 3.15 and show that in this case, the two definitions of
degree indeed agree, as pointed out in Remark 5.8. The following table uses
the definition arising from the $\mathsf{S}$-action on ${\rm
Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$.

$z_{i,j}$ | $z_{1,1}$ | $z_{1,2}$ | $z_{1,3}$ | $z_{2,1}$ | $z_{2,2}$
---|---|---|---|---|---
$\deg(z_{i,j})=w(j)-i$ | 2 | 3 | 1 | 1 | 2

Meanwhile the following table uses the definition in terms of $\psi_{w}^{-1}$.
Recall that $w_{0}(a)=n+1-a$ for all $a\in[n]$.

$z_{i,j}$ | $z_{1,1}$ | $z_{1,2}$ | $z_{1,3}$ | $z_{2,1}$ | $z_{2,2}$
---|---|---|---|---|---
$\psi_{w}^{-1}(z_{i,j})$ | $x_{1,2}$ | $x_{1,1}$ | $x_{1,3}$ | $x_{2,2}$ | $x_{2,1}$
$\deg(z_{i,j})=w_{0}(v_{w}(j))-i$ | 2 | 3 | 1 | 1 | 2

Moreover, notice that the generators $g^{w}_{k,\ell}$ are homogeneous with
respect to this grading. Indeed, $g^{w}_{4,1}$ has degree 1 and $g^{w}_{4,2}$
has degree 2. ∎

The following lemma shows that it is no accident that the generators
$g^{w}_{k,\ell}$ in the previous example are homogeneous. Indeed, the
generators $f^{w_{0}}_{k,\ell}$ for the $w_{0}$-chart are homogeneous with
respect to this grading, so our definition of the grading on
$\mathbb{C}[\mathbf{z}_{w}]$ via the $\psi_{w}$ map allows us to easily
translate the homogeneity to the ${\rm Hess}(\mathsf{N},h)\cap X^{w}_{\circ}$
setting.

######  Lemma 5.10.

Let $h$ be an indecomposable Hessenberg function and $w\in{\rm
Hess}(\mathsf{N},h)^{\mathsf{S}}$. Then, the above defines a positive
$\mathbb{Z}$-grading on $\mathbb{C}[\mathbf{z}_{w}]$ with respect to which the
generators $g^{w}_{k,\ell}$ for $J_{w,h}$ are homogeneous.

###### Proof.

It was shown in [6, Section 2.3] that the grading on
$\mathbb{C}[\mathbf{x}_{w_{0}}]$ is positive. Since
$\psi_{w}^{-1}(\mathbf{z}_{w})=\mathbf{x}_{w_{0}}\setminus D_{w}$ is a subset
of $\mathbf{x}_{w_{0}}$, it follows that the grading on
$\mathbb{C}[\mathbf{z}_{w}]$ is also positive. Moreover, it was shown in [6,
Lemma 2.18] that generators $f^{w_{0}}_{k,\ell}$ are homogeneous with respect
to the non-standard grading on $\mathbb{C}[\mathbf{x}_{w_{0}}]$. We showed in
Lemma 3.20 that $g^{w}_{k,\ell}=\psi_{w}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)})$.
Since $\psi_{w}$ is a homomorphism of rings, the image of the monomials in
$f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}$ under $\psi_{w}$ either retain their degree
or get mapped to zero. In either case, we conclude that $g^{w}_{k,\ell}$ is
homogeneous. ∎

In particular, from Lemma 5.10 we can conclude that the regular nilpotent
Hessenberg Schubert cell ideals $J_{w,h}$ are homogeneous with respect to this
non-standard grading.

We may now explicitly compute the Hilbert series of $R/J_{w,h}$ as follows.

######  Theorem 5.11.

Equip $R=\mathbb{C}[\mathbf{z}_{w}]$ with the positive $\mathbb{Z}$-grading
defined in Section 5.1. Suppose that $h$ is an indecomposable Hessenberg
function and $w\in{\rm Hess}(\mathsf{N},h)^{\mathsf{S}}$. Then
$\mathrm{in}_{<_{n}^{w}}(g^{w}_{k,\ell})=-z_{n+1-v_{w}(k),v_{w}^{-1}(v_{w}(\ell)+1)}$
has degree $v_{w}(k)-v_{w}(\ell)-1$ for all $k>h(\ell)$. Furthermore, the
Hilbert series of $R/J_{w,h}$ is

$H_{R/J_{w,h}}(t)=\frac{\displaystyle\prod_{\begin{subarray}{c}k>h(\ell)\\\
v_{w}(k)>v_{w}(\ell)+1\end{subarray}}(1-t^{v_{w}(k)-v_{w}(\ell)-1})}{\displaystyle\prod_{\begin{subarray}{c}i<w(j)\\\
j<w^{-1}(i)\end{subarray}}\left(1-t^{w(j)-i}\right)}.$

###### Proof.

Recall that the nonzero generators of $J_{w,h}$ are those $g^{w}_{k,\ell}$
satisfying $k>h(\ell)$ and $v_{w}(k)>v_{w}(\ell)+1$. By Lemmas 3.20 and 4.4,
$\mathrm{in}_{<_{n}^{w}}(g^{w}_{k,\ell})=\psi_{w}(\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)}))$.
This initial term was computed in Lemma 2.12(1) as
$\mathrm{in}_{<_{n}}(f^{w_{0}}_{v_{w}(k),v_{w}(\ell)})=-x_{n+1-v_{w}(k),v_{w}(\ell)+1}$.
Then by Definition 3.8 of $\psi_{w}$,

$\mathrm{in}_{<_{n}^{w}}(g^{w}_{k,\ell})=\psi_{w}(-x_{n+1-v_{w}(k),v_{w}(\ell)+1})=-z_{n+1-v_{w}(k),v_{w}^{-1}(v_{w}(\ell)+1)}.$

By definition of the $\mathbb{Z}$-grading given in Section 5.1, we have

$\deg(-z_{n+1-v_{w}(k),v_{w}^{-1}(v_{w}(\ell)+1)})=w_{0}(v_{w}(\ell)+1)-(n+1-v_{w}(k))=v_{w}(k)-v_{w}(\ell)-1,$

since $w_{0}(v_{w}(\ell)+1)=n+1-(v_{w}(\ell)+1)$. We argued in Lemma 5.10 that
the $g^{w}_{k,\ell}$ are homogeneous, so it follows that
$\deg(g^{w}_{k,\ell})=v_{w}(k)-v_{w}(\ell)-1$ for all $k>h(\ell)$. This proves
the first claim.

Next we compute the Hilbert series of $R/J_{w,h}$. Since
$\\{g^{w}_{k,\ell}\mid k>h(\ell)\\}$ is a Gröbner basis for the complete
intersection ideal $J_{w,h}$ with respect to $<_{n}^{w}$, by Lemma 5.7, we
conclude

$H_{R/J_{w,h}}(t)=H_{R}(t)\prod_{k>h(\ell)}\left(1-t^{\deg(g_{k,\ell}^{w})}\right).$

But
$H_{R}(t)=\displaystyle\prod_{z_{i,j}\in\mathbf{z}_{w}}(1-t^{\mu_{i,j}})^{-1}$,
where $\mu_{i,j}=\deg(z_{i,j})=w(j)-i$, so

$H_{R/J_{w,h}}(t)=\frac{\displaystyle\prod_{\begin{subarray}{c}k>h(\ell)\\\
v_{w}(k)>v_{w}(\ell)+1\end{subarray}}(1-t^{v_{w}(k)-v_{w}(\ell)-1})}{\displaystyle\prod_{\begin{subarray}{c}i<w(j)\\\
j<w^{-1}(i)\end{subarray}}\left(1-t^{w(j)-i}\right)}.$

where the product in the denominator is over those indices for which
$z_{i,j}\in\mathbf{z}_{w}$, as given in (3.1). ∎

######  Remark 5.12.

Since $R/J_{w,h}$ is isomorphic to a polynomial ring by the work in Section
5.2, we could have also computed the Hilbert series using the weights of the
$z_{i,j}$ remaining after taking the quotient of $R$ by $J_{w,h}$. In other
words, $H_{R/J_{w,h}}(t)$ is simply the series
$\displaystyle\prod_{z_{i,j}\text{ $\nmid$
}\mathrm{in}(g^{w}_{k,\ell})}(1-t^{\mu_{i,j}})^{-1}$, where the product is
over those indices $(i,j)$ for which $z_{i,j}\in\mathbf{z}_{w}$ does not
appear as an initial term of any $g_{k,\ell}$ with $k>h(\ell)$.

###  5.4. Geometric vertex decomposability

Knutson, Miller, and Yong related the notion of a vertex decomposition of a
simplicial complex to a classical algebraic geometry degeneration technique,
and called this a geometric vertex decomposition in the case that the
degeneration is reduced [16]. Recently, Klein and the fourth author expanded
this definition recursively to geometrically vertex decomposable ideals, which
provides a generalization of vertex decomposable simplicial complexes [15].
The condition that a simplicial complex is vertex decomposable can be
strengthened to require that the decomposition respects a fixed ordering of
the vertices. This strengthened notion is called compatibly vertex
decomposable. Klein and Rajchgot similarly introduced the notion of
$<$-compatibly geometrically vertex decomposable, where the degenerations
respect a lexicographic monomial order $<$. In particular, an ideal is
$<$-compatibly geometrically vertex decomposable if and only if its initial
ideal is the Stanley-Reisner ideal of a compatibly vertex decomposable
simplicial complex with respect to a corresponding vertex order [15,
Proposition 2.14].

While we refer the reader to [15] for formal definitions, we provide the
following sufficient condition for being geometrically vertex decomposable,
which is a straightforward generalization of [15, Proposition 2.14].

######  Lemma 5.13 ([6, Lemma 3.6]).

Let $I\subseteq\mathbb{C}[\mathbf{x}]$ be an ideal in a polynomial ring and
$<$ a lexicographic monomial order on $\mathbb{C}[\mathbf{x}]$. If
$\mathrm{in}_{<}(I)$ is an ideal of indeterminates, then $I$ is $<$-compatibly
geometrically vertex decomposable.

We saw in Theorem 4.6 that the initial ideal of the regular nilpotent
Hessenberg Schubert cell ideal $J_{w,h}$ is an ideal of indeterminates, so the
following is immediate from Lemma 5.13 upon choosing the lexicographic
monomial order $<_{n}^{w}$ from Definition 4.1.

######  Proposition 5.14.

Let $h$ be an indecomposable Hessenberg function and $w\in{\rm
Hess}(\mathsf{N},h)^{\mathsf{S}}$. Then $J_{w,h}$ is $<_{n}^{w}$-compatibly
geometrically vertex decomposable.

###  5.5. Frobenius splitting

Many well-studied subvarieties of the flag variety are known to be _Frobenius
split_ , including Schubert varieties and Richardson varieties. Furthermore,
each Schubert cell in the flag variety has a Frobenius splitting which
compatibly splits all opposite Schubert varieties intersected with that
Schubert cell.

In this subsection, we observe that the situation is similar for regular
nilpotent Hessenberg varieties. Namely, each Schubert cell in the flag variety
has a Frobenius splitting which compatibly splits all the regular nilpotent
Hessenberg varieties intersected with that Schubert cell. This extends [6,
Corollary 5.14].

Let $\mathbb{K}$ be a algebraically closed field of characteristic $p>0$. A
Frobenius splitting of a polynomial ring $R=\mathbb{K}[y_{1},\dots,y_{n}]$ is
a map $\phi:R\rightarrow R$ which satisfies the following three conditions:
(i) $\phi(a+b)=\phi(a)+\phi(b)$, for all $a,b\in R$, (ii)
$\phi(a^{p}b)=a\phi(b)$, for all $a,b\in R$, and (iii) $\phi(1)=1$. An ideal
$I\subseteq R$ is compatibly split by the Frobenius splitting
$\phi:R\rightarrow R$ if $\phi(I)\subseteq I$. We refer the reader to one of
the many references on Frobenius splitting for further information on the
subject (see e.g., [4, 18]).

Let $\mathbb{K}[{\bf z}_{w}]$ denote the coordinate ring of the Schubert cell
$X^{w}_{\circ}\subseteq GL_{n}(\mathbb{K})/B$. Consider each $g^{w}_{k,\ell}$
(which has integer coefficients) as an element of $\mathbb{K}[{\bf z}_{w}]$.
Let ${J}_{w,h}\subseteq\mathbb{K}[{\bf z}_{w}]$ be the ideal generated by
these polynomials $g^{w}_{k,\ell}\in\mathbb{K}[{\bf z}_{w}]$. Observe that
$J_{w,h}$ is the prime defining ideal of the regular nilpotent Hessenberg
Schubert cell $\text{Hess}(\mathsf{N},h)\cap X^{w}_{\circ}\subseteq
GL_{n}(\mathbb{K})/B$.

Since the coefficient of each $\text{in}_{<^{w}_{n}}(g^{w}_{k,\ell})$ is not
divisible by any prime $p>0$, we observe that the proof of Theorem 5.3 holds
in positive characteristic. That is, $J_{w,h}\subseteq\mathbb{K}[{\bf z}_{w}]$
is a triangular complete intersection, minimally generated by the
$g^{w}_{k,\ell}$. Let $G^{w}$ be the product of all of these minimal
generators. Let $Z$ denote the product of all indeterminates in the polynomial
ring $\mathbb{K}[{\bf z}_{w}]$. Define
$F^{w}:=\frac{Z}{\text{in}_{<^{w}_{n}}(G^{w})}G^{w}$, which is a polynomial
since $\text{in}_{<^{w}_{n}}(G^{w})$ is a squarefree monomial (up to sign),
and hence divides $Z$. By construction, $G^{w}$ divides $F^{w}$. Additionally,
$\text{in}_{<^{w}_{n}}(F^{w})$ is equal to $Z$, the product of all the
variables in $\mathbb{K}[{\bf z}_{w}]$.

Let $\mathrm{Tr}:\mathbb{K}[{\bf z}_{w}]\rightarrow\mathbb{K}[{\bf z}_{w}]$ be
the additive map defined on monomials as follows: given a monomial
$m\in\mathbb{K}[{\bf z}_{w}]$, define $\mathrm{Tr}(m)$ to be
$\frac{(mZ)^{1/p}}{Z}$ if $mZ$ is a $p^{\rm th}$ power and $0$ otherwise.
Given a polynomial $g\in\mathbb{K}[{\bf z}_{w}]$, define $\text{Tr}(g\hskip
2.84526pt\bullet)$ to be the map which sends $f\in\mathbb{K}[{\bf z}_{w}]$ to
$\mathrm{Tr}(gf)$.

######  Theorem 5.15.

The map $\mathrm{Tr}\left((F^{w})^{p-1}\bullet\right)$ is a Frobenius
splitting of $\mathbb{K}[{\bf z}_{w}]$ which compatibly splits each ideal
$J_{w,h}\subseteq\mathbb{K}[{\bf z}_{w}]$.

###### Proof.

This is immediate from [6, Theorem 5.8 and Corollary 5.9]. ∎

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