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\newcommand{\Hilb}{{\mathrm{Hilb}}}

\newcommand{\rev}{{\mathrm{rev}}}

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\newcommand{\Frob}{{\mathrm{Frob}}}
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\title[Equiv. Cohomology of Grass. Spanning Lines]{Equivariant Cohomology of Grassmannian Spanning Lines}

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\author[R. Chou, T. Matsumura, B. Rhoades]{\texorpdfstring{
Raymond Chou\thanks{\href{mailto:r2chou@ucsd.edu}{r2chou@ucsd.edu}}\addressmark{1}, 
Tomoo Matsumura\thanks{\href{tomoo.matsumura@gmail.com}{matsumura.tomoo@icu.ac.jp} TM was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 20K03571 and (B) 23K25772.}\addressmark{2},
\and
Brendon Rhoades\thanks{\href{bprhoades@ucsd.edu}{bprhoades@ucsd.edu}. BR was partially supported by NSF Grant DMS-2246846.}\addressmark{3} }{
Raymond Chou, 
Tomoo Matsumura,
and
Brendon Rhoades }}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California San Diego, San Diego, CA\\ 
\addressmark{2} International Christian University, Tokyo, Japan\\ 
\addressmark{2}Department of Mathematics, University of California San Diego, San Diego, CA}

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\abstract{For positive integers $d \leq k \leq n$, let $X_{n,k,d}$ be the moduli space of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\CC^k$ such that $\ell_1 + \cdots + \ell_n$ has vector space dimension $d$. The space $X_{n,k,d}$ carries an action of the rank $k$ torus $T = (\CC^*)^k$, and we present the $T$-equivariant cohomology of $X_{n,k,d}$. This solves a problem of Pawlowski and Rhoades. Our methods feature the orbit harmonics technique of combinatorial deformation theory and suggest a relationship between orbit harmonics and equivariant cohomology.}

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\keywords{equivariant cohomology, orbit harmonics, grassmannian line configurations}

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\section{Introduction}


Let $X$ be a topological space carrying an action of a complex torus $T$. The {\em $T$-equivariant cohomology ring} $H^*_T(X)$ is an enhancement of the ordinary cohomology $H^*(X)$ of $X$ which accounts for the action of $T$.\footnote{For simplicity, we use complex coefficients for cohomology throughout this extended abstract, but our main results hold over any coefficient ring.} The map $X \to \{\pt\}$ endows $H^*_T(X)$ with the structure of a $\CC[\tttt]$-module, where $\tttt$ is the Lie algebra of $T$. Under mild conditions, the ordinary cohomology $H^*(X)$ may be recovered from $H^*_T(X)$ by the relation
\begin{equation}
\label{eq:ordinary-equivariant-relation}
    H^*(X) = \CC \otimes_{\CC[\tttt]} H^*_T(X)
\end{equation}
where the generators of the polynomial ring $\CC[\tttt]$ act by zero on $\CC$. 

A {\em line} in $\CC^k$ is a 1-dimensional subspace $\ell \subseteq \CC^k$. We compute the equivariant cohomology of the following moduli space of line configurations.

\begin{defn}
    \label{def:x-variety}
    Let $d \leq k \leq n$ be positive integers. Let $X_{n,k,d}$ be the set of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\CC^k$ such that the vector space sum $\ell_1 + \cdots + \ell_n$ has  dimension $d$.
\end{defn}

Writing $\PP^{k-1}$ for the complex projective space of lines in $\CC^k$, we have a natural inclusion $X_{n,k,d} \subseteq (\PP^{k-1})^n$ which gives $X_{n,k,d}$ the structure of an algebraic variety.  The variety $X_{n,k,d}$ carries an action of the rank $k$ torus $T = (\CC^*)^k$ given by 
\begin{equation}
    t \cdot (\ell_1, \dots, \ell_n) := (t \cdot \ell_1, \dots, t \cdot \ell_n).
\end{equation}Special cases of $X_{n,k,d}$ have been considered before.

\begin{itemize}
    \item If $n = k = d$, the variety $X_{n,n,n}$ consists of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\CC^n$ which span $\CC^n$. This space is homotopy equivalent to the type A$_{n-1}$ complete flag variety $\mathrm{Fl}_n$.\footnote{The homotopy equivalence is the natural projection $G/T \twoheadrightarrow G/B$. This map is a fiber bundle with contractible fiber $\cong U$.} The geometry of $\mathrm{Fl}_n$ is governed by the combinatorics of permutations in the symmetric group $\symm_n$.
    \item If $k = d$, the variety $X_{n,k} := X_{n,k,k}$ consists of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\CC^k$ which span $\CC^k$. This variety of {\em spanning line configurations} was introduced by Pawlowski and Rhoades \cite{PR}. The geometry of $X_{n,k}$ is governed by the combinatorics of {\em Fubini words} in $\WWW_{n,k}$; these are surjective functions $w: [n] \twoheadrightarrow [k]$. Billey and Ryan \cite{BR}  gave combinatorial descriptions of the corresponding Bruhat order(s) on $\WWW_{n,k}$. 
\end{itemize}

Pawlowski and Rhoades proved \cite{PR} that the ordinary cohomology of $X_{n,k}$ is presented by the following quotient $R_{n,k}$ of $\CC[\xxx_n]$:
\begin{equation}
    H^*(X_{n,k}) = R_{n,k} := \CC[\xxx_n] / (x_1^k, \dots, x_n^k, e_n(\xxx_n), e_{n-1}(\xxx_n), \dots, e_{n-k+1}(\xxx_n)).
\end{equation}
Here $e_d(\xxx_n)$ is the degree $d$ elementary symmetric polynomial in the variable set $\xxx_n := \{x_1, \dots, x_n\}$. The ring $R_{n,k}$ was introduced by Haglund, Rhoades, and Shimozono \cite{HRS} to give an analogue of the coinvariant algebra in the context of the Haglund--Remmel--Wilson {\em delta conjecture} \cite{HRW}. Pawlowski and Rhoades asked \cite[Problem 9.8]{PR} for a presentation of the $T$-equivariant cohomology $H^*_T(X_{n,k})$; we solve the more general problem of presenting $H^*_T(X_{n,k,d})$.

To state our main result we need some notation. Write $\Gr(d,\CC^k)$ for the Grassmannian of $d$-dimensional subspaces $V \subseteq \CC^k$. 
 We have a surjection
\begin{equation}
    p: X_{n,k,d} \twoheadrightarrow \Gr(d,\CC^k)
\end{equation}
which sends an $n$-tuple $(\ell_1,\dots,\ell_n)$ of lines to $\ell_1 + \cdots + \ell_n$. The rank $k$ torus $T = (\CC^*)^k$ acts naturally on $\Gr(d,\CC^k)$, and the map $p$ is $T$-equivariant. We write $\VVV_d$ for the rank $d$ tautological vector bundle over $\Gr(d,\CC_k)$ whose fiber over a point $V \in \Gr(d,\CC^k)$ is the vector space $V$. The pullback $p^*(\VVV_d)$ is a vector bundle over $X_{n,k,d}$.

In addition to the length $n$ list $\xxx_n = \{x_1, \dots, x_n\}$ of $x$-variables, we consider a length $d$ list $\yyy_d = \{y_1, \dots, y_d\}$ of $y$-variables and a length $k$ list $\ttt_k = \{t_1, \dots, t_k\}$ of $t$-variables. Let $\CC[\xxx_n,\yyy_d,\ttt_k]$ be the rank $n+d+k$ polynomial ring over all of these variables. The symmetric group $\symm_d$ acts on the middle set of variables in $\CC[\xxx_n,\yyy_d,\ttt_k]$; we write $\CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}$ for the associated invariant subring. Our presentation of the $T$-equivariant cohomology of $X_{n,k,d}$ reads as follows, where $h_i$ stands for the complete homogeneous symmetric polynomial of degree $i$.


\begin{theorem}
    \label{thm:borel-presentation}
    For positive integers $n \geq k \geq d$, let $I_{n,k,d} \subseteq \CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}$ be the ideal generated by 
    \begin{itemize}
        \item $e_r(\ttt_k) - e_{r-1}(\ttt_k) h_1(\yyy_d) + \cdots + (-1)^r h_r(\yyy_d)$ for $r > k-d$, 
        \item $e_r(\xxx_n) - e_{r-1}(\xxx_n) h_1(\yyy_d) + \cdots + (-1)^r h_r(\yyy_d)$ for $r > n-d$, and
        \item $x_i^d - x_i^{d-1} e_1(\yyy_d) + \cdots + (-1)^d e_d(\yyy_d)$ for $i = 1, \dots, n$.
    \end{itemize}
    The $T$-equivariant cohomology of $X_{n,k,d}$ has presentation
    \begin{equation}
        H^*_T(X_{n,k,d}) = \CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}/I_{n,k,d}
    \end{equation}
    where 
    \begin{itemize}
        \item $x_i$ represents the first equivariant Chern class of the line bundle $\LLL_i$ over $X_{n,k,d}$ with fiber $\ell_i$ over $(\ell_1, \dots, \ell_n)$,
        \item $y_1, \dots, y_d$ represent equivariant Chern roots of $p^*(\VVV_d)$, and
        \item $t_1, \dots, t_n$ are the images in $H^*_T(X_{n,k,d})$ of the standard generators of $H^*_T(\mathrm{pt})$.
    \end{itemize}
\end{theorem}

%We have an identification $\CC[\ttt_k] = \CC[\tttt]$ where $\tttt \cong \CC^k$ is the Lie algebra of $T = (\CC^*)^k$. It will develop that $H^*_T(X_{n,k,d})$ is a free $\CC[\tttt]$-module of rank equal to the number of functions $w: [n] \to [k]$ whose image has size $d$.

When $k = d$, the variety $X_{n,k,d}$ specializes to the variety $X_{n,k}$ of spanning line configurations in $\CC^k$. Specializing \Cref{thm:borel-presentation} to $k = d$, our solution to the problem \cite[Problem 9.8]{PR} of Pawlowski and Rhoades is as follows.

\begin{corollary}
    \label{cor:PR}
    For positive integers $n \geq k$, the $T$-equivariant cohomology of $X_{n,k}$ has quotient presentation
    \begin{equation}
        H^*_T(X_{n,k}) = \CC[\xxx_n,\yyy_k]/I_{n,k}
    \end{equation}
    where $I_{n,k} \subseteq \CC[\xxx_n,\yyy_k]$ is the ideal generated by 
    \begin{itemize}
        \item $e_r(\xxx_n) - e_{r-1}(\xxx_n) h_1(\ttt_k) + \cdots + (-1)^r h_r(\ttt_k)$ for $r > n-k$, and
        \item $x_i^k - x_i^{k-1} e_1(\ttt_k) + \cdots + (-1)^k e_k(\ttt_k)$ for $i = 1, \dots, n$.
    \end{itemize}
\end{corollary}

The proof of \Cref{thm:borel-presentation} has two main parts: geometric and algebraic. Geometrically, we show that the relations in $I_{n,k,d}$ hold in the ring $H^*_T(X_{n,k,d})$ using the Whitney Sum Formula and calculate the rank of $H^*_T(X_{n,k,d})$ as a free $\CC[\tttt]$-module using a $T$-stable affine paving of $X_{n,k,d}$. Algebraically, we show that $\CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}/I_{n,k,d}$ is a free $\CC[\ttt_k]$-module of the appropriate rank. For this, we apply a technique in combinatorial deformation theory called {\em orbit harmonics}.

The remainder of this extended abstract is organized as follows. In \Cref{sec:Background} we give background on equivariant cohomology, affine pavings, and orbit harmonics. In \Cref{sec:Connection} we describe our general approach to equivariant cohomology via orbit harmonics and study the permutohedral variety from this point of view. In \Cref{sec:Grassmannian} we sketch the proof of \Cref{thm:borel-presentation}.




\section{Background}
\label{sec:Background}

\subsection{Equivariant Cohomology}


Let $X$ be a smooth complex variety equipped with an action of the rank $k$ torus $T = (\CC^*)^k$. As mentioned in the introduction, the equivariant cohomology ring $H^*_T(X)$ is an enhancement of the ordinary cohomology $H^*(X)$. We describe the basic features of $H^*_T(X)$, referring the reader to Anderson and Fulton's book \cite{AndersonFultonBook} for a detailed treatment.

Let $ET$ be a contractible space with a free left action of $T$. We define $ET \times_T X := ET \times X/\sim$ where $(e,x) \sim (e \cdot t^{-1}, t \cdot x)$ for all $e \in ET$, $x \in X$, and $t \in T$. The {\em equivariant cohomology ring} of $X$ is defined by
\begin{equation}
    H^*_T(X) := H^*(ET \times_T X)
\end{equation}
where $H^*(ET \times_T X)$ is the usual singular cohomology. For example, if $X = \{\pt\}$ is a single point, we may identify $ET \times_T \{ \pt \} = (\PP^{\infty})^k$, where $\PP^\infty$ is infinite-dimensional complex projective space. Since $H^*((\PP^\infty)^k) = \CC[\ttt_k]$, the map $X \rightarrow \{\pt\}$ gives the ring $H^*_T(X)$  the structure of a $\CC[\ttt_k]$-module. 

Let $Y \subseteq X$ be a closed subvariety which is closed under the $T$-action. If $Y$ has codimension $c$ in $X$, we have a class $[Y] \in H^*_T(X)$ of cohomological degree $2c$.  An {\em affine paving} of $X$ is a filtration
\begin{equation}
    \varnothing = X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots \subseteq X_m = X
\end{equation}
of $X$ by closed subvarieties such that each difference $X_i \smallsetminus X_{i-1}$ is isomorphic to a disjoint union of affine spaces (possibly of varying dimensions). These affine spaces are referred to as {\em cells}, and the affine paving is {\em $T$-invariant} if the cells are closed under the action of $T$. If $X$ has a $T$-invariant affine paving, the equivariant cohomology ring $H^*_T(X)$ is a free $\CC[\ttt_k]$-module with basis given by the classes $[\overline{C}]$ of the closures $\overline{C}$ of these cells.

Let $\EEE \twoheadrightarrow X$ be a $T$-equivariant vector bundle of rank $r$. For $1 \leq i \leq r$ we have the {\em equivariant Chern class} $c_i^T(\EEE) \in H^{2i}_T(X)$. The {\em total equivariant Chern class} is the sum $c^T(\EEE) := 1 + c_1^T(\EEE) + \cdots + c_r^T(\EEE)$. If we have a short exact sequence
\begin{equation}
    0 \to \EEE' \to \EEE \to \EEE'' \to 0
\end{equation}
of $T$-equivariant bundles, there holds the relation $c^T(\EEE) = c^T(\EEE') \cdot c^T(\EEE'')$. It follows that if $\EEE$ splits as a direct sum of equivariant line bundles $\EEE = \LLL_1 \oplus \cdots \oplus \LLL_r$, we have the factorization
\begin{equation}
    c^T(\EEE) = c^T(\LLL_1) \cdots c^T(\LLL_r) = (1 + c^T_1(\LLL_1)) \cdots (1 + c^T_1(\LLL_r)).
\end{equation}
Even if $\EEE$ does not split as a direct sum of equivariant line bundles, by choosing an appropriate flag extension $X' \twoheadrightarrow X$ we may still factor $c^T(\EEE) = (1 + \alpha_1) \cdots (1 + \alpha_r)$ where $\alpha_1, \dots, \alpha_r \in H^2_T(X')$. The classes $\alpha_1, \dots, \alpha_r$ are the {\em equivariant Chern roots} of $\EEE$; any symmetric polynomial in $\alpha_1, \dots, \alpha_r$ lies in $H^*_T(X)$. See \cite[Section 2.3]{AndersonFultonBook} for more information on these facts.




%Computing equivariant cohomology is a central theme in Schubert Calculus, [references]

\subsection{Orbit Harmonics}

Let $\Zpoints \subseteq \CC^n$ be a finite locus of points in affine $n$-space $\CC^n$. We have the vanishing ideal
\begin{equation}
    \II(\Zpoints) := \{ f \in \CC[\xxx_n] \,:\, f(\zzz) = 0 \text{ for all $\zzz \in \Zpoints$} \} \subseteq \CC[\xxx_n].
\end{equation}
If $I \subseteq \CC[\xxx_n]$ is an ideal, recall that the {\em associated graded ideal} $\gr \, I \subseteq \CC[\xxx_n]$ is the homogeneous ideal
\begin{equation}
    \gr \, I := ( \tau(f) \,:\, f \in I, \, f \neq 0) \subseteq \CC[\xxx_n]
\end{equation}
where $\tau(f)$ is the top-degree homogeneous component of a nonzero polynomial $f$.
The {\em orbit harmonics} deformation associates to $\Zpoints$ the graded quotient ring $\CC[\xxx_n]/ \gr \, \II(\Zpoints)$ where $\gr \, \II(\Zpoints) \subseteq \CC[\xxx_n]$. 
 We have a vector space isomorphism
\begin{equation}
    \label{eq:orbit-harmonics-isomorphism}
    \CC[\Zpoints] := \CC[\xxx_n]/\II(\Zpoints) \cong_\CC \CC[\xxx_n]/\gr \, \II(\Zpoints)
\end{equation}
where $\CC[\xxx_n]/\gr \, \II(\Zpoints)$ is a graded vector space. If the locus $\Zpoints$ is a stable under the action of a finite matrix group $G \subseteq GL_n(\CC)$, \eqref{eq:orbit-harmonics-isomorphism} is an isomorphism of $G$-modules, where $\CC[\xxx_n]/\gr \, \II(\Zpoints)$ is a graded $G$-module.


In geometric terms, orbit harmonics is a flat family which linearly deforms the reduced locus $\Zpoints \subseteq \CC^n$ to a subscheme of degree $\# \Zpoints$ supported at the origin. This deformation is shown schematically below in the case of a locus of size $|\Zpoints| = 6$ in $\CC^2$ carrying an action of $G \cong \symm_3$ via reflection in the three displayed lines. Orbit harmonics quotients $\CC[\xxx_n]/\gr \, \II(\Zpoints)$ have been applied to Donaldson--Thomas theory \cite{RRT}, increasing subsequence combinatorics \cite{RhoadesIncreasing}, Ehrhart theory \cite{RR}, and (importantly for us) cohomology presentation.

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\section{Equivariant Cohomology and Orbit Harmonics}
\label{sec:Connection}


Let $X$ be a variety equipped with an action of a rank $k$ torus $T \cong (\CC^*)^k$, and let $\tttt \cong \CC^k$ be the Lie algebra of $T$. In many combinatorially interesting situations \cite{GP, Griffin, GLW, HRS, PR, RhoadesSpanning}, there is a nonempty Zariski-open subset $U \subseteq \tttt$ such that 
\begin{itemize}
    \item for each $\aalpha \in U$, we have a finite locus $\Zpoints(\aalpha) \subseteq \CC^n$ depending on $\aalpha$,
    \item the homogeneous ideal $\gr \, \II(\Zpoints(\aalpha)) \subseteq \CC[\xxx_n]$ does not depend on $\aalpha$, and
    \item the ordinary cohomology ring $H^*(X)$ has presentation
    \begin{equation}
        H^*(X) = \CC[\xxx_n]/\gr \, \II(\Zpoints(\aalpha)).
    \end{equation}
\end{itemize}
The loci $\Zpoints(\alpha)$ may be put into a family $\Zpoints \subseteq \CC^n \times \tttt$ given by
\begin{equation}
    \Zpoints := \text{Zariski closure of } \bigcup_{\aalpha \in U} \Zpoints(\aalpha) \times \{\aalpha\} \text{ in } \CC^n \times \tttt.
\end{equation}
We have the vanishing ideal $\II(\Zpoints) \subseteq \CC[\CC^n \times \tttt] \cong \CC[\xxx_n, \ttt_k]$. It turns out that the equivariant cohomology of $X$ often has quotient presentation
\begin{equation}
\label{eq:orbit-harmonics-prediction}
    H^*_T(X) = \CC[\xxx_n,\ttt_k]/\II(\Zpoints)
\end{equation}
where the $t$-variables come from the torus action. The formula $H^*(X) = \CC \otimes_{\CC[\tttt]} H^*_T(X)$ may be interpreted in terms of \eqref{eq:orbit-harmonics-prediction} as taking the scheme-theoretic fiber $\pi^{-1}(0)$ where $\pi: \Zpoints \to \tttt$ is the natural projection. Going in the other direction, orbit harmonics can sometimes {\em predict} equivariant cohomology rings, a theme we explore. We give two examples of this phenomenon (one old and one new) involving previously studied varieties before turning to the new variety $X_{n,k,d}$.



\subsection{Type A Springer Fibers}

Recall that the complete flag variety $\mathrm{Fl}_n$ is the moduli space
\begin{equation}
    \mathrm{Fl}_n = \{ V_\bullet = (0 = V_0 \subset V_1 \subset \cdots \subset V_n = \CC^n) \,:\, \dim V_i = i \}
\end{equation}
of maximal chains of nested subspaces of $\CC^n$. The rank $n$ torus $(\CC^*)^n \subseteq GL_n(\CC)$ of diagonal matrices acts naturally on $\CC^n$, and induces an action on $\mathrm{Fl}_n$.

Let $\mu = (\mu_1, \dots, \mu_k) \vdash n$ be a partition of $n$ with $k$ parts and let $X: \CC^n \to \CC^n$ be a nilpotent matrix of Jordan type $\mu$. The {\em Springer fiber} is the closed subvariety
\begin{equation}
    \BBB_\mu := \{ V_\bullet \in \mathrm{Fl}_n \,:\, X V_i \subseteq V_i \text{ for all $i$} \}
\end{equation}
of $\mathrm{Fl}_n$. The Springer fiber $\BBB_\mu$ is stable under the action of the rank $k$ subtorus $T \subseteq (\CC^*)^n$ consisting of matrices of the form $\lambda_1 I_{\mu_1} \oplus \cdots \oplus \lambda_k I_{\mu_k}$ for $\lambda_1, \dots, \lambda_k \in \CC^*$. This gives rise to identifications $T = (\CC^*)^k$ and $\tttt = \CC^k$.

Let $U \subseteq \tttt = \CC^k$ be the Zariski-open set
\begin{equation}
    U := \{ \aalpha = (\alpha_1, \dots, \alpha_k) \in \CC^k \,:\, \text{$\alpha_1, \dots, \alpha_k$ are distinct} \}.
\end{equation}
For any $\aalpha = (\alpha_1, \dots, \alpha_k) \in U$, we have a locus $\Zpoints(\aalpha) \subset \CC^n$ given by
\begin{equation}
    \Zpoints(\aalpha) := \{ (z_1, \dots, z_n) \in \CC^n \,:\, \text{$\alpha_i$ appears $\mu_i$ times among $z_1, \dots, z_n$} \}.
\end{equation}
For example, if $\mu = (2,1)$ we have $\Zpoints(\alpha) = \{(\alpha_1, \alpha_1, \alpha_2), (\alpha_1, \alpha_2, \alpha_1), (\alpha_2, \alpha_1, \alpha_1) \}.$ Garsia and Procesi proved \cite{GP} that the ideal $\gr \, \II(\Zpoints(\aalpha)) \subseteq \CC[\xxx_n]$ does not depend on $\aalpha \in U$. Combining their result with the presentation of $H^*(\BBB_\mu)$ given by Hotta and Springer \cite{HS}, we arrive at the identification
\begin{equation}
    H^*(\BBB_\mu) = \CC[\xxx_n]/\gr \, \II(\Zpoints(\aalpha)) \text{ for any $\aalpha \in U$.}
\end{equation}
Let $\Zpoints \subseteq \CC^n \times \tttt = \CC^{n+k}$ be the Zariski closure of $\bigcup_{\aalpha \in U} \Zpoints(\aalpha) \times \{\aalpha\}$ inside $\CC^{n+k}$. We have $\II(\Zpoints) \subseteq \CC[\xxx_n,\ttt_k]$. Kumar and Procesi derived \cite{KP} the presentation
\begin{equation}
    H^*_T(\BBB_\mu) = \CC[\xxx_n,\ttt_k]/\II(\Zpoints)
\end{equation}
of the equivariant cohomology of $\BBB_\mu$.

\subsection{Permutohedral variety}

Let $S: \CC^n \to \CC^n$ be a diagonal matrix with distinct entries. The {\em permutohedral variety} $\Perm_n$ is the closed subvariety of $\mathrm{Fl}_n$ given by
\begin{equation}
    \Perm_n := \{ V_\bullet \in \mathrm{Fl}_n \,:\, S V_i \subseteq V_{i+1} \text{ for $i = 1, 2, \dots, n-1$}\}.
\end{equation}
This is an important example of a {\em toric variety} and a {\em regular semisimple Hessenberg variety}. The variety $\Perm_n$ is stable under the action of the rank  $n-1$ torus $T \subset GL_n(\CC)$ given by
\begin{equation}
    T = \{ \mathrm{diag}(\lambda_1, \dots, \lambda_n) \,:\, \lambda_1 \cdots \lambda_n = 1 \}.
\end{equation} 
We describe how known presentations of the (equivariant) cohomology of $\Perm_n$ may be interpreted via orbit harmonics.

Let $\FFF$ be the family of $N := 2^n - 2$ nonempty and proper subsets $I \subset [n]$ and let $\CC^\FFF$ be the $N$-dimensional complex vector space with basis $\FFF$. Let $\xxx_\FFF := \{ x_I \,:\, I \in \FFF \}$ be a family of variables indexed by $\FFF$ and let $\CC[\xxx_\FFF] = \CC[ x_I \,:\, I \in \FFF]$ be the rank $N$ polynomial ring over these variables.

The Lie algebra $\tttt$ may be identified with the $(n-1)$-dimensional vector space
\begin{equation}
    \tttt = \{ \mathrm{diag}(\alpha_1, \dots, \alpha_n) \,:\, \alpha_1 + \cdots + \alpha_n = 0 \}
\end{equation} and we identify its coordinate ring as $\CC[\tttt] = \CC[\ttt_n]/(t_1 + \cdots + t_n)$. Let $U \subset \tttt$ be the Zariski-open subset of points $\aalpha = (\alpha_1, \dots, \alpha_n)$ with distinct coordinates. For  $\aalpha \in U$ and any permutation $w \in \symm_n$, we define a point $$p(w,\aalpha) = ( p(w,\aalpha)_I )_{I \in \FFF} \in \CC^\FFF$$ as follows. The permutation $w$ determines a flag $\varnothing = I_0(w) \subset I_1(w) \subset \cdots \subset I_n(w) = [n]$ of subsets of $[n]$ where $I_j(w) := \{w(1), \dots, w(j) \}$. For a subset $I \in \FFF$, we define the $I$-th component $p(w,\aalpha)_I$ of $p(w,\aalpha)$ by
\begin{equation}
    p(w,\aalpha)_I := \begin{cases}
        \alpha_{w(j)} - \alpha_{w(j+1)} & \text{if $I = I_j(w)$ for some $0 < j < n$}, \\
        0 & \text{otherwise.}
    \end{cases}
\end{equation}
We let $\Zpoints(\aalpha) \subseteq \CC^\FFF$ be the locus
\begin{equation}
    \Zpoints(\aalpha) := \{ p(w,\aalpha) \,:\, w \in \symm_n \}
\end{equation}
so that $\II(\Zpoints(\aalpha)) \subseteq \CC[\CC^\FFF] = \CC[\xxx_\FFF]$ for each $\aalpha \in U$. We also have the family
\begin{equation}
    \Zpoints := \text{Zariski closure of } \bigcup_{\aalpha \in U} \Zpoints(\aalpha) \times \{\aalpha\} \text{ in $\CC^\FFF \times \tttt$}.
\end{equation}
The ideal $\II(\Zpoints)$ is a subset of $\CC[\CC^\FFF \times \tttt] = \CC[\xxx_\FFF] \otimes \CC[\ttt_n]/(t_1 + \cdots + t_n)$.

\begin{theorem}
    \label{thm:permutohedral}
    The associated graded ideal $\gr \, \II(\Zpoints(\aalpha)) \subseteq \CC[\CC^\FFF]$ does not depend on $\aalpha \in U$. For any $\aalpha \in U$, the ordinary cohomology of $\Perm_n$ has presentation
    \begin{equation}
        H^*(\Perm_n) = \CC[\CC^\FFF]/ \gr \, \II(\Zpoints(\aalpha)).
    \end{equation}
    Furthermore, the $T$-equivariant cohomology of $\Perm_n$ has presentation
    \begin{equation}
        H^*_T(\Perm_n) = \CC[\CC^\FFF \times \tttt]/\II(\Zpoints).
    \end{equation}
\end{theorem}

\begin{proof}
    (Sketch) Danilov proved \cite{Danilov} that the cohomology of $\Perm_n$ has presentation
    \begin{equation}
        H^*(\Perm_n) = \CC[\xxx_\FFF]/J
    \end{equation}
    where $J \subseteq \CC[\xxx_\FFF]$ is the ideal generated by 
    \begin{itemize}
    \item
    all products $x_I \cdot x_{I'}$ for $I, I' \in \FFF$ where $I \not\subseteq I'$ and $I' \not\subseteq I$, and 
    \item all differences of the form
    $$\sum_{i \in I} x_I - \sum_{i+1 \in I'} x_{I'}$$
    for $i = 1, 2, \dots, n-1$.
    \end{itemize}
   % In fact, Danilov's proved a more general result which presents $H^*(X_P)$ where $X_P$ is the toric variety associated to any simple lattice polytope $P$.
    Let $\aalpha \in U$. To show $J \subseteq \gr \, \II(\Zpoints(\aalpha))$, we prove that each generator of $J$ is the top-degree homogeneous component of a polynomial in $\CC[\xxx_\FFF]$ which vanishes on $\Zpoints(\aalpha)$. If $I, I' \in \FFF$ satisfy $I \not\subseteq I'$ and $I' \not\subseteq I$, the product $x_I \cdot x_{I'}$ vanishes on $\Zpoints(\aalpha)$ since for any $w \in \symm_n$ the nonzero components of $p(w,\aalpha)$ are indexed by a flag of subsets of $[n]$. Furthermore, for each $i = 1, 2, \dots, n-1$ it can be checked that the polynomial
    \begin{equation}
        \left(\sum_{i \in I} x_I - \sum_{i+1 \in I'} x_{I'} \right) - (\alpha_i - \alpha_{i+1})
    \end{equation}
    vanishes on $\Zpoints(\aalpha)$, and the top degree component of this polynomial is the generator $\sum_{i \in I} x_I - \sum_{i+1 \in I'} x_{I'}$ of $J$. This proves that  $J \subseteq \gr \, \II(\Zpoints(\aalpha))$. We have a canonical surjection
    \begin{equation}
    \label{eq:surjection}
         H^*(\Perm_n) = \CC[\xxx_\FFF]/J \twoheadrightarrow  \CC[\xxx_\FFF]/\gr \, \II(\Zpoints(\aalpha)).
    \end{equation}
    The orbit harmonics isomorphism \eqref{eq:orbit-harmonics-isomorphism} implies that $\CC[\xxx_\FFF]/\gr \, \II(\Zpoints(\aalpha))$ has dimension $\# \Zpoints(\aalpha) = n!$. It is well-known that $H^*(\Perm_n)$ also has vector space dimension $n!$, which forces the surjection \eqref{eq:surjection} to be an isomorphism and $\gr \, \II(\Zpoints(\aalpha)) = J$ for any $\aalpha \in U$. 

    Let $L \subseteq \CC[\xxx_\FFF]$ be the ideal generated by all products $x_I \cdot x_{I'}$ for $I, I' \in \FFF$ for which $I \not\subseteq I'$ and $I' \not\subseteq I$. Bifet, De Concini, and Procesi proved \cite{BDP} that the equivariant cohomology ring of $\Perm_n$ has presentation
    \begin{equation}
        H^*_T(\Perm_n) = \CC[\xxx_\FFF]/L.
    \end{equation}
    We have a $\CC$-algebra homomorphism
    \begin{equation}
        \widetilde{\varphi}: \CC[\CC^\FFF \times \tttt] = \CC[\xxx_\FFF] \otimes \CC[\ttt_n]/(t_1 + \cdots + t_n) \longrightarrow \CC[\xxx_\FFF]/L
    \end{equation}
    characterized by $\widetilde{\varphi}: x_I \mapsto x_I$ for $I \in \FFF$ and $\widetilde{\varphi}: t_i - t_{i+1} \mapsto \sum_{i \in I} x_I - \sum_{i+1 \in I'} x_I$ for $i = 1, 2, \dots, n-1$. It can be checked that $\II(\Zpoints) \subseteq \mathrm{Ker}(\widetilde{\varphi})$, so we have an induced homomorphism
    \begin{equation}
        \varphi: \CC[\CC^\FFF \times \tttt]/\II(\Zpoints) \longrightarrow \CC[\xxx_\FFF]/L. 
    \end{equation}
    On the other hand, since $L \subseteq \II(\Zpoints)$ we have another $\CC$-algebra homomorphism
    \begin{equation}
        \psi: \CC[\xxx_\FFF]/L \longrightarrow \CC[\CC^\FFF \times \tttt]/\II(\Zpoints)
    \end{equation}
    characterized by $\psi: x_I \mapsto x_I$ for $I \in \FFF$. It is not hard to see that $\varphi$ and $\psi$ are mutually inverse.
\end{proof}

\section{Grassmannian Line Configurations}
\label{sec:Grassmannian}

%In this section, we will show that $I_{n,k,d}$ is the vanishing ideal of a certain point locus $\Zpoints_{n,k,d}(\aalpha_k)$. We use this to show that the quotient...

%\subsection{Point loci and orbit harmonics} 

%We begin our study of the equivariant cohomology of $X_{n,k,d}$ by calculating the orbit harmonics quotient ring of a strategically chosen finite point locus. This locus carries an action of $\symm_d$, and taking $\symm_d$-invariants will yield the ordinary cohomology of $X_{n,k,d}$.

This section outlines the main ideas used to prove \Cref{thm:borel-presentation}. Recall that we aim to show  $H^*_T(X_{n,k,d}) = \CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}/I_{n,k,d}$ where $I_{n,k,d}$ has generators as described in that theorem. We define
\begin{equation}
    \WWW_{n,k,d} := \{ w: [n] \to [k] \,:\, \text{the image of $w$ has size $d$} \}.
\end{equation}
It is not difficult to see that $\WWW_{n,k,d}$ is counted by
$\# \WWW_{n,k,d} = \frac{k!}{(k-d)!} \cdot \Stir(n,d)$
where $\Stir(n,d)$ is the {\em Stirling number of the second kind} counting $d$-block set partitions of $[n]$. 

\begin{lemma}
    \label{lem:free-module}
    The quotient ring $\CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}/I_{n,k,d}$ is a free $\CC[\ttt_k]$-module of rank equal to $\# \WWW_{n,k,d}$.
\end{lemma}

\Cref{lem:free-module} is established using orbit harmonics arguments related to those in \Cref{sec:Connection}; see \cite{CMR} for details.
With this result in hand, the cohomology presentation in \Cref{thm:borel-presentation} is established as follows.

\begin{proof} (of \Cref{thm:borel-presentation}, sketch) 
The map $p: X_{n,k,d} \to \Gr(d,\CC^k)$ given by $p: (\ell_1, \dots, \ell_n) \mapsto \ell_1 + \cdots + \ell_n$ is a fiber bundle with fiber isomorphic to $X_{n,d}$. The Leray--Hirsch Theorem and results of Pawlowski--Rhoades \cite{PR} may be used to show that $H^*_T(X_{n,k,d})$ is generated by
\begin{itemize}
    \item the equivariant Chern classes $c_1^T(\LLL_i)$ where $\LLL_i \twoheadrightarrow X_{n,k,d}$ has fiber $\ell_i$ over $(\ell_1, \dots, \ell_n)$,
    \item the equivariant Chern classes $c_i^T(p^*(\VVV_d))$ where $\VVV_d \twoheadrightarrow \Gr(d,\CC^k)$ is the vector bundle with fiber $V$ over $V \in \Gr(d,\CC^k)$ and $i = 1, 2, \dots, d$, and
    \item the standard generators $t_1, \dots, t_k$ of $H^*_T(\pt)$.
\end{itemize}
We therefore have a surjective $\CC[\ttt_k]$-algebra homomorphism
\begin{equation}
    \widetilde{\varphi}: \CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d} \twoheadrightarrow H^*_T(X_{n,k,d})
\end{equation}
characterized by
$\widetilde{\varphi}: x_i \mapsto c_1^T(\LLL_i)$ and $\widetilde{\varphi}: e_i(\yyy_d) \mapsto c_i^T(p^*(\VVV_d))$.

We show that each generator of $I_{n,k,d}$ lies in the kernel of $\widetilde{\varphi}$. Let $\CC^k$ be the trivial rank $k$ bundle over a variety with the standard action of $T = (\CC^*)^k$ on each fiber. The bundle $\CC^k$ has total Chern class $c^T(\CC^k) = (1 + t_1) \cdots (1 + t_k)$ over any variety. We have a short exact sequence
\begin{equation}
    0 \to \VVV_d \to \CC^k \to \CC^k/\VVV_d \to 0
\end{equation}
of bundles over $\Gr(d,\CC^k)$. Since $\CC^k/\VVV_d$ has rank $k-d$, we have $c_r^T(\CC^k/\VVV_d) = 0$ for $r > k-d$. Since $c^T(\CC^k) = c^T(\VVV_d) \cdot c^T(\CC^k/\VVV_d)$ we have 
\begin{center}
$\sum_{a+b = r}(-1)^a e_a(\ttt_k) h_b(\yyy_d) \in \mathrm{Ker}(\widetilde{\varphi})$ whenever $r > k-d$.
\end{center}
Similarly, we have a short exact sequence
\begin{equation}
    0 \to \KKK_{n-d} \to \LLL_1 \oplus \cdots \oplus \LLL_n \xrightarrow{ \, \psi \, } p^*(\VVV_d) \to 0
\end{equation}
of bundles over $X_{n,k,d}$ where $\psi$ is induced by vector addition $(v_1, \dots, v_n) \mapsto v_1 + \cdots + v_n$ and $\KKK_{n-d}$ is the kernel of $\psi$. Since $\KKK_{n-d}$ has rank $n-d$, we have $c^T_r(\KKK_{n-d}) = 0$ whenever $r > n-d$. The relation
$c^T(\LLL_1) \cdots c^T(\LLL_n) = c^T(\LLL_1 \oplus \cdots \oplus \LLL_n) = c^T(\KKK_{n-d}) \cdot c^T(p^*(\VVV_d))$ implies
\begin{center}
$\sum_{a+b = r}(-1)^a e_a(\xxx_n) h_b(\yyy_d) \in \mathrm{Ker}(\widetilde{\varphi})$ whenever $r > n-d$.
\end{center}
Finally, for each $i = 1, 2, \dots, n$ we have a short exact sequence
\begin{equation}
    0 \to \LLL_i \to \CC^k \to \CC^k/\LLL_i \to 0
\end{equation}
of vector bundles over $X_{n,k,d}$. Since $\CC^k/\LLL_i$ has rank $k-1$, we have $c_k^T(\CC^k/\LLL_i) = 0$. The relation $c^T(\CC^k) = c^T(\LLL_i) \cdot c^T(\CC^k/\LLL_i)$ implies
\begin{center}
$x_i^k - x_i^{k-1} e_1(\ttt_k) + \cdots + (-1)^k e_k(\ttt_k) \in \mathrm{Ker}(\widetilde{\varphi})$ for $i = 1, 2, \dots, n$.
\end{center}
We conclude that $I_{n,k,d} \subseteq \mathrm{Ker}(\widetilde{\varphi})$, so we have an induced surjection of $\CC[\ttt_k]$-algebras
\begin{equation}
    \varphi: \CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}/I_{n,k,d} \twoheadrightarrow H^*_T(X_{n,k,d}).
\end{equation}

We want to prove that the surjection $\varphi$ is in fact an isomorphism. The fiber bundle
\begin{equation}
    X_{n,d} \hookrightarrow X_{n,k,d} \xrightarrow{ \, \, p \, \,}  \Gr(d,\CC^k)
\end{equation}
gives rise (see \cite[Lemma 4.3]{CMR}) to an isomorphism of $H^*_T(\Gr(d,\CC^k))$-modules
\begin{equation}
    H^*_T(X_{n,k,d}) \cong H^*_T(\Gr(d,\CC^k)) \otimes_\CC H^*(X_{n,d}).
\end{equation}
The standard Schubert cell decomposition of $\Gr(d,\CC^k)$ gives rise to a $T$-invariant affine paving of $\Gr(d,\CC^k)$ with $\binom{k}{d}$ cells. On the other hand, Pawlowski--Rhoades \cite{PR} proved that $H^*(X_{n,d})$ has vector space dimension $d! \cdot \Stir(n,d)$.  It follows that $H^*_T(X_{n,k,d})$ is a free $\CC[\ttt_k]$-module of rank
\begin{equation}
    \binom{k}{d} \times d! \cdot \Stir(n,d) = \frac{k!}{(k-d)!} \cdot \Stir(n,d) = \# \WWW_{n,k,d}.
\end{equation}
By \Cref{lem:free-module}, we conclude that $\varphi$ is an epimorphism between free $\CC[\ttt_k]$-modules of the same rank $\# \WWW_{n,k,d}$. It follows that $\varphi$ is an isomorphism.
\end{proof}

%\begin{defn}
%    \label{def:z-locus-specialized}
%    For $\aalpha = (\alpha_1, \dots, \alpha_k) \in U$, define $\Zpoints_{n,k,d}(\aalpha) \subseteq \CC^{n+d}$ to be the locus of points $(z_1, \dots, z_n; z_{n+1}, \dots, z_{n+d})$ such that
%   \begin{itemize}
%       \item every coordinate $z_i$ lies in $\{\alpha_1, \dots, \alpha_k \}$,
%       \item the last coordinates $z_{n+1}, \dots, z_{n+d}$ are distinct, and
%       \item we have $\{z_1, \dots, z_n\} = \{z_{n+1}, \dots, z_{n+d} \}.$
%    \end{itemize}
%\end{defn}




%The points in $\Zpoints_{n,k,d}(\aalpha_k)$ may be enumerated as follows. We have $\frac{k!}{(k-d)!}$ choices for the last $d$ coordinates  $z_{n+1}, \dots, z_{n+d}$. Once these terminal coordinates are fixed, the first $n$ coordinates correspond to a surjective map $[n] \twoheadrightarrow [d]$; the number of such maps is $d! \cdot \Stir(n,d)$, where $\Stir(n,d)$ is the Stirling number of the second kind counting $d$-block set partitions of $[n]$. We conclude that $|\Zpoints_{n,k,d}(\aalpha_k)| = \frac{k! \cdot d!}{(k-d)!} \cdot \Stir(n,d).$ 
%Observe that the locus $\Zpoints_{n,k,d}(\aalpha)$ carries a free action of $\symm_d$ on the last $d$ coordinates. For any $\aalpha \in U$, the homogeneous ideal $\gr \, \II(\Zpoints_{n,k,d}(\aalpha))$ has the following presentation.

%\begin{lemma}
%    \label{lem:specialized-presentation}
%    Let $J_{n,k,d} \subseteq \CC[\xxx_n,\yyy_d]$ be the ideal generated by
%    \begin{itemize}
%        \item $h_r(\yyy_d)$ for all $r > k-d$,
%        \item $e_r(\xxx_n) - e_{r-1}(\xxx_n) h_1(\yyy_d) + \cdots + (-1)^r h_r(\yyy_d)$ for all $r > n-d$, and
%        \item $x_i^d - x_i^{d-1} e_1(\yyy_d) + \cdots + (-1)^d e_d(\yyy_d)$ for $i = 1,2, \dots, n$.
%    \end{itemize}
%    For any $\aalpha \in U$ we have $\gr \, \II(\Zpoints_{n,k,d}(\aalpha)) = J_{n,k,d}$.
%\end{lemma}



% Given \Cref{lem:specialized-presentation}, it is natural to put the loci $\Zpoints_{n,k,d}(\aalpha)$ into a single infinite family.
%Let $\Zpoints_{n,k,d} \subseteq \CC^{n+d} \times \tttt = \CC^{n+d+k}$ be the variety
%    \begin{equation}
%        \Zpoints_{n,k,d} := \text{Zariski closure of } \bigcup_{\aalpha \in U} \Zpoints_{n,k,d}(\aalpha) \times \{\aalpha\} \text{ in $\CC^{n+d} \times \tttt$.}
%    \end{equation}
%The vanishing ideal $\II(\Zpoints_{n,k,d})$ is a subset of $\CC[\xxx_n,\yyy_d,\ttt_k]$. We will prove that the ideal $\II(\Zpoints_{n,k,d})$ has the following set of generators.

%\begin{defn}
%    \label{def:j-t-ideal}
%    Let $J_{n,k,d}^{\QQ,\ttt} \subseteq \QQ[\xxx_n,\yyy_d,\ttt_k]$ be the ideal generated by 
%    \begin{itemize}
%    \item $e_r(\ttt_k) - e_{r-1}(\ttt_k) h_1(\yyy_d) + \cdots + (-1)^r h_r(\yyy_d)$ for all $r > k-d$,
%    \item $e_r(\xxx_n) - e_{r-1}(\xxx_n) h_1(\yyy_d) + \cdots + (-1)^r h_r(\yyy_d)$ for all $r > n-d$, and
%    \item $x_i^d - x_i^{d-1} e_1(\yyy_d) + \cdots + (-1)^d e_d(\yyy_d)$ for $i = 1, \dots , n$.
%    \end{itemize}
%\end{defn}

%The generators of $\II(\Zpoints_{n,k,d}(\aalpha_k))$ are obtained from those in \Cref{def:j-t-ideal} by setting $t_i \to \alpha_i$, and it can be shown that setting the $t_i$ variables to $0$ recovers the associated graded ideal $\gr(\II(Z_{n,k,d}(\aalpha_k))$. The following containment of ideals is true, and the proof is particularly illuminating, so we present it here:

%\begin{lemma}
%    \label{lem:j-t-containment}
%    We have the containment of ideals $J_{n,k,d}^{\QQ,\ttt} \subseteq \II(\Zpoints_{n,k,d})$.
%\end{lemma}

%\begin{proof}
%    Let $u$ be a new variable. If $r > k-d$, the rational function
%    $$ \frac{(1 + t_1 u) \cdots (1 + t_k u)}{(1 + y_1 u) \cdots (1 + y_d u)}$$
%    when restricted to $\Zpoints_{n,k,d}$ is a polynomial in $u$ of degree $k-d$. Since the coefficient of $u^r$ in this rational function is $e_r(\ttt_k) - e_{r-1}(\ttt_k) h_1(\yyy_d) + \cdots + (-1)^r h_r(\yyy_d)$, we see that the generators in the first bullet point of \Cref{def:j-t-ideal} lie in $\II(\Zpoints_{n,k,d})$. The generators in the second and third bullet points are handled in a similar fashion.
%\end{proof}

%To establish the equality of ideals, we present a monomial basis for the quotient by $J_{n,k,d}^{\QQ,\ttt}$ as follows.

%An $(n,d)$-staircase is a sequence $(s_1,\dots,s_n)$ that contains a subsequence $(0,1,\dots,d-1)$, and all other entries are $d-1$. A length $n$ sequence $(e_1, \dots, e_n)$ of nonnegative integers is {\em $(n,d)$-substaircase} if it is componentwise $\leq$ at least one $(n,d)$-staircase. A monomial $x_1^{e_1} \cdots x_n^{e_n}$ in the variables $\xxx_n = (x_1, \dots, x_n)$ is {\em $(n,d)$-substaircase} if its exponent sequence $(e_1,\dots,e_n)$ has the corresponding property. The number of $(n,d)$-stubstaircase monomials coincides with the number of surjections $[n] \to [d]$, namely $d! \cdot \Stir(n,d)$ (\cite{HRS},Lemma 4.8).

%\begin{defn}
%    \label{defn:a-monomials}
%    For positive integers $n \geq k \geq d$, let $\AAA_{n,k,d}$ be the following set of monomials in $\QQ[\xxx_n,\yyy_d]$:
%    \begin{equation}
%    \AAA_{n,k,d} := \{ x_1^{a_1} \cdots x_n^{a_n} \cdot y_1^{b_1} \cdots y_d^{b_d} \,:\, (a_1, \dots, a_n) \text{ is $(n,d)$-substaircase and } b_i < k-d+i \}.
%    \end{equation}
%\end{defn}
%The number of such monomials is given by $|\AAA_{n,k,d}| = d! \cdot Stir(n,d)\times \frac{k!}{(k-d)!} = \frac{k! \cdot d!}{(k-d)!} \cdot |\Stir(n,d)| = |\Zpoints_{n,k,d}(\aalpha_k)|$.

%\begin{proposition}
%    \label{prop:t-ideal-equality}
%    We have the equality of ideals $J_{n,k,d}^{\QQ,\ttt} = \II(\Zpoints_{n,k,d})$ inside $\QQ[\xxx_n,\yyy_d,\ttt_k]$. The set $\AAA_{n,k,d}^\ttt$ descends to a $\QQ$-basis for the common quotient ring
%    $$\QQ[\xxx_n,\yyy_d,\ttt_k]/J_{n,k,d}^{\QQ,\ttt} = \QQ[\xxx_n,\yyy_d,\ttt_k]/\II(\Zpoints_{n,k,d}).$$
%    In particular, this quotient is a free $\QQ[\ttt_k]$-module of rank $\frac{k! \cdot d!}{(k-d)!} \cdot \Stir(n,d)$.
%\end{proposition}

%%[sketch the argument here if there is still room but I doubt it]

%\subsection{A fiber bundle} Our geometric analysis of $X_{n,k,d}$ makes heavy use of the fact that $X_{n,k,d}$ is a fiber bundle over the Grassmannian $\Gr(d,\CC^k)$. The fibers of $X_{n,k,d} \to \Gr(d,\CC^k)$ are isomorphic to the following variety.

%\begin{defn}
%    \label{def:xnd}
%    For $n \geq d$, let $X_{n,d}$ be the moduli space of $n$-tuples $(\ell_1, \dots, \ell_n)$ of lines in $\CC^d$ which satisfy $\ell_1 + \cdots + \ell_n = \CC^d$.
%\end{defn}

%The space $X_{n,d}$ was first defined in \cite{PR}; we have $X_{n,d,d} = X_{n,d}$. The general relationship between the spaces $X_{n,k,d}$ and $X_{n,d}$ is as follows.

%\begin{observation}
%    \label{obs:fiber-bundle} The map $p: X_{n,k,d} \to \Gr(d,\CC^k)$ sending $(\ell_1, \dots, \ell_n)$ to $\ell_1 + \cdots + \ell_n$ is a fiber bundle with fiber isomorphic to $X_{n,d}$. That is, we have the fiber bundle
%\end{observation}

%\begin{equation}\label{eq:fiberbundle}
%    X_{n,d} \hookrightarrow X_{n,k,d} \xrightarrow{p} \Gr(d,\CC^k)
%\end{equation}

%The rank $k$ torus $T = (\CC^*)^k$ acts on both $X_{n,k,d}$ and $\Gr(d,\CC^k)$.
%Observation~\ref{obs:fiber-bundle} and the Leray--Hirsch Theorem have the following consequences for ordinary and equivariant cohomology.

%\begin{lemma}
%    \label{lem:leray-hirsch-decomposition}
%    We have an isomorphism of $H^*(\Gr(d,\CC^k);\ZZ)$-modules
%    $$H^*(X_{n,k,d};\ZZ) \cong H^*(\Gr(d,\CC^k);\ZZ) \otimes_\ZZ H^*(X_{n,d};\ZZ)$$
%    and an isomorphism of $H^*_T(\Gr(d,\CC^k);\ZZ)$-modules
%    $$H^*_T(X_{n,k,d};\ZZ) \cong H^*_T(\Gr(d,\CC^k);\ZZ) \otimes_\ZZ H^*(X_{n,d};\ZZ)$$
%\end{lemma}

%\subsection{Presentation for the equivariant cohomology ring}

%We first begin by describing an affine paving of $X_{n,k,d}$. For $d \leq k \leq n$, we write
%\begin{equation}
%    \WWW_{n,k,d} := \{ w: [n] \to [k] \,:\, |\im(w)| = d \}
%\end{equation}
%for the set of length $n$ words $[w(1), \dots, w(n)]$ over the alphabet $[k]$ in which exactly $d$ letters appear. We have $|\WWW_{n,k,d}| = \frac{k!}{(k-d)!} \cdot \Stir(n,d)$.

%We may embed $X_{n,k,d} \subseteq (\PP^{k-1})^n$, and there is a canonical identification $(\PP^{k-1})^n \cong \MMM_{n,k}/(\CC^*)^n$, the set of $k \times n$ matrices with no zero columns. We may then identify $X_{n,k,d}$ as the collection of (equivalence classes of) rank $d$ matrices. A sort of Gaussian elimination in [Pawlowski--Rhoades] gives the following cells:

%\begin{equation}
%    C_w := U(w) \cdot  \widehat{C}_w \cdot (\CC^*)^n / (\CC^*)^n.
%\end{equation}

%where $U(w)$ is a certain subgroup of the unipotent group $U(n)$, and $\widehat{C}_w$ are the \emph{pattern matrices} from \cite{PawlowskiRhoades}. These cells form an affine paving, which determines the module rank of $H^*_T(X_{n,k,d};\ZZ)$:

%\begin{lemma}\cite{PR}
%    \label{lem:affine-paving}
%    The set $\{C_w \,:\, w \in \WWW_{n,k,d} \}$ forms the cells of an affine paving of $X_{n,k,d}$. The cells $C_w$ are closed under the left action of $T = (\CC^*)^k$.
%\end{lemma}

%\begin{lemma}
%    \label{lem:cohomology-ring-ranks} Let $d \leq k \leq n$.
%    The ordinary cohomology ring $H^*(X_{n,k,d};\ZZ)$ is a free $\ZZ$-module of rank $\frac{k!}{(k-d)!} \cdot \Stir(n,d)$. The equivariant cohomology ring $H^*_T(X_{n,k,d};\ZZ)$ is a free $\ZZ[\ttt_k]$-module of rank $\frac{k!}{(k-d)!} \cdot \Stir(n,d)$.
%\end{lemma}

%%Not sure if the below is appropriate for the FPSAC audience; if necessary, we can give an abridged version of the geometric argument. However, if possible I'd like to show the parallels (generating function argument) in the Whitney sum argument with the orbit harmonics bit, I think it really highlights the theme of the paper

%To realize $H^*_T(X_{n,k,d};\ZZ)$ as a quotient of $\ZZ[\xxx_n,\yyy_d,\ttt_k]$ by the ideal $I_{n,k,d}$, we have the following lemmata:

%\begin{lemma}\label{lem:ring-generation}
%    Let $\UUU_d$ be the rank $d$ tautological vector bundle over $\Gr(d,\CC^k)$ whose fiber over a subspace $V \in \Gr(d,\CC^k)$ is the space $V$ itself. Also, for $1 \leq i \leq n$, let $\LLL_i$ be the line bundle over $X_{n,k,d}$ whose fiber over $\ell_\bullet = (\ell_1, \dots, \ell_n)$ is the line $\ell_i$. The equivariant cohomology ring $H^*_T(X_{n,k,d};\ZZ)$ is generated as a $\ZZ[\ttt_k]$-algebra by the equivariant Chern classes 
%    $$p^*(c_1^T(\UUU_d)), p^*(c_2^T(\UUU_d)), \dots, p^*(c_d^T(\UUU_d)) \quad \text{and} \quad c_1^T(\LLL_1), c_1^T(\LLL_2), \dots, c_1^T(\LLL_n).$$
%\end{lemma}

%\begin{lemma}\label{lem:chern-class-relations}
%    For $1 \leq i \leq n$, let $x_i := c_1^T(\LLL_i) \in H^2_T(X_{n,k,d};\ZZ)$. In the equivariant cohomology ring $H^*_T(X_{n,k,d};\ZZ)$ we have the relations
%    \begin{itemize}
%        \item $e_r(\ttt_k) - e_{r-1}(\ttt_k) h_1(p^*(\UUU_d)) + \cdots + (-1)^r h_r(p^*(\UUU_d)) = 0$ for all $r > k-d$,
%        \item $e_r(\xxx_n) - e_{r-1}(\xxx_n) h_1(p^*(\UUU_d)) + \cdots + (-1)^r h_r(p^*(\UUU_d)) = 0$ for all $r > n-d$, and
%        \item $x_i^d - x_i^{d-1} e_1(p^*(\UUU_d)) + \cdots + (-1)^r e_d(p^*(\UUU_d)) = 0$ for $i = 1, \dots, n$.
%    \end{itemize}
%\end{lemma}

%We briefly explain the first bullet point, and remark that the other two follow from similar Whitney sum arguments. There is a short exact sequence of vector bundles over $\Gr(d,\CC^k)$
%\begin{equation}
%    0 \to \UUU_d \to \CC^k \to \CC^k/\UUU_d \to 0
%\end{equation}
%where $\CC^k$ is the rank $k$ trivial bundle over $\Gr(d,\CC^k)$ with fiber $\CC^k$, and $\CC^k/\UUU_d$ is the rank $k-d$ tautological quotient bundle. This gives the following relation of equivariant total Chern classes
%\begin{equation}
%\label{eqn:equivariant-rational-expression-one}
%    c^T(\CC^k/\UUU_d) = \frac{c^T(\CC^k)}{c^T(\UUU_d)} = \frac{(1 + t_1) \cdots (1 + t_k)}{(1 + y_1) \cdots (1 + y_d)}
%\end{equation}
%where the $y_i$ are the equivariant Chern roots, and we have $e_i(\yyy_d) = c_i^T(\UUU_d)$. Since $\CC^k/\UUU_d$ is rank $k-d$, we must have $c_r^T(\CC^k/\UUU_d)$ if $r > k-d$; taking the degree $r$ part of both sides of \ref{eqn:equivariant-rational-expression-one} and pulling back via $p^*$ establishes the first bullet point. Note the similarity in the expression for \ref{eqn:equivariant-rational-expression-one} and the generating function in \ref{lem:j-t-containment}. We are now ready to prove the main theorem:

%%modify the coefficients as necessary
%\begin{proof}[Proof of \Cref{thm:borel-presentation}]
%    It follows from \Cref{lem:ring-generation} that $H^*_T(X_{n,k,d})$ is concentrated in even degrees. The Universal Coefficient Theorem and \Cref{lem:cohomology-ring-ranks} imply that $H^*_T(X_{n,k,d};\QQ)$ is a free $\QQ[\ttt_k]$-module of rank $\frac{k!}{(k-d)!} \cdot \Stir(n,d)$. 

%    \Cref{lem:chern-class-relations} implies that we have a well-defined $\QQ[\ttt_k]$-algebra homomorphism
%    \begin{equation}
%        \varphi: \CC[\xxx_n,\yyy_d,\ttt_k]^{\symm_d}/I_{n,k,d} \to H^*_T(X_{n,k,d};\QQ)
%    \end{equation}
%    which sends $x_i$ to $c_1^T(\LLL_i)$ and the elementary symmetric polynomial $e_i(\yyy_d)$ to $c_i^T(p^*(\UUU_d))$. \Cref{lem:ring-generation} implies that $\varphi$ is a surjection. 
    
%    We know that the domain of $\varphi$ is a free $\QQ[\ttt_k]$-module of rank $\frac{k!}{(k-d)!} \cdot \Stir(n,d)$.  \Cref{lem:t-invariant-free-module} implies that the target of $\varphi$ is a free $\QQ[\ttt_k]$-module of the same rank. A consequence of the Cayley--Hamilton theorem implies that the surjection $\varphi$ is in fact an isomorphism.
%\end{proof}

%\section{Future Directions and Questions}

%\subsection{Classes of Cell Closures}
%The cells $[\overline{C_w}], w \in W_{n,k,d}$ were instrumental in the proof of theorem \ref{thm:borel-presentation}. In the case of $k = d$, these classes were computed in \cite{PR} to be certain \emph{double Schubert polynomials} using the technique of \emph{degeneracy loci}. In the case of $k \neq d$, these methods fail; this leaves the following open problem:

%\begin{problem}
%    Compute the classes $[\overline{C_w}]$, $w \in W_{n,k,d}$, in terms of the generators $\xxx_n,\yyy_d,\ttt_k$.
%\end{problem}

%\subsection{GKM Rings}
%A powerful technique for computing equivariant cohomology rings stems from the work of Goresky--Kottwitz--MacPherson \cite{GKM}. Given a scheme $X$ with a torus action $T$, the inclusion of a torus fixed point $\{\bullet\} \hookrightarrow X$ induces a map in equivariant cohomology $H^*_T(X) \to H^*_T(\bullet)$. We may patch the maps together for the torus fixed points $X^T$ together to obtain a map:

%$$
%H^*_T(X) \xrightarrow{\iota} \bigoplus_{u \in X^T} H^*_T(\bullet)
%$$
%The authors of \cite{GKM} give a set of sufficient conditions (known as the \emph{GKM conditions}) for when $\iota$ is injective, as well as give a set of relations which generate $\im(\iota)$. Such a presentation is useful for computations, as one need not work with equivalence classes of polynomials, as in Borel-style presentations.

%In the case of $X_{n,k,d}$, one of the GKM conditions is not satisfied, yet it can be shown that the map $\iota$ is injective with an algebraic argument. This suggests the following problem:

%\begin{problem}
%    Find a set of defining relations for the image of $\iota$ for $X_{n,k,d}$.
%\end{problem}

%Many analogues of the GKM relations are satisfied by $\iota(H^*_T(X_{n,k,d}))$, yet these relations are insufficient to describe the image exactly. This suggests a more general combinatorial problem for determining the image:

%\begin{problem}
%    Given an ideal $I \subset \CC[\xxx_n,\ttt_k]$, and for each $u \in S$ an evaluation map $f_u: \CC[\xxx_n,\ttt_k] \to \CC[\ttt_k]$, we may patch these maps together to obtain a map $\CC[\xxx_n,\ttt_k] / I \xrightarrow{\iota} \bigoplus_{u \in S} \CC[\ttt_k]$. When is a tuple of polynomials $(p_1(\ttt_k),\dots,p_{|S|}(\ttt_k)) \in \im(\iota)$?
%\end{problem}



\acknowledgements{The authors thank Leonardo Mihalcea, Ed Richmond, and Travis Scrimshaw for helpful conversations.}

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