# The Geometry and Combinatorics of Some Hessenberg Varieties Related to the
Permutohedral Variety

Jan-Li Lin  Department of Mathematics and Statistics, Washington University in
St. Louis, One Brookings Drive, St. Louis, Missouri 63130, U.S.A.  jan-
<EMAIL_ADDRESS>
###### Abstract.

We construct a concrete isomorphism from the permutohedral variety to the
regular semisimple Hessenberg variety associated to the Hessenberg function
$h_{+}(i)=i+1$, $1\leq i\leq n-1$. In the process of defining the isomorphism,
we introduce a sequence of varieties which we call the prepermutohedral
varieties. We first determine the toric structure of these varieties and
compute the Euler characteristics and the Betti numbers using the theory of
toric varieties. Then, we describe the cohomology of these varieties. We also
find a natural way to encode the one-dimensional components of the cohomology
using the codes defined by Stembridge [Ste]. Applying the isomorphisms we
constructed, we are also able to describe the geometric structure of regular
semisimple Hessenberg varieties associated to the Hessenberg function
represented by $h_{k}=(2,3,\cdots,k+1,n,\cdots,n)$, $1\leq k\leq n-3$. In
particular, we are able to write down the cohomology ring of the variety.
Finally, we determine the dot representation of the permutation group
$\mathfrak{S}_{n}$ on these varieties.

##  1\. Introduction

There are several different ways to describe the permutohedral variety
$\mathcal{X}$ of dimension $n-1$, for instances:

  1. (1)

It is the toric variety associated to the normal fan of the permutohedron.

  2. (2)

It is the graph of the Cremona involution

$\displaystyle J:\mathbb{P}^{n-1}$
$\displaystyle\dashrightarrow\mathbb{P}^{n-1}$
$\displaystyle[z_{1}:\cdots:z_{n}]$
$\displaystyle\longmapsto[z_{1}^{-1}:\cdots:z_{n}^{-1}]$

  3. (3)

It is an iterated blowup of $\mathbb{P}^{n-1}$ at all (strict transforms of)
coordinate linear subspaces in a certain order, as follows.

$\textstyle{\mathcal{X}_{n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n-2}}$$\textstyle{\mathcal{X}_{n-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n-3}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{\mathcal{X}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}\
\ \ \ \ }$$\textstyle{\mathcal{X}_{0}=\mathbb{P}^{n-1}}$

Each $\mathcal{X}_{k+1}$ is the blowup of $\mathcal{X}_{k}$ at all the strict
transform of the $k$-dimensional coordinate spaces ($0\leq k\leq n-3$).
Further details are in Section 2.

  4. (4)

It is the regular semisimple Hessenberg variety ${\mathcal{H}ess}(\SS,h_{+})$
associated to the Hessenberg function $h_{+}$ defined by $h_{+}(i)=i+1$ for
$1\leq i\leq n-1$ and $h_{+}(n)=n$.

The isomorphisms among the first three descriptions are well-known. On the
other hand, to the author’s knowledge, the isomorphism between the
permutohedral variety and the Hessenberg variety is proved by general theory
of toric varieties [DMPS, Lemma 10 and Theorem 11]. The first goal of this
article is to construct a concrete isomorphism from the iterated blowups on
$\mathbb{P}^{n-1}$ to the Hessenberg variety. In the process, we further
obtain isomorphisms from $\mathcal{X}_{k}$, the $k$-th step in the iterated
blowups, to a “Hessenberg type” subvariety of a partial flag variety which is
denoted by ${\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$ (for details, see Section 3).

Next, in Section 4, we apply the theory of toric varieties to explore certain
geometric properties of the varieties $\mathcal{X}_{k}$. In particular, we
show that the Euler characteristic of $\mathcal{X}_{k}$ is equal to the
permutation number $P(n,k+1)=\frac{n!}{(n-k-1)!}$. In fact, there is a basis
for the homology $H_{*}(\mathcal{X}_{k})$ whose elements are in one-to-one
correspondence with permutations of $(k+1)$ different numbers chosen from
$[n]:=\\{1,\cdots,n\\}$. As a consequence, the question of finding Betti
numbers of $\mathcal{X}_{k}$ turns into a counting question of permutations
with certain property. More precisely, if we make the following definition:

######  Definition.

For a permutation $a_{1},\cdots,a_{k+1}$ of $k+1$ different numbers in $[n]$,
set $\alpha_{0}=[n]\setminus\\{a_{1},\cdots,a_{k+1}\\}$, a descent (resp.
ascent) for the permutation $a_{1},\cdots,a_{k+1}$ is either $a_{j}>a_{j+1}$
(resp. $a_{j}<a_{j+1}$) for some $j=1,\cdots,k$, or $a>a_{1}$ (resp.
$a<a_{1}$) for some $a\in\alpha_{0}$.

Then we can calculate the even Betti numbers of $\mathcal{X}_{k}$ as follows
(note that the odd Betti numbers of $\mathcal{X}_{k}$ are all $0$.)

######  Proposition 1.1 (Proposition 4.2).

The $2i$-th Betti number of $\mathcal{X}_{k}$ is given by

$\displaystyle\beta_{2i}(\mathcal{X}_{k})$
$\displaystyle=\\#(\text{permutations of $k+1$ different numbers in $[n]$ with
$i$ descents})$ $\displaystyle=\\#(\text{permutations of $k+1$ different
numbers in $[n]$ with $n-1-i$ ascents}).$

These Betti numbers are quite natural generalization of the Eulerian numbers.
In fact, for $k=n-2$, $\mathcal{X}_{n-2}$ is the permutohedral variety and it
is well known that $\beta_{2i}(\mathcal{X}_{n-2})=A(n,i+1)$ are the Eulerian
numbers. However, the author’s cannot find information about them in the
literature. It might be interesting to study further properties of these
numbers.

In Section 5, we first describe $H^{*}(\mathcal{X}_{k})$ using the blowup
structure (Proposition 5.2). There is a system of codes defined by Stembridge
in [Ste]. He also proved in the article that the representation of the
symmetric group $\mathfrak{S}_{n}$ on $H^{*}(\mathcal{X})$ is isomorphic to
the permutation representation induced by the action on codes [Ste,
Proposition 4.1]. In Proposition 5.3, we realize the isomorphism by defining a
natural one-to-one correspondence between codes of length $n$ and one-
dimensional components of $H^{*}(\mathcal{X}^{n-1})$. The correspondence can
also be restricted to $H^{*}(\mathcal{X}^{n-1}_{k})$, and is compatible with
the $\mathfrak{S}_{n}$ action. This immediately gives us a way to concretely
construct permutation basis for the $\mathfrak{S}_{n}$ representation on each
$H^{*}(\mathcal{X}^{n-1}_{k})$, $0\leq k\leq n-2$.

The question of finding a permutation basis for the dot action on the
cohomology of Hessenberg varieties has drawn people’s attention recently [AHM,
CHL, Chow, HPT] because of it’s relation with the Stanley-Stembridge
conjecture. The relation was observed by Shareshian and Wahcs [SW]. They also
announced an important conjecture that was later proved by Brosnan and Chow
[BC], and independently by Guay-Paquet [GP]. In the case of the permutohedral
variety, the Stanley-Stembridge conjecture is known to be true, and a
permutation basis is also known. The known basis was first conjectured by Chow
[Chow] and then proved by Cho, Hong, and Lee [CHL]. It is based on the theory
on equivariant cohomology, then pass it on to the usual cohomology. The basis
constructed in this paper is based on the geometric structure of
${\mathcal{H}ess}^{(k)}(\SS,h_{+})$ and the combinatorics of the codes, and is
for the usual cohomology. It might be interesting to compare these two kinds
of basis.

In Section 6, we use the isomorphisms at each step of the blowups to
investigate semisimple Hessenberg varieties ${\mathcal{H}ess}(\SS,h_{k})$
associated to the Hessenberg function $h_{k}=(2,3,\cdots,k+1,n,\cdots,n)$,
$1\leq k\leq n-3$. We observe that ${\mathcal{H}ess}(\SS,h_{k})$ has a fiber
bundle structure over $\mathcal{X}_{k}$ with fibers isomorphic to the flag
variety ${\mathcal{F}lag}(\mathbb{C}^{n-k-1})$. We use this structure to
obtain a description of the cohomology of ${\mathcal{H}ess}(\SS,h_{k})$.

######  Proposition 1.2 (Proposition 6.1).

The cohomology ring $H^{*}({\mathcal{H}ess}(\SS,h_{k}))$ is generated over
$H^{*}(\mathcal{X}_{k})$ as

$H^{*}({\mathcal{H}ess}(\SS,h_{k}))\cong
H^{*}(\mathcal{X}_{k})[X_{k+2},\cdots,X_{n}]/(e_{1}(X_{k+2},\cdots,X_{n}),\cdots,e_{n-k-1}(X_{k+2},\cdots,X_{n})),$

where $e_{j}$ is the $j$-th elementary symmetric polynomials. In addition, for
the images $x_{i}$’s of the $X_{i}$’s, the classes $x_{k+2}^{i_{k+2}}\cdots
x_{n}^{i_{n}}$, with exponents $0\leq i_{j}\leq n-j$, form a basis for
$H^{*}(\mathcal{Y})$ over $H^{*}(\mathcal{X}_{k})$.

Finally, we determine the dot representation on the cohomologies of
${\mathcal{H}ess}(\SS,h_{k})$.

######  Proposition 1.3 (Proposition 6.2).

The dot representation on ${\mathcal{H}ess}(\SS,h_{k})$ is isomorphic to the
representation on

$H^{*}(\mathcal{X}_{k})[X_{k+2},\cdots,X_{n}]/(e_{1}(X_{k+2},\cdots,X_{n}),\cdots,e_{n-k}(X_{k+2},\cdots,X_{n}))$

which acts on $H^{*}(\mathcal{X}_{k})$ as described in Section 5, and acts
trivially on the $H^{*}(\mathcal{X}_{k})$-basis $x_{k+2}^{i_{k+2}}\cdots
x_{n}^{i_{n}}$, $0\leq i_{j}\leq n-j$.

Our proof is based on the characteristic series of the representation, and the
conclusion is up to an isomorphism. It would be interesting to compute the
$\mathfrak{S}_{n}$ action on the basis elements; presumably first on the
equivariant cohomology then pass to the usual cohomology, as was done in
[Tymo2].

##  2\. The permutohedral variety as iterated blowups of $\mathbb{P}^{n-1}$

One can obtain the permutohedral variety by performing a sequence of blowups
on $\mathbb{P}^{n-1}$, as follows.

  1. (1)

First, we blowup the $n$ points $Z_{1}=[1:0:\cdots:0]$,
$Z_{2}=[0:1:0\cdots:0]$, $\cdots$, $Z_{n}=[0:\cdots:0:n]$. We denote the
resulting variety and the projection map by
$\pi_{1}:\mathcal{X}_{1}\to\mathbb{P}^{n-1}:=\mathcal{X}_{0}$. We also have
the exceptional divisors $E_{i}=\pi_{1}^{-1}(Z_{i})\subset\mathcal{X}_{1}$.

  2. (2)

Next, we blowup the strict transforms (in $\mathcal{X}_{1}$) of all the lines
$Z_{\\{i,j\\}}\subset\mathbb{P}^{n-1}$ connecting $Z_{i}$ and $Z_{j}$ for
$1\leq i<j\leq n$. This gives us the second level space
$\pi_{2}:\mathcal{X}_{2}\to\mathcal{X}_{1}$ and the exceptional divisors
$E_{\\{i,j\\}}\subset\mathcal{X}_{2}$. Notice that, although the lines
$Z_{\\{i,j\\}}$ and $Z_{\\{i,k\\}}$ intersect at $Z_{i}$ in
$\mathbb{P}^{n-1}$, the blowups in step 1 would separate their strict
transforms. Therefore, the resulting space $\mathcal{X}_{2}$ is independent of
the order of blowups.

  3. (3)

We repeat the above process until we reach codimension $2$. More precisely, in
the $k$-th step ($1\leq k\leq n-2$) we do the following. For
$\alpha\subset[n]$ a subset of $k$ elements, let $Z_{\alpha}$ denote the
linear subvariety of $\mathbb{P}^{n-1}$ generated by $\\{Z_{i}|i\in\alpha\\}$,
and $\overline{Z}_{\alpha}$ the strict transform of $Z_{\alpha}$ in
$\mathcal{X}_{k-1}$. The blowups in the previous steps have the effect of
blowing up all coordinate linear subspaces on $Z_{\alpha}$, thus
$\overline{Z}_{\alpha}$ is a permutohedral variety of dimension $k-1$. We blow
up all the $\overline{Z}_{\alpha}$ in this step. The $\overline{Z}_{\alpha}$’s
intersect with each other along coordinate subspaces of lower dimensions,
hence are separated by previous blowups. Thus, the order to perform blowups in
this step does not matter. This produces the space $\mathcal{X}_{k}$, the map
$\pi_{k}:\mathcal{X}_{k}\to\mathcal{X}_{k-1}$, as well as the exceptional
divisors $E_{\alpha}\subset\mathcal{X}_{k}$

The end result is a sequence of spaces and projection maps:

$\textstyle{\mathcal{X}_{n-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n-2}}$$\textstyle{\mathcal{X}_{n-3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n-3}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{2}}$$\textstyle{\mathcal{X}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1}\
\ \ \ \ }$$\textstyle{\mathcal{X}_{0}=\mathbb{P}^{n-1}}$

The variety $\mathcal{X}=\mathcal{X}_{n-2}$ is the permutohedral variety.

######  Definition.

We call the varieties $\mathcal{X}_{k}$ ($0\leq k\leq n-2$) the
prepermutohedral variety of order $k$.

##  3\. Permutohedral varieties as Hessenberg varieties

We consider regular semisimple Hessenberg varieties of type A. To set the
notations, let $\SS$ be an $n\times n$ complex diagonal matrix with different
diagonal entries $s_{1},\cdots,s_{n}$ and $h:[n]\to[n]$ be a Hessenberg
function, i.e., $h(i)\geq h(j)\text{ for all $n\geq i>j\geq 1$}$ ($h$ is non-
decreasing) and $h(i)\geq i$ for $i=1,\cdots,n$.

${\mathcal{H}ess}(\SS,h)=\left\\{V_{\bullet}=(\\{0\\}=V_{0}\subset
V_{1}\subset\cdots\subset V_{n}=\mathbb{C}^{n})\ |\ \SS(V_{i})\subset
V_{h(i)}\text{ for $i=1,\cdots,n$}\right\\}.$

We also use the list notation $h=(h(1),\cdots,h(n))$ to denote a Hessenberg
function. In this section, we focus on the specific Hessenberg function
$h_{+}=(2,3,\cdots,n-1,n,n)$, i.e. $h_{+}(i)=\min(i+1,n)$ for $1\leq i\leq n$.
We start with an observation in linear algebra.

######  Lemma 3.1.

Let $\SS$ be an $n\times n$ complex diagonal matrix with different diagonal
entries $s_{1},\cdots,s_{n}$ and
$\mathbf{v}=\vphantom{(z_{1},\cdots,z_{n})}^{\mathrm{t}}\\!{(z_{1},\cdots,z_{n})}\in\mathbb{C}^{n}$
be a (column) vector. If at least $k$ of the $z_{i}$’s are nonzero (i.e. there
are no more than $n-k$ of the $z_{i}=0$), then the vectors

$\mathbf{v},\ \SS\mathbf{v},\ \cdots,\ \SS^{k-1}\mathbf{v}$

are linearly independent. If exactly $k$ of the $z_{i}$’s are nonzero, then
the vectors

$\mathbf{v},\ \SS\mathbf{v},\ \cdots,\ \SS^{k}\mathbf{v}$

are linearly dependent.

###### Proof.

Assuming $z_{i}\neq 0$, then a linear relation

$a_{0}\mathbf{v}+a_{1}\SS\mathbf{v}+\cdots+a_{k-1}\SS^{k-1}\mathbf{v}=0$

implies the relation (on the first coordinate)

$a_{0}+a_{1}s_{i}+\cdots+a_{k-1}s_{i}^{k-1}=0.$

That is, $s_{i}$ is a root of the polynomial $\sum_{i=0}^{k-1}a_{i}z^{i}$. The
first part of the lemma is then a consequence of the fact that a polynomial
equation of degree $k-1$ cannot have $k$ or more distinct roots.

For the second part of the lemma, assume that $z_{1},\cdots,z_{k}$ are the
non-zero $z_{i}$’s. Then the coefficients of the polynomial
$(z-s_{1})\cdots(z-s_{k})$ give a non-trivial linear relation among
$\mathbf{v},\ \SS\mathbf{v},\ \cdots,\ \SS^{k}\mathbf{v}$. ∎

In particular, if $\mathbf{0}=(0,\cdots,0)$ denotes the origin, and
$\mathbf{v}\in\mathbb{C}^{n}\setminus\\{\mathbf{0}\\}$, then $\mathbf{v}$
gives rise to a point $[\mathbf{v}]\in\mathbb{P}^{n-1}$. We further denote

${\mathcal{I}nd}:=\bigcup_{1\leq i<j\leq n}(z_{i}=z_{j}=0).$

The set ${\mathcal{I}nd}$ is the indeterminate set of the Cremona involution
$J$ defined in the introduction. If
$\mathbf{v}\in\mathbb{P}^{n-1}\setminus{\mathcal{I}nd}$, then at least $n-1$
of the $x_{i}$ are nonzero. Thus, by the lemma, the vectors

$\mathbf{v},\ \SS\mathbf{v},\ \cdots,\ \SS^{n-1}\mathbf{v}$

are linearly independent, and the following is a well-defined flag (i.e. the
dimension of the vector spaces are correct).

$V_{\bullet}=\left(\langle\mathbf{0}\rangle\subset\langle\mathbf{v}\rangle\subset\langle\mathbf{v},\SS\mathbf{v}\rangle\subset\cdots\subset\langle\mathbf{v},\SS\mathbf{v},\cdots\SS^{n-2}\mathbf{v}\rangle\subset\mathbb{C}^{n}\right).$

Moreover, it is obvious that $V_{\bullet}\in{\mathcal{H}ess}(\SS,h_{+})$.
Conversely, suppose that $V_{\bullet}=\left(\\{0\\}\subset
V_{1}\subset\cdots\subset V_{n-1}\subset\mathbb{C}^{n}\right)$ is a flag in
${\mathcal{H}ess}(\SS,h_{+})$, and the one-dimensional space $V_{1}$, as an
element of $\mathbb{P}^{n-1}$, satisfies $V_{1}\not\in{\mathcal{I}nd}$, then
$V_{\bullet}$ must be in the form

$V_{\bullet}=\left(\langle\mathbf{0}\rangle\subset\langle\mathbf{v}\rangle\subset\langle\mathbf{v},\SS\mathbf{v}\rangle\subset\cdots\subset\langle\mathbf{v},\SS\mathbf{v},\cdots\SS^{n-2}\mathbf{v}\rangle\subset\mathbb{C}^{n}\right)$

for any nonzero $\mathbf{v}\in V_{1}$. This defines an isomorphism

$\displaystyle\mathbb{P}^{n-1}\setminus{\mathcal{I}nd}$
$\displaystyle\longrightarrow\mathcal{U}\subset{\mathcal{H}ess}(\SS,h_{+})$
$\displaystyle\mathbf{v}$
$\displaystyle\longmapsto\left(\langle\mathbf{0}\rangle\subset\langle\mathbf{v}\rangle\subset\langle\mathbf{v},\SS\mathbf{v}\rangle\subset\cdots\subset\langle\mathbf{v},\SS\mathbf{v},\cdots\SS^{n-2}\mathbf{v}\rangle\subset\mathbb{C}^{n}\right),$

where $\mathcal{U}$ is the open subset of ${\mathcal{H}ess}(\SS,h_{+})$
consists of all flags $V_{\bullet}=\left(\\{0\\}\subset
V_{1}\subset\cdots\subset V_{n-1}\subset\mathbb{C}^{n}\right)$ such that
$V_{1}\not\in{\mathcal{I}nd}$. We will extend this isomorphism to isomorphisms
between blowups of $\mathbb{P}^{n-1}$ and Hessenberg type varieties defined in
the next paragraph.

In order to do so, we introduce a type of partial flag variety. For $0\leq
k\leq n-2$, define

${\mathcal{F}lag}^{(k+1)}(\mathbb{C}^{n}):=\left\\{V_{\bullet}=(V_{0}:=\langle\mathbf{0}\rangle\subset
V_{1}\subset\cdots\subset V_{k+1})\ |\ \dim_{\mathbb{C}}(V_{i})=i\text{ for
$i=1,\cdots,k+1$}\right\\}.$

We also define the corresponding Hessenberg type variety

${\mathcal{H}ess}^{(k+1)}(\SS,h_{+}):=\left\\{V_{\bullet}\in{\mathcal{F}lag}^{(k+1)}(\mathbb{C}^{n})\
|\ \SS V_{i}\subset V_{i+1}\text{ for $i=0,\cdots,k$}\right\\}.$

Notice that
${\mathcal{F}lag}^{(n-1)}(\mathbb{C}^{n})={\mathcal{F}lag}(\mathbb{C}^{n})$
and ${\mathcal{H}ess}^{(n-1)}(\SS,h_{+})={\mathcal{H}ess}(\SS,h_{+})$. The
main goal of this section is to show the following.

######  Proposition 3.2.

There is a natural isomorphism

$\mathcal{X}_{k}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$

for $k=0,\cdots,n-2$, where $\mathcal{X}_{k}$ is the prepermutohedral variety.

###### Proof.

It is clear that
$\mathcal{X}_{0}:=\mathbb{P}^{n-1}\cong{\mathcal{H}ess}^{(1)}(\SS,h_{+})$. For
$V_{\bullet}=(V_{0}:=\langle\mathbf{0}\rangle\subset V_{1}\subset
V_{2})\in{\mathcal{H}ess}^{(2)}(\SS,h_{+})$, suppose that
$\mathbf{0}\neq\mathbf{v}=(z_{1},\cdots,z_{n})\in V_{1}$. By lemma 3.1, if at
least two of the $x_{i}$’s are non-zero, then $\mathbf{v}$ and $\SS\mathbf{v}$
are linearly independent, and $V_{2}=\langle\mathbf{v},\SS\mathbf{v}\rangle$
is determined. If only one of the $z_{i}\neq 0$, then $V_{1}=Z_{i}$ (see
Section 2 for the definition of $Z_{i}$) for some $i=1,\cdots,n$ and thus $\SS
V_{1}=V_{1}$. To determine $V_{2}$, we need an extra piece of information,
which is the direction that give us the second dimension of $V_{2}$. This can
be specified as a point in $\mathbb{P}(\mathbb{C}^{n}/V_{1})$, which is
canonically isomorphic to the exceptional divisor $E_{i}$ when we blowup
$\mathbb{P}^{n-1}$ at $Z_{i}$. Therefore, after we blowup all the $Z_{i}$’s,
then for those $V_{1}$ such that $\SS V_{1}=V_{1}$, we also know what $V_{2}$
is. This gives the isomorphism
$\mathcal{X}_{1}\to{\mathcal{H}ess}^{(2)}(\SS,h_{+})$.

We continue the blowup process inductively as follows. First, we introduce the
“forgetful” morphism

$f^{(k)}:{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})\to{\mathcal{H}ess}^{(k)}(\SS,h_{+})\text{
for $k=1,\cdots,n-3$,}$

which sends a flag of $k+1$ vector spaces to the first $k$ of them, and
“forgets” the last vector space.

######  Claim.

The morphism $f^{(k)}$ is a birational map.

###### Proof.

Given $V_{\bullet}=(V_{0}:=\langle\mathbf{0}\rangle\subset
V_{1}\subset\cdots\subset V_{k})\in{\mathcal{H}ess}^{(k)}(\SS,h_{+})$, since
$\SS V_{k-1}\subset V_{k}$, we know from linear algebra that
$\dim_{\mathbb{C}}(\langle V_{k}\cup\SS V_{k}\rangle)=k\text{ or }k+1$.
Moreover, $\dim_{\mathbb{C}}(\langle V_{k}\cup\SS V_{k}\rangle)=k+1$ is the
generic situation in ${\mathcal{H}ess}^{(k)}(\SS,h_{+})$ and the set of all
$V_{\bullet}$ such that $\dim_{\mathbb{C}}(\langle V_{k-1}\cup\SS
V_{k}\rangle)=k$ is a closed subset of ${\mathcal{H}ess}^{(k)}(\SS,h_{+})$. If
$\dim_{\mathbb{C}}(\langle V_{k}\cup\SS V_{k}\rangle)=k+1$, then sending

$(V_{0}\subset V_{1}\subset\cdots\subset V_{k})\longmapsto(V_{0}\subset
V_{1}\subset\cdots\subset V_{k}\subset\langle V_{k}\cup\SS V_{k}\rangle)$

gives the inverse of $f^{(k)}$ for a generic flag in
${\mathcal{H}ess}^{(k)}(\SS,h_{+})$. ∎

If $\dim_{\mathbb{C}}(\langle V_{k}\cup\SS V_{k}\rangle)=k$, then $\SS
V_{k}=V_{k}$, and a reflection of Lemma 3.1 tells us that, via the isomorphism
$\mathcal{X}_{k}\cong{\mathcal{H}ess}^{(k)}(\SS,h_{+})$, $V_{\bullet}$ lies in
the strict transform of $Z_{\alpha}$, denoted by $\overline{Z}_{\alpha}$, for
some $\alpha\subset[n]$ with $k$ elements.

If we blowup $\mathcal{X}_{k-1}$ along $\overline{Z}_{\alpha}$, then the
exceptional divisor $E_{\alpha}$ is canonically identified with the projective
normal bundle $\mathbb{P}(N_{Z_{\alpha}\subset X_{k-1}})$. A point on
$E_{\alpha}$ carries the information of $V_{\bullet}$, together with a
(projective) normal direction of $Z_{\alpha}$ in $\mathbb{C}^{n}$, i.e. an
element in $\mathbb{P}(\mathbb{C}^{n}/Z_{\alpha})$. This assigns a unique flag
in ${\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$. More precisely, if
$\mathbf{v}\in\mathbb{C}^{n}/Z_{\alpha}$ represents the direction in
$\mathbb{P}(\mathbb{C}^{n}/Z_{\alpha})$, then one sets $V_{k+1}=\langle
V_{k},\mathbf{v}\rangle$. Once we blowup $\mathcal{X}_{k-1}$ along the strict
transforms of all $Z_{\alpha}$ for $\alpha\subset[n]$ with $k$ elements, we
obtain the isomorphism
$\mathcal{X}_{k}\cong{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$. ∎

######  Example 3.1.

Suppose we have $\mathbf{v}_{1}=(1,1,0,0,0)\in\mathbb{C}^{5}$, then
$V_{1}=\langle\mathbf{v}_{1}\rangle\subset
V_{2}=\langle\mathbf{v}_{1},\SS\mathbf{v}_{1}\rangle$ form the first two
spaces in the flag but $\SS V_{2}=V_{2}$. Thus we need to blowup
$Z_{\\{1,2\\}}$. Notice that $V_{2}=Z_{\\{1,2\\}}\subset\mathbb{C}^{5}$ and
elements in $E_{\\{1,2\\}}$ over $\mathbf{v}_{1}$ are in one-to-one
correspondence to projectivized normal vectors for the embedding
$\mathbb{C}^{2}\hookrightarrow\mathbb{C}^{5}$ at $\mathbf{v}_{1}$, i.e.,
$\mathbb{P}(\mathbb{C}^{5}/\mathbb{C}^{2})$. Suppose we pick
$\mathbf{v}_{2}=(1,1,1,0,0)$ as a representative for an element in
$\mathbb{P}(\mathbb{C}^{5}/\mathbb{C}^{2})$, then that gives us $V_{3}=\langle
V_{2},\mathbf{v}_{2}\rangle$ (one can check that $V_{3}$ is independent of the
choice of $\mathbf{v}_{2}$), but then we would have $\SS V_{3}=V_{3}$ again.
This means that the element represented by $\mathbf{v}_{2}$ in $E_{\\{1,2\\}}$
is in the strict transform of the set $Z_{\\{1,2,3\\}}$. When we blowup
$Z_{\\{1,2,3\\}}$, elements of $E_{\\{1,2,3\\}}$ over $\mathbf{v}_{2}$ will
then corresponds to $\mathbb{P}(\mathbb{C}^{5}/\mathbb{C}^{3})$. Picking a
representative, say $\mathbf{v}_{3}=(1,1,1,1,1)$, we will have $V_{4}=\langle
V_{3},\mathbf{v}_{3}\rangle$ and $V_{5}=\mathbb{C}^{5}$. This gives a flag
$V_{\bullet}=\left(\langle\mathbf{0}\rangle\subset V_{1}\subset\cdots\subset
V_{5}\right)\in{\mathcal{H}ess}(\SS,h_{+})$.

##  4\. Prepermutohedral varieties as toric varieties

From the isomorphism constructed in the previous section, we can discover how
the torus $(\mathbb{C}^{*})^{n-1}$ sits inside
${\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$, as follows. The point
${\bf{z}}=(z_{1},\cdots,z_{n-1})\in(\mathbb{C}^{*})^{n-1}$ corresponds to the
flag

$V_{\bullet}:=\left(\langle\mathbf{0}\rangle\subset\langle\mathbf{v}\rangle\subset\langle\mathbf{v},\SS\mathbf{v}\rangle\subset\cdots\subset\langle\mathbf{v},\SS\mathbf{v},\cdots\SS^{k}\mathbf{v}\rangle\subset\mathbb{C}^{n}\right)$

where $\mathbf{v}=(1,z_{1},\cdots,z_{n-1})$. We denote this correspondence as
an injective map
$\phi_{k+1}:(\mathbb{C}^{*})^{n-1}\hookrightarrow{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$
by $\phi({\bf{z}})=V_{\bullet}$.

One can also observe the algebraic group structure of
$(\mathbb{C}^{*})^{n-1}\subset{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$. Given
$(z^{\prime}_{1},\cdots,z^{\prime}_{n-1})\in(\mathbb{C}^{*})^{n-1}$ and
$\mathbf{v}^{\prime}=(1,z^{\prime}_{1},\cdots,z^{\prime}_{n-1})$, the product
of $(z_{1},\cdots,z_{n-1})$ and $(z^{\prime}_{1},\cdots,z^{\prime}_{n-1})$
corresponds to the flag

$\left(\langle\mathbf{0}\rangle\subset\langle\mathbf{v}\mathbf{v}^{\prime}\rangle\subset\langle\mathbf{v}\mathbf{v}^{\prime},\SS(\mathbf{v}\mathbf{v}^{\prime})\rangle\subset\cdots\subset\langle\mathbf{v}\mathbf{v}^{\prime},\SS(\mathbf{v}\mathbf{v}^{\prime}),\cdots,\SS^{k}(\mathbf{v}\mathbf{v}^{\prime})\rangle\subset\mathbb{C}^{n}\right),$

where $\mathbf{v}\mathbf{v}^{\prime}$ is the coordinate-wise product of
$\mathbf{v}$ and $\mathbf{v}^{\prime}$. That is,
$\phi({\bf{z}})\phi({\bf{z}}^{\prime}):=\phi({\bf{z}}{\bf{z}}^{\prime})$.
Moreover, one can discover the action of $(\mathbb{C}^{*})^{n-1}$ on
${\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$ similarly.

###  4.1. Fan structure for the prepermutohedral varieties

Given that $\mathcal{X}_{k}\cong{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$ is a
toric variety, we would like to know the structure of the corresponding fan,
and the geometry properties we can conclude from the fan structure. The
standard reference for this part is [Fulton, CLS].

First, recall the structure of the fan $\Delta_{\mathbb{P}^{n-1}}$
corresponding to $\mathbb{P}^{n-1}$. Let $e_{1},\cdots,e_{n-1}$ be the
standard basis of $\mathbb{Z}^{n-1}$ and $e_{n}=-(e_{1}+\cdots+e_{n-1})$. The
cones in $\Delta_{\mathbb{P}^{n-1}}$ are generated by proper subsets of
$\\{e_{1},\cdots,e_{n}\\}$. More precisely, for any proper subset
$\alpha\subset[n]$, we denote $\sigma_{\alpha}$ to be the cone generated by
$\\{e_{i}|i\in\alpha\\}$, i.e.,
$\sigma_{\alpha}=\\{\sum_{i\in\alpha}a_{i}e_{i}|a_{i}\geq 0\\}$. Then

$\Delta_{\mathbb{P}^{n-1}}=\left\\{\sigma_{\alpha}\ |\
\alpha\subset[n]\right\\},$

with the convention that $\sigma_{\emptyset}=\\{\mathbf{0}\\}$.

As we showed earlier, the prepermutohedral variety $\mathcal{X}_{k}$ ($1\leq
k\leq n-2$) is obtained from $\mathbb{P}^{n-1}$ by blowing up the torus
invariant subvarieties (in the order of dimensions) up to dimensions $k-1$.
For the fan structure, this means we perform the star subdivisions (see [CLS,
Definition 3.3.17]) on all cones of codimensions $0,1,$ and so on, up to cones
of codimension $k-1$. We will describe the fan $\Delta_{\mathcal{X}_{k}}$
corresponds to the prepermutohedral variety $\mathcal{X}_{k}$ after setting up
some notations.

For a non-empty proper subset $\alpha\subset[n]$, we define the vector
$e_{\alpha}:=\sum_{i\in\alpha}e_{i}$. The rays (i.e. one-dimensional cones) of
$\Delta_{\mathcal{X}_{k}}$ are generated by $e_{\alpha}$ for non-empty proper
subset $\alpha\subset[n]$ such that $|\alpha|\geq n-k$ or $|\alpha|=1$. To
simplify notation, we write $e_{i}$ instead of $e_{\\{i\\}}$ for $i\in[n]$.

A chain of subsets of $[n]$ is a sequence of strict inclusions

$\mathcal{C}=(\alpha_{0}\subset\cdots\subset\alpha_{p})$

of proper subsets of $[n]$. It is also allowed that $p=0$, i.e.,
$\mathcal{C}=(\alpha_{0})$ is a chain consisting of only one set. We will also
use an alternative list notation to denote a chain. We use special bracket
symbols $\llbracket\ \rrbracket$ to enclose the list: first list the numbers
in $\alpha_{0}$, then list the numbers in $\alpha_{1}\setminus\alpha_{0}$
(separated by a bar), and so on, all the way to $[n]\setminus\alpha_{p}$. For
example, suppose $n=9$ and we have

$\mathcal{C}=(\\{1,4\\}\subset\\{1,2,3,4\\}\subset\\{1,2,3,4,6,7,9\\}),$

then in the list notation, $\mathcal{C}=\llbracket
1,4|2,3|6,7,9|5,8\rrbracket$.

Cones in $\Delta_{\mathcal{X}_{k}}$ are in one-to-one correspondence with
chains $\mathcal{C}$ such that $|\alpha_{0}|\leq n-k-1$ and $|\alpha_{j}|\geq
n-k$ for $j>0$. The correspondence is given as follows. For each such chain
$\mathcal{C}$, we associate it with the cone $\sigma_{\mathcal{C}}$ generated
by the vectors $e_{i}$, $i\in\alpha_{0}$ and $e_{\alpha_{j}}$, $j\geq 1$. The
dimension of the cone $\sigma_{\mathcal{C}}$ is equal to $|\alpha_{0}|+p$.

In particular, the top dimensional cones of $\Delta_{\mathcal{X}_{k}}$ are in
one-to-one correspondence with chains of the form

$\mathcal{C}=(\alpha_{0}\subset\alpha_{n-k}\subset\cdots\subset\alpha_{n-1})$

such that $|\alpha_{0}|=n-k-1$ and $|\alpha_{j}|=j$ for all
$j=n-k,\cdots,n-1$. (For $k=0$, this means $\mathcal{C}=(\alpha_{0})$ with
$|\alpha_{0}|=n-1$). For $k\geq 2$, such chains are determined by the sequence
of numbers

$\displaystyle a_{n}$ $\displaystyle:=[n]\setminus\alpha_{n-1},$
$\displaystyle a_{j}$ $\displaystyle:=\alpha_{j}\setminus\alpha_{j-1},\ \
n-k+1\leq j\leq n-1$ $\displaystyle a_{n-k}$
$\displaystyle:=\alpha_{n-k}\setminus\alpha_{0}.$

For $k=0$, such chains are determined by $a_{n}=[n]\setminus\alpha_{0}$; and
for $k=1$ the chains are determined by
$a_{n},a_{n-1}=\alpha_{n-1}\setminus\alpha_{0}$. In the list notation, by a
slightly abuse of notation, we would write
$\mathcal{C}=\llbracket\alpha_{0}|a_{n-k}|\cdots|a_{n}\rrbracket$.

Notice that, since $\alpha_{0}$ can be written as
$\alpha_{0}=[n]\setminus\\{a_{j}|n-k\leq j\leq n\\}$, the information on the
list of numbers $a_{n-k},\cdots,a_{n}$ is sufficient to determine the chain
$\mathcal{C}$. Therefore, the top dimensional cones in
$\Delta_{\mathcal{X}_{k}}$ are in one-to-one correspondence with permutations
of $k+1$ different numbers from $[n]$. The number of top dimensional cones in
$\Delta_{\mathcal{X}_{k}}$ is then given by the permutation number

$P(n,k+1)=\frac{n!}{(n-k-1)!}.$

A result in toric varieties [Fulton, Section 3.2] states that the Euler
characteristic of a toric variety is equal to the number of top dimensional
cones in its associated fan. Therefore, we conclude the following.

######  Proposition 4.1.

The Euler characteristic of the variety $\mathcal{X}_{k}$ is given by

$\chi(\mathcal{X}_{k})=P(n,k+1).$

We remark here that one can also calculate the Euler characteristics using the
structure of the cohomology groups in the next section.

###  4.2. Betti numbers of the prepermutohedral varieties

In order to calculate the Betti numbers for $\mathcal{X}_{k}$, we need to
study more on the cone structure of the fan associated to $\mathcal{X}_{k}$.

The intersection of two cones is given by the following operation on chains:
Given $\mathcal{C}=(\alpha_{0}\subset\cdots\subset\alpha_{p})$ and
$\mathcal{C}^{\prime}=(\alpha^{\prime}_{0}\subset\cdots\subset\alpha^{\prime}_{p^{\prime}})$,
define $\mathcal{C}\cap\mathcal{C}^{\prime}$ to be the following.

  * •

The first set in the chain is $\alpha_{0}\cap\alpha^{\prime}_{0}$.

  * •

The rest of the chain is the greatest common subchain of the chains of sets
$\alpha_{1}\subset\cdots\subset\alpha_{p}$ and
$\alpha^{\prime}_{1}\subset\cdots\subset\alpha^{\prime}_{p^{\prime}}$

For instance, suppose $n=10$, $\mathcal{C}=\llbracket
1,4,10|2,3|6,7|9|5,8\rrbracket$, and $\mathcal{C}^{\prime}=\llbracket
1,4,6|7,2,3|5,9|8\rrbracket$, then
$\mathcal{C}\cap\mathcal{C}^{\prime}=\llbracket 1,4|2,3,6,7|5,8,9\rrbracket$.
With this definition on the intersection of chains, we then have
$\sigma_{\mathcal{C}}\cap\sigma_{\mathcal{C}^{\prime}}=\sigma_{\mathcal{C}\cap\mathcal{C}^{\prime}}$.

Also, with the same notations as above, for the chains
$\mathcal{C},\mathcal{C}^{\prime}$, the corresponding cones
$\sigma_{\mathcal{C}}\subset\sigma_{\mathcal{C}^{\prime}}$ if
$\alpha_{0}\subset\alpha^{\prime}_{0}$ and for all $j>0$,
$\alpha_{j}=\alpha^{\prime}_{j^{\prime}}$ for some $j^{\prime}$.

We would like to assign a total order ‘$>$’ on the top dimensional cones of
$\Delta_{\mathcal{X}_{k}}$ with some desired property. Equivalently, we will
define the order on the set of chains. Set $\tau_{\mathcal{C}}$ to be the
intersection of $\sigma_{\mathcal{C}}$ with all
$\sigma_{\mathcal{C}^{\prime}}$ that comes after $\sigma_{\mathcal{C}}$ (i.e.
$\mathcal{C}^{\prime}>\mathcal{C}$) and such that
$\dim(\sigma_{\mathcal{C}}\cap\sigma_{\mathcal{C}^{\prime}})=n-2$. The desired
property is that

($*$) If $\tau_{\mathcal{C}}\subset\sigma_{\mathcal{C}^{\prime}}$, then
$\mathcal{C}^{\prime}\geq\mathcal{C}$

This is also the condition ($*$) in [Fulton, Section 5.2]. It implies that the
classes $[V(\tau_{\mathcal{C}})]$ form a basis for $H_{*}(X;\mathbb{Z})$ (see
[Fulton, Theorem on p.102]). We claim that the reversed lexicographic order in
the list notation for chains will satisfy ($*$ ‣ 4.2).

To define the order, suppose
$\mathcal{C}=\llbracket\alpha_{0}|a_{n-k}|\cdots|a_{n}\rrbracket$ and
$\mathcal{C}^{\prime}=\llbracket\alpha^{\prime}_{0}|a^{\prime}_{n-k}|\cdots|a^{\prime}_{n}\rrbracket$.
We define $\mathcal{C}<\mathcal{C}^{\prime}$ if for the greatest $j$ such that
$a_{j}\neq a^{\prime}_{j}$, we have $a_{j}<a^{\prime}_{j}$.

Under the above notations and definitions, there are two cases for which we
can have $\mathcal{C}^{\prime}>\mathcal{C}$ and
$\dim(\sigma_{\mathcal{C}}\cap\sigma_{\mathcal{C}^{\prime}})=n-2$:

  * Case 1:

$\alpha_{0}=\alpha^{\prime}_{0}$. In this case, we have
$a^{\prime}_{j+1}=a_{j}>a^{\prime}_{j}=a_{j+1}$ for some $n-k\leq j\leq n-1$
and $a_{i}=a^{\prime}_{i}$ for all other $n-k\leq i\leq n$, $i\neq j,j+1$.

  * Case 2:

$\alpha_{0}\neq\alpha^{\prime}_{0}$. In this case, we have
$a^{\prime}_{n-k}>a_{n-k}$ and $a_{i}=a_{i}^{\prime}$ for all $i>n-k$.

We need the following definition to describe the intersection
$\tau_{\mathcal{C}}$.

######  Definition.

Given a chain
$\mathcal{C}=\llbracket\alpha_{0}|a_{n-k}|\cdots|a_{n}\rrbracket$ in the list
notation, a descent (resp. ascent) in $\mathcal{C}$ is either $a_{j}>a_{j+1}$
(resp. $a_{j}<a_{j+1}$) for some $j=n-k,\cdots,n-1$, or $a>a_{n-k}$ (resp.
$a<a_{n-k}$) for some $a\in\alpha_{0}$.

We can now describe $\tau_{\mathcal{C}}$. It is obtained from
$\sigma_{\mathcal{C}}$ through the following process.

  * •

Whenever there is a descent $a_{j}>a_{j+1}$, one removes the set $\alpha_{j}$
in the chain $\mathcal{C}$.

  * •

Whenever there is a descent $a>a_{n-k}$, one removes the number $a$ from the
set $\alpha_{0}$.

For instance, suppose $n=9,k=4$ and $\mathcal{C}=\llbracket
1,2,5,8|4|3|6|9|7\rrbracket$, then there are $4$ descents in $\mathcal{C}$:
$5>4,8>4,4>3$, and $9>7$. The cone of intersections
$\tau_{\mathcal{C}}=\llbracket 1,2|5,8,4,3|6|9,7\rrbracket$, and
$\dim(\tau_{\mathcal{C}})=5$.

One observes that every time there is an descent in $\mathcal{C}$, the
dimension of $\tau_{\mathcal{C}}$ would go down by $1$. Therefore, the
dimension of $\tau_{\mathcal{C}}$ is given by

$\dim(\tau_{\mathcal{C}})=n-1-\\#(\text{descents in
$\mathcal{C}$)}=\\#(\text{ascents in $\mathcal{C}$}).$

The complex dimension of corresponding orbit closure $V[\tau_{\mathcal{C}}]$
will then be

$\displaystyle\dim_{\mathbb{C}}(V[\tau_{\mathcal{C}}])$
$\displaystyle=n-1-\dim(\tau_{\mathcal{C}})$ $\displaystyle=\\#(\text{descents
in $\mathcal{C}$)}=n-1-\\#(\text{ascents in $\mathcal{C}$}).$

Notice that $\mathcal{C}$ is the chain of a top dimensional cone with the
property that for each block (numbers between two bars) in the chain
corresponds to $\tau_{\mathcal{C}}$, numbers in the block are descending in
$\mathcal{C}$. Thus, it is the smallest in the reverse lexicographic order
among the longest chains that refine the chain for $\tau_{\mathcal{C}}$, and
we can conclude that $\tau_{\mathcal{C}}\subset\sigma_{\mathcal{C}^{\prime}}$
implies $\mathcal{C}^{\prime}\geq\mathcal{C}$. Therefore, by [Fulton, Theorem
on p.102], the homology classes of orbit closures $V[\tau_{\mathcal{C}}]$, as
$\mathcal{C}$ runs through all top dimensional cones, will form a basis of
$H_{*}(X;\mathbb{Z})$. This immediately implies the following.

######  Proposition 4.2.

The $2i$-th Betti number of $\mathcal{X}_{k}$ is given by

$\displaystyle\beta_{2i}(\mathcal{X}_{k})$
$\displaystyle=\\#(\text{permutations of $k+1$ different numbers in $[n]$ with
$i$ descents})$ $\displaystyle=\\#(\text{permutations of $k+1$ different
numbers in $[n]$ with $n-1-i$ ascents}).$

We remark here that it is possible to find some recursive relations among
these Betti numbers using the recursive structure of the cohomology groups
discuss in the next section (5.1).

##  5\. The dot action on cohomology groups

We investigate the dot action on the cohomology groups of the pre-
permutohedral varieties. We work on the usual cohomology groups. In this
section and the next, we sometimes need to specify the dimension of a
prepermutohedral variety. In such situation, we will use
$\mathcal{X}^{n-1}_{k}$ to denote a prepermutohedral variety of dimension
$n-1$ and of order $k$.

###  5.1. The cohomology groups of prepermutohedral varieties

First, we recall the standard result on the cohomology of blowup spaces. For
more details, see [Voisin, §7.3.3]. Let $X$ be a Kähler manifold of dimension
$n$, $Z\subset X$ be a submanifold, and
$\tilde{X}_{Z}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}X$ be the blowup
of $X$ along $Z$. Then $\tilde{X}_{Z}$ is still a Kähler manifold. Let
$E=\tau^{-1}(Z)$ be the exceptional divisor, then $E=\mathbb{P}(N_{Z\subset
X})$ is the projective bundle of the normal bundle of $Z$ in $X$. Thus $E$ is
of rank $r-1$, where $r=\operatorname{codim}(Z)$, and
$E\stackrel{{\scriptstyle j}}{{\hookrightarrow}}X$ is a hypersurface.

######  Theorem 5.1.

[Voisin, Theorem 7.31] Let $h=c_{1}({\mathcal{O}}_{E}(1))\in
H^{2}(E;\mathbb{Z})$, we have isomorphism

$\textstyle{H^{p}(X;\mathbb{Z})\oplus\left(\bigoplus_{i=0}^{r-2}H^{p-2i-2}(Z;\mathbb{Z})\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau^{*}+\sum_{i}j_{*}\circ(\cup
h^{i})\circ\tau|_{E}^{*}}$$\textstyle{H^{p}(\tilde{X}_{Z};\mathbb{Z}).}$

The map on the second component is decomposed as follows.

$\textstyle{H^{p-2i-2}(Z;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau|_{E}^{*}}$$\textstyle{H^{p-2i-2}(E;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cup
h^{i}}$$\textstyle{H^{p-2}(E;\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{*}}$$\textstyle{H^{p}(\tilde{X}_{Z};\mathbb{Z}).}$

Here $j_{*}$ is the Gysin morphism which is defined as the Poincare dual of
the map

$j_{*}:H_{2n-p}(E;\mathbb{Z})\to H_{2n-p}(\tilde{X}_{Z};\mathbb{Z}).$

Some remarks:

  * •

The theorem is also valid for cohomology groups with $\mathbb{C}$
coefficients. In this paper, we always use complex coefficients, thus we will
write $H^{p}(X)$ for $H^{p}(X;\mathbb{C})$ from now on.

  * •

We denote $H^{*}(X)=\oplus_{p}H^{p}(X)$ as a graded complex vector space.
Moreover, define $H^{p}_{q}(X):=H^{p+q}(X)$, i.e. the degree $p$ part of
$H_{q}^{*}(X)$ is equal to the degree $p+q$ part of $H^{*}(X)$. The theorem
can be written in term of the graded $\mathbb{C}$ vector spaces as

$H^{*}(\tilde{X}_{Z})\cong
H^{*}(X)\oplus\left(\bigoplus_{i=1}^{r-1}H^{*}_{-2i}(Z)\right).$

In the process of obtaining $\mathcal{X}_{k}$, we blowup strict transforms of
coordinate linear subvarieties $Z_{\alpha}$. The strict transforms
$\overline{Z}_{\alpha}$ are themselves permutohedral varieties of lower
dimension. We will use $\mathcal{X}^{d}$ to denote the permutohedral variety
of dimension $d$. A more compact form of the following description is in
[Procesi].

######  Theorem 5.2.

The cohomology, as a graded complex vector space, of the pre-permutohedral
variety $\mathcal{X}_{k}^{n-1}$, $1\leq k\leq n-2$, is given by

(5.1) $H^{*}(\mathcal{X}_{k}^{n-1})\cong
H^{*}(\mathbb{P}^{n-1})\oplus\left(\bigoplus_{j=1}^{k}\bigoplus_{\stackrel{{\scriptstyle\alpha\subset[n],}}{{|\alpha|=j}}}\bigoplus_{i=1}^{n-j-1}H^{*}_{-2i}(\mathcal{X}^{j-1})\right).$

###### Proof.

We apply Theorem 5.1 with our setting:

  * •

$\mathcal{X}_{0}^{n-1}=\mathbb{P}^{n-1}$.

  * •

For $1\leq k\leq n-2$, $\mathcal{X}_{k}^{n-1}$ is the blowup of
$\mathcal{X}_{k-1}^{n-1}$ along all $\overline{Z}_{\alpha}$ with
$\alpha\subset[n]$ and $|\alpha|=k$.

  * •

$\overline{Z}_{\alpha}$ is isomorphic to $\mathcal{X}^{k-1}$, thus the
codimension of $\overline{Z}_{\alpha}$ is $n-k$.

This gives us

$H^{*}(\mathcal{X}_{k}^{n-1})\cong
H^{*}(\mathcal{X}_{k-1}^{n-1})\oplus\left(\bigoplus_{\stackrel{{\scriptstyle\alpha\subset[n],}}{{|\alpha|=k}}}\bigoplus_{i=1}^{n-k-1}H^{*}_{-2i}(\overline{Z}_{\alpha})\right)\cong
H^{*}(\mathcal{X}_{k-1}^{n-1})\oplus\left(\bigoplus_{\stackrel{{\scriptstyle\alpha\subset[n],}}{{|\alpha|=k}}}\bigoplus_{i=1}^{n-k-1}H^{*}_{-2i}(\mathcal{X}^{k-1})\right).$

The isomorphism in the theorem can be obtained by applying the above
isomorphism inductively on $k$. ∎

In the following, we compute some examples in low dimensions.

######  Example 5.1.

For the base case, $H^{*}(\mathcal{X}_{0}^{n-1})\cong
H^{*}(\mathbb{P}^{n-1})\cong\mathbb{C}[\xi]/(\xi^{n})$ where $\xi\in
H^{2}(\mathbb{P}^{n-1})$ is the first Chern class of the hyperplane bundle.
Next, let us consider $\mathcal{X}_{1}^{n-1}$. This is the space obtained in
the first step of the blowups. Here, we blowup the points
$Z_{1},\cdots,Z_{n}$. The cohonology groups become

$\displaystyle H^{0}(\mathcal{X}_{1}^{n-1})$ $\displaystyle\cong
H^{0}(\mathbb{P}^{n-1})\cong\mathbb{C}$ $\displaystyle
H^{2j}(\mathcal{X}_{1}^{n-1})$ $\displaystyle\cong
H^{0}(\mathbb{P}^{n-1})\oplus\bigoplus_{i=1}^{n}H^{0}(Z_{i})\cong\mathbb{C}\oplus\mathbb{C}^{n},\text{
$1\leq j\leq n-2$}$ $\displaystyle H^{2n-2}(\mathcal{X}_{1}^{n-1})$
$\displaystyle\cong H^{2n-2}(\mathbb{P}^{n-1})\cong\mathbb{C}.$

In particular, if $n=3$, then $\mathcal{X}_{1}^{2}$ is the blowup of
$\mathbb{P}^{2}$ at the three points $[1:0:0],[0:1:0]$ and $[0:0:1]$.
$H^{0}(\mathcal{X}_{1}^{2})\cong\mathbb{C}$;
$H^{1}(\mathcal{X}_{1}^{2})\cong\mathbb{C}\oplus\mathbb{C}^{2}$ and
$H^{2}(\mathcal{X}_{1}^{2})\cong\mathbb{C}$.

######  Example 5.2.

We investigate the cohomology of $\mathcal{X}^{n-1}_{2}$ next. in order to
obtain the space, we blowup all the lines $Z_{\\{i,j\\}}$ where
$\\{i,j\\}\subset[n]$ and $i\neq j$. Notice $Z_{\\{i,j\\}}$ is isomorphic to
its strict transform $\overline{Z}_{\\{i,j\\}}$ since its dimension is one. We
have $H^{0}(Z_{\\{i,j\\}})\cong\mathbb{C}$ and
$H^{2}(Z_{\\{i,j\\}})\cong\mathbb{C}$. Therefore, by Theorem 5.2, we have

$\displaystyle H^{0}(\mathcal{X}_{2}^{n-1})$ $\displaystyle\cong
H^{0}(\mathbb{P}^{n-1})\cong\mathbb{C}$ $\displaystyle
H^{2}(\mathcal{X}_{2}^{n-1})$ $\displaystyle\cong
H^{0}(\mathbb{P}^{n-1})\oplus\bigoplus_{\stackrel{{\scriptstyle
i,j\in[n],}}{{i\neq
j}}}H^{0}(Z_{\\{i,j\\}})\cong\mathbb{C}\oplus\mathbb{C}^{\binom{n}{2}}$
$\displaystyle H^{2j}(\mathcal{X}_{2}^{n-1})$ $\displaystyle\cong
H^{0}(\mathbb{P}^{n-1})\oplus\bigoplus_{i=1}^{n}H^{0}(Z_{i})\oplus\bigoplus_{\stackrel{{\scriptstyle
i,j\in[n],}}{{i\neq j}}}\left(H^{0}(Z_{\\{i,j\\}})\oplus
H^{0}(Z_{\\{i,j\\}})\right)$
$\displaystyle\cong\mathbb{C}\oplus\mathbb{C}^{n}\oplus\mathbb{C}^{\binom{n}{2}}\oplus\mathbb{C}^{\binom{n}{2}},\
\ \ \text{ $2\leq j\leq n-3$}.$ $\displaystyle
H^{2n-4}(\mathcal{X}_{2}^{n-1})$ $\displaystyle\cong
H^{0}(\mathbb{P}^{n-1})\oplus\bigoplus_{\stackrel{{\scriptstyle
i,j\in[n],}}{{i\neq
j}}}H^{2}(Z_{\\{i,j\\}})\cong\mathbb{C}\oplus\mathbb{C}^{\binom{n}{2}}$
$\displaystyle H^{2n-2}(\mathcal{X}_{2}^{n-1})$ $\displaystyle\cong
H^{2n-2}(\mathbb{P}^{n-1})\cong\mathbb{C}.$

###  5.2. Encoding the cohomology groups and the $\mathfrak{S}_{n}$
representation

There is a nice way to encode the one-dimensional components of the cohomology
groups of the permutohedral variety by the codes defined by Stembridge [Ste].
We recall the notations first. For a sequence
${\mathbf{a}}=(a_{1},\cdots,a_{n})$ of nonnegative integers, we will call $n$
the length of ${\mathbf{a}}$. Let $S^{+}({\mathbf{a}})=\\{a_{i}|1\leq i\leq
n,a_{i}>0\\}$ denote the set of positive integers in ${\mathbf{a}}$. For a
positive integer $k$, the sequence ${\mathbf{a}}$ is called $k$-admissible if
$S^{+}({\mathbf{a}})=\\{1,\cdots,k\\}$; ${\mathbf{a}}$ is $0$-admissible if
$S^{+}({\mathbf{a}})=\emptyset$, i.e. ${\mathbf{a}}$ consists of all $0$’s.
The sequence ${\mathbf{a}}$ is called admissible if it is $k$-admissible for
some $k\geq 0$.

Let $m_{j}({\mathbf{a}})$ denote the number of occurrences of the integer $j$
in the sequence ${\mathbf{a}}$. A marked sequence is a pair $({\mathbf{a}},f)$
where ${\mathbf{a}}$ is a sequence of nonnegative integers and
$f:S^{+}({\mathbf{a}})\to\mathbb{N}$ be a map such that $1\leq
f(j)<m_{j}({\mathbf{a}})$ for all $j\in S^{+}({\mathbf{a}})$. We will adapt
the notation in [Ste] and represent a marked sequence $({\mathbf{a}},f)$ by
putting a hat notation on top of $j$ at the $[f(j)+1]$-st occurrence of $j$.
The index of a marked sequence is defined as
$\operatorname{ind}({\mathbf{a}},f):=\sum_{j\in S^{+}({\mathbf{a}})}f(j)$.

A code is defined to be an admissible marked sequence, i.e. a marked sequence
$({\mathbf{a}},f)$ such that ${\mathbf{a}}$ is admissible. There is one code
$((0,\cdots,0),\emptyset)$ consists of all $0$’s and the empty function as the
marking function. We define the index of this code to be $0$.

For a code ${\mathbf{a}}$, let
$\max({\mathbf{a}})=\max\\{a_{i}|i=1,\cdots,n\\}$ be the maximum number in
${\mathbf{a}}$, and let
$\mu({\mathbf{a}}):=m_{\max({\mathbf{a}})}({\mathbf{a}})$ be the number of
occurrences of $\max({\mathbf{a}})$ in ${\mathbf{a}}$. For example, if
${\mathbf{a}}=1201\hat{2}\hat{1}2$, then $\max({\mathbf{a}})=2$ and
$\mu({\mathbf{a}})=3$. We further define ${\mathbf{a}}^{\prime}$ to be the
sequence obtained after removing all the $\max({\mathbf{a}})$ from
${\mathbf{a}}$, but keep everything else unchanged. For the example
${\mathbf{a}}=1201\hat{2}\hat{1}2$ above, one would get
${\mathbf{a}}^{\prime}=101\hat{1}$. It is possible to have
${\mathbf{a}}^{\prime}$ equals to the empty sequence, in which case
${\mathbf{a}}$ consists all $0$’s or all $1$’s. Notice that if ${\mathbf{a}}$
is a code, then ${\mathbf{a}}^{\prime}$ is either a code or an empty sequence.

Recall, from Theorem 5.2, that we have the decomposition of the cohomology of
the permutohedral variety, by setting $k=n-2$ in (5.1), as follows.

(5.2) $H^{*}(\mathcal{X}^{n-1})\cong
H^{*}(\mathbb{P}^{n-1})\oplus\left(\bigoplus_{j=1}^{n-2}\bigoplus_{\stackrel{{\scriptstyle\alpha\subset[n],}}{{|\alpha|=j}}}\bigoplus_{i=1}^{n-j-1}H^{*}_{-2i}(\overline{Z}_{\alpha})\right).$

The main result of this section is the following.

######  Proposition 5.3.

Let $n\geq 2$ be an integer. There is a natural one-to-one correspondence
between codes of length $n$ and one-dimensional components of
$H^{*}(\mathcal{X}^{n-1})$.

###### Proof.

We will construct the correspondence inductively. The base case is a code
${\mathbf{a}}$ with $\mu({\mathbf{a}})=n$. That means, ${\mathbf{a}}$ consists
of either all $0$’s or all $1$’s. In this case, suppose
$\operatorname{ind}({\mathbf{a}},f)=i$, we assign $({\mathbf{a}},f)$ to the
$2i$-th cohomology of $H^{*}(\mathbb{P}^{n-1})$ in the above decomposition.

Next, suppose that $\mu({\mathbf{a}})<n$. Then we must have
$\max({\mathbf{a}})>0$ and thus $2\leq\mu({\mathbf{a}})\leq n-1$. We first set
the integers $i,j$ and the subset $\alpha\subset[n]$ that are used as indices
in the above decomposition.

  * •

$j=n-\mu({\mathbf{a}})$,

  * •

$\alpha=\left\\{i\,|\,a_{i}<\max({\mathbf{a}})\right\\}$,

  * •

$i=f(\max({\mathbf{a}}))$.

Notice that, since $\max({\mathbf{a}})>0$, $i=f(\max({\mathbf{a}}))$ is well-
defined; moreover, we have $1\leq i\leq\mu({\mathbf{a}})-1=n-j-1$ by the
definition of $f$. Concretely, what these indices record are the following:

  * •

$\mu({\mathbf{a}})$ record the codimension of the subspace which is blown up
in the process of constructing $\mathcal{X}^{n-1}$.

  * •

$\overline{Z}_{\alpha}$ is the subspace getting blown up in the process of
constructing $\mathcal{X}^{n-1}$. Equivalently, the set

$[n]\setminus\alpha=\left\\{i\,|\,a_{i}=\max({\mathbf{a}})\right\\}$

encodes the coordinates of the subspace that are set to be $0$. For example,
suppose $n=6$ and ${\mathbf{a}}=12\hat{1}01\hat{2}$, then
$\mu({\mathbf{a}})=2$, $\alpha=\\{1,3,4,5\\}$, and the subspace getting blown
up is defined by the equations $z_{2}=z_{6}=0$ in $\mathbb{P}^{5}$ (then
taking the strict transform to the proper space).

  * •

$i=f(\max({\mathbf{a}}))$ denotes the shifting of degrees when we map
$H^{*}(\overline{Z}_{\alpha})$ into $H^{*}(\mathcal{X}^{n-1})$.

Next, ${\mathbf{a}}^{\prime}$ is a code of length $j=n-\mu({\mathbf{a}})$.
Therefore, by induction hypothesis, it corresponds to a unique component in
$H^{*}_{-2i}(\overline{Z}_{\alpha})\cong H^{*}_{-2i}(\mathcal{X}^{j-1})$.

Finally, since one can recover ${\mathbf{a}}$ uniquely from
${\mathbf{a}}^{\prime}$, $\alpha$, and $i$, this correspondence is one-to-one.
One can also observe that the ranges of $j$ and $i$ we defined from codes of
length $n$ are the same as the range for the indices in the decomposition
(5.2). Therefore the correspondence is surjective. ∎

Recall that, for integers $n\geq 2$ and $0\leq k\leq n-2$, the
prepermutohedral variety $\mathcal{X}^{n-1}_{k}$ is obtained from
$\mathcal{X}^{n-1}_{0}=\mathbb{P}^{n-1}$ by blowing up points, lines, …, all
the way to the coordinate subspaces of dimension $k-1$. Since
$\mu({\mathbf{a}})$ records the codimension of the subspace getting blown up,
one can conclude the following coding for the cohomologies of
$\mathcal{X}^{n-1}_{k}$ as well.

######  Corollary 5.4.

For integers $n\geq 2$ and $0\leq k\leq n-2$, there is a natural one-to-one
correspondence between one-dimensional components of
$H^{*}(\mathcal{X}^{n-1}_{k})$ and the set of codes ${\mathbf{a}}$ of length
$n$ such that $\mu({\mathbf{a}})\geq n-k$.

######  Example 5.3.

For $n=4$, we can read the cohomologies of $\mathcal{X}^{3}_{k}$, $k=0,1,2$,
and the corresponding codes, from the following table:

$\begin{array}[]{|c|cc|cc|cc|}\hline\cr\mbox{}&\vrule\lx@intercol\hfil\mathcal{X}^{3}_{0}\hfil\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil\mathcal{X}^{3}_{1}\hfil\lx@intercol\vrule\lx@intercol&\vrule\lx@intercol\hfil\mathcal{X}^{3}_{2}\hfil\lx@intercol\vrule\lx@intercol\\\
\hline\cr&&\ \text{codes}&&\ \text{codes}&&\ \text{codes}\\\ \hline\cr
H^{0}(\mathcal{X}^{3}_{\bullet})\cong&H^{0}(\mathbb{P}^{3})&0000&&&&\\\
\hline\cr
H^{2}(\mathcal{X}^{3}_{\bullet})\cong&H^{2}(\mathbb{P}^{3})&1\hat{1}11&\oplus_{j=1}^{4}H^{0}(Z_{j})&01\hat{1}1&\oplus_{\stackrel{{\scriptstyle
i,j\in[n],}}{{i\neq j}}}H^{0}(Z_{\\{i,j\\}})&001\hat{1}\\\ \hline\cr
H^{4}(\mathcal{X}^{3}_{\bullet})\cong&H^{4}(\mathbb{P}^{3})&11\hat{1}1&\oplus_{j=1}^{4}H^{0}(Z_{j})&011\hat{1}&\oplus_{\stackrel{{\scriptstyle
i,j\in[n],}}{{i\neq j}}}H^{2}(Z_{\\{i,j\\}})&1\hat{1}2\hat{2}\\\ \hline\cr
H^{6}(\mathcal{X}^{3}_{\bullet})\cong&H^{6}(\mathbb{P}^{3})&111\hat{1}&&&&\\\
\hline\cr\end{array}$

Entries in the table record the direct summands needed to obtain the
cohomology of the corresponding spaces. For example, one can read from the
table that $H^{2}(\mathcal{X}^{3}_{1})\cong
H^{2}(\mathbb{P}^{3})\oplus_{j=1}^{4}H^{0}(Z_{j})$ and
$H^{4}(\mathcal{X}^{3}_{2})\cong
H^{4}(\mathbb{P}^{3})\oplus_{j=1}^{4}H^{0}(Z_{j})\oplus_{\stackrel{{\scriptstyle
i,j\in[n],}}{{i\neq j}}}H^{2}(Z_{\\{i,j\\}})$. Notice that $Z_{j}$ here are
points and $Z_{\\{i,j\\}}$ are lines. The codes recorded here are the
representatives for all those in the direct sum (i.e. in an $\mathfrak{S}_{4}$
orbit). For instance, the code $01\hat{1}1$ represents the four codes
$01\hat{1}1,10\hat{1}1,1\hat{1}01$, and $1\hat{1}10$. We pick the
representative to be the one that is increasing in numbers.

The convenience of using the codes to encode components of cohomology comes in
two ways. First, it give a nice compact way to write
$H^{*}(\mathcal{X}^{n-1}_{k})$. Let ${\mathcal{C}ode}(n)$ denote the set of
all codes of length $n$, then

$H^{*}(\mathcal{X}^{n-1}_{k})=\bigoplus_{\stackrel{{\scriptstyle{\mathbf{a}}\in{\mathcal{C}ode}(n)}}{{\mu({\mathbf{a}})\geq
n-k}}}\mathbb{C}_{\mathbf{a}}.$

Second, the $\mathfrak{S}_{n}$-representation on
$H^{*}(\mathcal{X}^{n-1}_{k})$ is compatible with the $\mathfrak{S}_{n}$
action on ${\mathcal{C}ode}(n)$, in the sense that for any
$w\in\mathfrak{S}_{n}$, we have
$w\,\mathbb{C}_{\mathbf{a}}=\mathbb{C}_{w{\mathbf{a}}}$ in the above
decomposition of $H^{*}(\mathcal{X}^{n-1}_{k})$. We explain this assertion in
the next paragraph.

For $w\in\mathfrak{S}_{n}$, the action of $\mathfrak{S}_{n}$ on
${\mathcal{C}ode}(n)$ is given by $w\cdot({\mathbf{a}},f)=(w{\mathbf{a}},f)$
where $(w{\mathbf{a}})_{i}=a_{w^{-1}(i)}$, $1\leq i\leq n$. The action on
$\mathcal{X}^{n-1}_{k}$ is induced from the action on $\mathbb{P}^{n-1}$ given
by permuting the coordinates, i.e.
$w\cdot[z_{1}:\cdots:z_{n}]=[z_{w(1)}:\cdots:z_{w(n)}]$. Then the action on
the cohomology is given by the pullback of the action on
$\mathcal{X}^{n-1}_{k}$. For the coordinate hyperplane $(z_{i}=0)$, its
pullbak is $(z_{w^{-1}(i)}=0)$. Recall that, in the encoding, the locations of
the maximal numbers corresponds to coordinates to set to be $0$. Moreover,
permuting the coordinates also results in permuting the normal directions to
the coordinate linear subspaces, and the exceptional divisor corresponds to
the projective bundle associated to the normal bundle. Hence we know that $w$
maps $H^{*}(\overline{Z}_{\alpha})$ to $H^{*}(\overline{Z}_{w^{-1}\alpha})$.
Then, we can conclude $w\,\mathbb{C}_{\mathbf{a}}=\mathbb{C}_{w{\mathbf{a}}}$
by induction on $n$.

Therefore, the $\mathfrak{S}_{n}$ representation on $\mathcal{X}^{n-1}_{k}$ is
the permutation representation induced by the permutation on codes of length
$n$ with $\mu({\mathbf{a}})\geq n-k$. An immediate consequence of this fact is
the following.

######  Proposition 5.5.

There is a permutation basis for the dot representation on the cohomology of
the prepermutohedral varieties $\mathcal{X}_{k}^{n-1}$, $1\leq k\leq n-2$.

###### Proof.

For each $\mathfrak{S}_{n}$ orbit of codes under the encoding, pick a
representative ${\mathbf{a}}$ and pick an element
$\xi\in\mathbb{C}_{{\mathbf{a}}}$. Setting all other components to be $0$
gives us an element in $H^{*}(\mathcal{X}_{k})$. Then the $w\xi$’s, as $w$
runs through $\mathfrak{S}_{n}$, form a linearly independent set. As we pick
these elements for all orbits, we obtain a permutation basis. ∎

###  5.3. The characteristic series for the dot action on prepermutohedral
varieties

We follow the notation in [Procesi] and denote

$A_{n-1}(t)=\sum_{j=0}^{n-1}\operatorname{ch}H^{2j}(\mathcal{X}^{n-1})t^{j},$

where $H^{2j}(\mathcal{X}^{n-1})$ denotes the
$\mathfrak{S}_{n}$-representation, and ‘$\operatorname{ch}$’ is the Frobenius
characteristic map. In [Procesi], Procesi observed the following recursive
formula

$A_{n-1}(t)=s_{n}\sum_{i=0}^{n-1}t^{i}+\sum_{i=0}^{n-3}s_{n-1-i}A_{i}(t)\left(\sum_{l=1}^{n-i-2}t^{l}\right)$

from the iterated blowup structure of $\mathcal{X}^{n-1}$. Here $s_{n}$ and
$s_{n-1-i}$ are the Schur symmetric functions. The first term in the recursive
relation corresponds to the representation on base space $\mathbb{P}^{n-1}$,
and for each $i=0,\cdots,n-3$, the term
$s_{n-1-i}A_{i}(t)\sum_{l=1}^{n-i-2}t^{l}$ corresponds to the representation
on the blowup of $i$-dimensional coordinate space.

For a prepermutohedral variety $\mathcal{X}^{n-1}_{k}$, we denote

$A_{n-1,k}(t):=\sum_{j=0}^{n-1}\operatorname{ch}H^{2j}(\mathcal{X}^{n-1}_{k})t^{j}.$

Since we also have the iterated blowup structure for $\mathcal{X}^{n-1}_{k}$,
we have a similar recursive formula for $A_{n-1,k}(t)$, with the second
summation only goes from $0$ to $k-1$. With the more compact notation
$[n]_{t}=\sum_{i=0}^{n-1}t^{i}$, and with the identity $s_{j}=h_{j}$ (the
complete homogeneous symmetric functions), the recursive formula can be
written as

(5.3)
$A_{n-1,k}(t)=h_{n}[n]_{t}+\sum_{i=0}^{k-1}h_{n-1-i}A_{i}(t)t[n-i-2]_{t}.$

Notice that for $k=0$, the second term disappears. This recursive formula is
sufficient for our purpose, but it might be interesting to derive a close form
for $A_{n-1,k}(t)$.

##  6\. The geometry of certain Hessenberg varieties

In this section, we consider Hessenberg varieties associated with Hessenberg
functions of the type $h_{k}=(2,3,\cdots,k+1,n,\cdots,n)$, for some $k\leq
n-3$. That is,

$h_{k}(j)=\begin{cases}j+1,&j=1,\cdots,k\\\ n,&j=k+1,\cdots,n.\end{cases}$

We denote the corresponding Hessenberg variety
$\mathcal{Y}={\mathcal{H}ess}(\SS,h_{k})$. It is a smooth complex variety of
dimension $k+\frac{1}{2}{(n-k)(n-k-1)}$. There is a morphism from
$\mathcal{Y}$ to the Hessenberg type variety
$\mathcal{X}_{k}\cong{\mathcal{H}ess}^{(k+1)}(\SS,h_{+})$ which remembers only
the flags up to dimension $k+1$. More precisely, we define

$f:\mathcal{Y}\to\mathcal{X}_{k}$

as follows:

$(V_{0}\subset V_{1}\subset\cdots\subset V_{k}\subset\cdots\subset
V_{n})\longmapsto(V_{0}\subset V_{1}\subset\cdots\subset V_{k+1}).$

Since $h_{k}(j)=n$ for $j\geq k+2$, there is no condition imposed from $h_{k}$
on $V_{j}$ for $j\geq k+2$. (Notice that $h_{k}(k)=k+1$ implies $\SS
V_{k}\subset V_{k+1}$, so there is still condition from $h_{k}$ for
$V_{k+1}$.) Therefore, the fiber of $f$ over a partial flag
$(V_{0}\subset\cdots\subset V_{k+1})$ can be identified with the flag variety
${\mathcal{F}lag}(\mathbb{C}^{n}/V_{k+1})\cong{\mathcal{F}lag}(\mathbb{C}^{n-k-1})$.
Hence $\mathcal{Y}$ has a fiber bundle structure over $\mathcal{X}_{k}$.

Over $\mathcal{Y}$, there is a tautological filtration

$\mathcal{V}_{k+1}\subset\mathcal{V}_{k+2}\subset\cdots\subset\mathcal{V}_{n}\cong\mathcal{Y}\times\mathbb{C}^{n}$

of vector subbundles of the trivial bundle over $\mathcal{Y}$: the fiber of
$\mathcal{V}_{j}$ over a flag $(V_{0}\subset\cdots\subset V_{n})$ is the
vector space $V_{j}$. We then have the line bundles
$\mathcal{L}_{j}:=\mathcal{V}_{j}/\mathcal{V}_{j-1}$, $k+2\leq j\leq n$. Set
$x_{j}=-c_{1}(\mathcal{L}_{j})$ to be the negative of the first Chern class of
the line bundle $\mathcal{L}_{j}$, then we have the following description of
the cohomology ring $H^{*}(\mathcal{Y})$.

######  Proposition 6.1.

The cohomology ring $H^{*}(\mathcal{Y})=H^{*}(\mathcal{Y};\mathbb{C})$ is
generated over $H^{*}(\mathcal{X}_{k})$ by the classes $x_{k+2},\cdots,x_{n}$,
subject to the relations $e_{i}(x_{k+2},\cdots,x_{n})=0$ for $1\leq i\leq
n-k-1$. That is,

$H^{*}(\mathcal{Y})\cong
H^{*}(\mathcal{X}_{k})[X_{k+2},\cdots,X_{n}]/(e_{1}(X_{k+2},\cdots,X_{n}),\cdots,e_{n-k-1}(X_{k+2},\cdots,X_{n})),$

and the classes $x_{k+2},\cdots,x_{n}$ can be identified with the images of
$X_{k+2},\cdots,X_{n}$ under the quotient. In addition, the classes
$x_{k+2}^{i_{k+2}}\cdots x_{n}^{i_{n}}$, with exponents $0\leq i_{j}\leq n-j$,
form a basis for $H^{*}(\mathcal{Y})$ over $H^{*}(\mathcal{X}_{k})$.

###### Proof.

The proof mimics the proof for the cohomology of the flag variety [Fulton,
Proposition 10.2.3] and is based on basic facts about projective bundles. For
more details, see [GrHa, p. 606] or [Fulton2, Appendix B.4]. For a vector
bundle $\mathcal{V}$ over a variety $\mathcal{X}$, let
$\rho:\mathbb{P}(\mathcal{V})\to\mathcal{X}$ denote the corresponding
projective bundle. There is a tautological bundle
$\mathcal{L}\subset\rho^{*}(\mathcal{V})$. Set $\xi=-c_{1}(\mathcal{L})$, then

(6.1) $H^{*}(\mathbb{P}(\mathcal{V}))\cong
H^{*}(\mathcal{X})[\xi]/(\xi^{r}+a_{1}\xi^{r-1}+\cdots+a_{r}),$

where $a_{i}=c_{i}(\mathcal{V})\in H^{2i}(\mathcal{X})$.

In this proof we suppress all the notions of pullbacks of bundles. One can
construct $\mathcal{Y}$ from $\mathcal{X}_{k}$ as a sequence of projective
bundles. First, over $\mathcal{X}_{k}$ there is a bundle
$\mathcal{U}\to\mathcal{X}_{k}$ of rank $n-k-1$ whose fiber over the flag
$(V_{0}\subset\cdots\subset V_{k+1})$ is the vector space
$\mathbb{C}^{n}/V_{k+1}$. The projective bundle $\mathbb{P}(\mathcal{U})$ is
the first bundle in the sequence. It gives the direction of the extra
dimension of $V_{k+2}$ over the flag $(V_{0}\subset\cdots\subset V_{k+1})$.
The tautological bundle $\mathcal{U}_{1}$ of $\mathbb{P}(\mathcal{U})$ pulls
back to the line bundle $\mathcal{L}_{k+2}$ on $\mathcal{Y}$.

Next, over $\mathbb{P}(\mathcal{U})$, we have the bundle
$\mathcal{U}/\mathcal{U}_{1}$ of rank $n-k-2$, and we construct the second
projective bundle
$\mathbb{P}(\mathcal{U}/\mathcal{U}_{1})\to\mathbb{P}(\mathcal{U})$. The
tautological bundle of $\mathbb{P}(\mathcal{U}/\mathcal{U}_{1})$ is of the
form $\mathcal{U}_{2}/\mathcal{U}_{1}$ for some vector bundle
$\mathcal{U}_{2}$ of rank $2$ and
$\mathcal{U}_{1}\subset\mathcal{U}_{2}\subset\mathcal{U}$ as bundles over
$\mathbb{P}(\mathcal{U}/\mathcal{U}_{1})$. Moreover, the tautological bundle
of $\mathbb{P}(\mathcal{U}/\mathcal{U}_{1})$ pulls back to $\mathcal{L}_{k+3}$
on $\mathcal{Y}$. One can continue this process and construct
$\mathbb{P}(\mathcal{U}/\mathcal{U}_{2})$ as a projective bundle over
$\mathbb{P}(\mathcal{U}/\mathcal{U}_{1})$, with tautological bundle of the
form $\mathcal{U}_{3}/\mathcal{U}_{2}$, and so on. At the end, one arrive at
the space $\mathbb{P}(\mathcal{U}/\mathcal{U}_{n-k-1})$ which is isomorphic to
the Hessenberg variety $\mathcal{Y}$. Therefore, by the formula 6.1 and the
fact that the tautological bundle of $\mathbb{P}(\mathcal{U}/\mathcal{U}_{j})$
pulls back to the line bundle $\mathcal{L}_{k+j+1}$ on $\mathcal{Y}$, we
obtain the conclusion that $H^{*}(\mathcal{Y})$ is generated by
$x_{k+2},\cdots,x_{n}$ over $H^{*}(\mathcal{X}_{k})$.

The rest of the proof is also very similar to the proof of [Fulton,
Proposition 10.2.3]. The reason for $e_{i}(x_{k+2},\cdots,x_{n})=0$ is because
it is the $i$-th Chern class of the trivial bundle $\mathcal{V}_{n}$ defined
above; and the isomorphism in the proposition is based on the same algebraic
fact stated on [Fulton, p.163]. ∎

The dot action on $H^{*}(\mathcal{X}_{k})$ was described in Section 5.
Motivated by the fact that the dot action acts trivially on the usual
cohomology of the flag variety, it is natural to conjecture that the action on
the basis classes $x_{k+2}^{i_{k+2}}\cdots x_{n}^{i_{n}}$ are trivial. While
we can not prove this result for the classes, we can prove the result on the
isomorphism level with an argument using characteristics series of the
representation. The author would like to thank John Shareshian for bringing
the article [HNY] to his attention.

######  Proposition 6.2.

The dot representation on $\mathcal{Y}$ is isomorphic to the representation on

$H^{*}(\mathcal{X}_{k})[X_{k+2},\cdots,X_{n}]/(e_{1}(X_{k+2},\cdots,X_{n}),\cdots,e_{n-k}(X_{k+2},\cdots,X_{n}))$

which acts on $H^{*}(\mathcal{X}_{k})$ as described in Section 5, and acts
trivially on the $H^{*}(\mathcal{X}_{k})$-basis $x_{k+2}^{i_{k+2}}\cdots
x_{n}^{i_{n}}$, $0\leq i_{j}\leq n-j$, where the $x_{i}$’s are the images of
the $X_{i}$’s under the quotient map.

###### Proof.

We have the recursive relation for the characteristic series of
$\mathcal{X}_{k}$ (5.3), with some terms rearranged:

(6.2)
$A_{n-1,k}(t)=h_{n}[n]_{t}+\sum_{i=0}^{k-1}t[n-i-2]_{t}A_{i}(t)h_{n-1-i}.$

Suppose that $S_{n}$ acts trivially on $X_{k+2}^{i_{k+2}}\cdots
X_{n}^{i_{n}}$, $0\leq i_{j}\leq n-j$, then on the characteristic series, what
happen would be a degree shifting for each basis element, i.e. multiplying by
a power of $t$. For $X_{k+2}^{i_{k+2}}$, $0\leq i_{k+2}\leq n-k-2$, it
corresponds to multiplying $1+t+\cdots+t^{n-k-2}=[n-k-1]_{t}$. Finally, the
effect of taking the basis $X_{k+1}^{i_{k+2}}\cdots X_{n}^{i_{n}}$, $0\leq
i_{j}\leq n-j$ in to consideration on the characteristic series, assuming
$S_{n}$ acts trivially on them, would be multiplying by

$[n-k-1]_{t}[n-k-2]_{t}\cdots[1]_{t}=[n-k-1]_{t}!$

The incomparability graph of the Hessenberg function $h_{k}$ is the lollipop
graph $L_{n-k,k}$. For the chromatic quasisymmetric function $X_{Ln-k,k}({\bf
x},t)$, there is a recursive formula [HNY, Proposition 4.4]

(6.3) $X_{L_{n-k,k}}({\bf
x},t)=[n-k-1]_{t}!\left([n]_{t}e_{n}+\sum_{i=0}^{k-1}t[n-k+i-1]_{t}X_{P_{k-i}}({\bf
x},t)e_{n-k+i}\right)$

Setting $i^{\prime}=k-1-i$ in the sum, we have

(6.4) $X_{L_{n-k,k}}({\bf
x},t)=[n-k-1]_{t}!\left([n]_{t}e_{n}+\sum_{i^{\prime}=0}^{k-1}t[n-i^{\prime}-2]_{t}X_{P_{i^{\prime}+1}}({\bf
x},t)e_{n-1-i^{\prime}}\right)$

Notice that the incomparability graph of the Hessenberg function $h_{+}$ for
$\mathcal{X}^{i}$ is $P_{i+1}$, and it is known that $\omega X_{P_{i+1}}({\bf
x},t)=A_{i}(t)$, where $\omega$ is the involution on the ring of symmetric
functions. By definition, $\omega e_{i}=h_{i}$, therefore, comparing the right
sides of (6.2) and (6.4), we obtain the following.

$\omega X_{L_{n-k,k}}({\bf x},t)=[n-k-1]_{t}!A_{n-1,k}(t)$

Finally, by the theorem that was originally conjectured by Shareshian and
Wachs [SW, Conjecture 1.4] and proved independently by Brosnan and Chow [BC],
and by Guay-Paquet [GP], we have the identity.

$\sum_{j=0}^{\dim(\mathcal{Y})}\operatorname{ch}H^{2j}(\mathcal{Y})t^{j}=\omega
X_{L_{n-k,k}}({\bf x},t)$

Therefore we can conclude that
$\sum_{j=0}^{\dim(\mathcal{Y})}\operatorname{ch}H^{2j}(\mathcal{Y})t^{j}=[n-k-1]_{t}!A_{n-1,k}(t)$.
Since the representation is determined by the characteristic series up to
isomorphism, this concludes the proof. ∎

We remark that the proposition also implies the dot representation is a
permutation representation on $H^{*}(\mathcal{Y})$.

## References