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\title[Standard Monomials for Positroid Varieties]{Standard Monomials for Positroid Varieties}

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\author{\texorpdfstring{Ayah Almousa\thanks{\href{mailto:aalmousa@sc.edu}{aalmousa@sc.edu}}\addressmark{1}, Shiliang Gao\thanks{\href{mailto:shiliang.gao@cornell.edu}{shiliang.gao@cornell.edu}. Supported by the NSF MSPRF under grant No. DMS-2402285.}\addressmark{2}, \and Daoji Huang\thanks{\href{mailto:dhuang@ias.edu}{dhuang@ias.edu}. Supported by NSF-DMS2202900.}\addressmark{3}}{Ayah Almousa, Shiliang Gao, and Daoji Huang}}

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\address{\addressmark{1} Department of Mathematics, University of South Carolina, Columbia, SC \\ \addressmark{2} Department of Mathematics, Cornell University, Ithaca, NY \\ 
\addressmark{3} School of Mathematics, Institute for Advanced Study, Princeton, NJ
}

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\abstract{We give an explicit characterization of the standard monomials for positroid varieties with respect to the Hodge degeneration and give a Gr\"obner basis. 
 Furthermore, we 
 show that promotion and evacuation 
 biject standard monomials of a positroid variety with those of its cyclic shifts and $w_0$-reflection, respectively.
The connection to promotion allows us to identify standard monomials of a positroid variety with Lam's cyclic Demazure crystal. Using a recurrence on the Hilbert series, we give an inductive formula for the characters of cyclic Demazure modules, solving a problem posted by Lam.}

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\keywords{positroid varieties, promotion, evacuation, Gr\"obner basis, standard monomial theory.}


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\begin{document}

\maketitle
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\section{Introduction}\label{section: intro}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Positroid varieties are subvarieties of the Grassmannian $\gr(k,n)$ introduced by Knutson--Lam--Speyer \cite{KLS13juggling, KLS2014projections}, motivated by questions in total positivity \cite{Lus98,post}. 
% These varieties also arise in Poisson geometry \cite{BGY06} and in the study of scattering amplitudes \cite{ABCGPT}.
% Recent work of Galashin--Lam \cite{GL21a, galashin2021positroids} studied mixed Hodge structure on the cohomology of open positroid varieties and its relation with Khovanov--Rozansky link homology and $q,t$-Catalan numbers.
% Cohomologically, computing the classes of positroid varieties relates to the famous Schubert-times-Schur problem in $\mathrm{H}^*(Fl(n))$ (see e.g. \cite[Section 6.5]{speyer2023richardson}). 

Classical work of Hodge \cite{Hodge} described a particular set of bases for the homogeneous coordinate rings of the Grassmannian $\gr(k,n)$ and its Schubert varieties under the Pl\"ucker embedding.
Building on Hodge's ideas, Seshadri initiated the study of standard monomial theory (SMT), with the aim of giving standard monomial bases for the space of global sections of line bundles over a (generalized) flag variety $G/P$.
The foundation of SMT was built in the works of Lakshmibai, Musili, and Seshadri,  \cite{GmodP1, GmodP2, GmodP3, GmodP4}.
The tools developed in SMT yield a wide range of applications, such as derivations of geometric properties (Cohen--Macaulayness, normality, singularities, cohomological vanishings, etc.)\! of Schubert varieties and character formulas for Demazure modules.

The main goal of our paper is to extend the work of Hodge to \emph{positroid varieties} in $\gr(k,n)$, based on works of Knutson--Lam--Speyer. 
Positroid varieties are the closed strata of the positroid stratification, the common refinement of the cyclically permuted Bruhat decompositions.
These are also exactly the projections of Richardson varieties from the complete flag variety to $\gr(k,n)$, and are
 the only compatibly split subvarieties with respect to the standard Frobenius splitting on the Grassmannian \cite{KLS2014projections}.
%Since their introduction, positroid varieties have been of great interest in the study of Schubert calculus \cite{K14interval}, cluster algebras \cite{GL23cluster}, and positive geometries \cite{AHBL17}.

Using the standard monomial theory of Richardson varieties \cite{ lakshmibai2003richardson,BrionLak},
Knutson--Lam--Speyer \cite{KLS2014projections} described the Stanley--Reisner complexes of positroid varieties 
under the \emph{Hodge degeneration}, a special kind of Gr\"obner degeneration.
However, they did not give an explicit description (i.e., without referring to the chains in the Bruhat order of $G/B$) of the standard monomials. In this extended abstract, we present
 an explicit description of standard monomials for positroid varieties, analogous to Hodge's original description based on semistandard Young tableaux (Theorem~A).
This approach yields several applications in algebra, combinatorics, and representation theory. More specifically:
\begin{itemize}
\itemsep0em
    \item an explicit Gr\"obner basis for positroid varieties with respect to the Hodge degeneration, analogous to the classical ``straightening relations'' (Theorem~B).
    \item a connection between the \emph{promotion} (resp. \emph{evacuation}) on rectangular semistandard Young tableaux and  rotations  (resp. reflections) of positroid varieties (Theorem~C).
    \item a formula for characters of \emph{cyclic Demazure modules}, resolving a problem posted by Lam \cite{Lam19} (Theorem~D).
\end{itemize}

\section{Background}\label{section: positroid-background}

\subsection{Grassmannians and Pl\"ucker embeddings}\label{subsec:plucker}
For $k\leq n\in \mathbb{Z}_{>0}$ and an algebraically closed field $\k$ of characteristic $0$, the \textbf{Grassmannian} $\gr(k,n)$ is 
\[\gr(k,n) = \{W\subseteq \k^n: \dim(W) = k\}.\]
For $W\in \gr(k,n)$ and $\{w_1,\dots,w_k\}$ a chosen basis of $W$. The \textbf{Pl\"ucker embedding} is the map 
    $\iota:\gr(k,n)\rightarrow \mathbb{P}(\Lambda^k(\k^n))$ sending 
    $W$ to $[w_1\wedge \dots \wedge w_k]$.

Set $R(k,n):= \k[[\aa]:\aa\in \binom{[n]}{k}]$ to be the homogeneous coordinate ring of the projective space $\mathbb{P}(\Lambda^k(\k^n))$. 
We extend the notation of $[\aa]$ to all sequences 
$\aa = (a_1,\dots,a_k)\in [n]^k$, 
where we use the convention that for any permutation $\sigma\in \symm_n$, 
\begin{equation}\label{eqn:extendedplucker}
    [a_{\sigma(1)},a_{\sigma(2)},\dots, a_{\sigma(k)}] = \sgn(\sigma)\cdot [a_1,\dots, a_k].
\end{equation}
In particular, this implies that $[\aa] = 0$ if $a_i = a_j$ for some $i\neq j\in [n]$.


The defining ideal $\cJ$ of $\gr(k,n)$ as a projective subvariety of $\mathbb{P}(\Lambda^k(\k^n))$ is generated by the following \textbf{straightening relations}:

\begin{definition}[\cite{SturmfelsWhite}]\label{def:Pluckerrelations}
    Let $s\in [k], \alpha \in \binom{[n]}{s-1},\beta\in \binom{[n]}{k+1}$ and $\gamma\in \binom{[n]}{k-s}$ where elements in $\alpha,\beta,\gamma$ are in increasing order. The straightening relation attached to $\alpha,\beta,\gamma$ is 
    \[\sum_{I\in \binom{[k+1]}{s}}(-1)^{\sgn(I)}[\alpha_1,\dots,\alpha_{s-1},\beta_{i'_1},\dots,\beta_{i'_{k-s+1}}]\cdot [\beta_{i_1},\dots,\beta_{i_s},\gamma_{1},\dots,\gamma_{k-s}]\]
    where $I = \{i_1<\dots<i_s\}\subset [k+1]$, $\{i'_1<\dots<i'_{k-s+1}\}:= [k+1]\setminus I$,  and $\sgn(I):= \sum_{j = 1}^s i_j - \binom{s+1}{2}$.
\end{definition}



% \begin{example}
%     The Grassmannian $\gr(2,4)$ is defined by one equation in $\mathbb{P}^5$:
%     $$
% \k[\gr(2,4)] \cong \frac{R(2,4)}{\cJ} = \frac{\k[[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]}{\langle [1,4][2,3] - [1,3][2,4] + [1,2][3,4]\rangle}.
% $$
% \end{example}

We will often\footnote{Sometimes we need to work with tableaux with permuted entries. In these cases, it is important to keep in mind that we may sort the columns as long as we keep track of signs.} represent a monomial $\mm = \prod_{i = 1}^d [\aa^{(i)}]$ as a $k\times d$ tableau, where column $i$ is strictly increasing with entries in $\aa^{(i)}$ for all $i$. 

\begin{example}
% The straightening relation generating the defining ideal for $\gr(2,4)$ could be represented  as
% \begin{equation}\label{eq: plucker-gr-24}
% \yshort{1{\rr{2}}, {\rr{4}}{\rr{3}}} \; - \; 
% \yshort{1{\rr{2}}, {\rr{3}}{\rr{4}}} \; + \; 
% \yshort{1{\rr{3}}, {\rr{2}}{\rr{4}}}
% \end{equation}
% where in this case, $\alpha = \{1\}$, \textcolor{red}{$\beta = \{2,3,4\}$}, and $\gamma = \emptyset$.
An example of a straightening relation in $\gr(4,8)$  arises from the product $[1,2,6,7]\cdot [3,4,5,8]$. Here $\alpha = \{1,2\}$, \textcolor{red}{$\beta = \{3,4,5,6,7\}$}, and $\gamma = \{8\}$. The monomials are obtained by permuting the elements in $\beta$ (colored in red). 
$$
\yshort{1{\rr{3}}, 2{\rr{4}}, {\rr{6}}{\rr{5}},{\rr{7}}8} \; - \; \yshort{1{\rr{3}}, 2{\rr{4}}, {\rr{5}}{\rr{6}},{\rr{7}}8} \; + \; \yshort{1{\rr{3}}, 2{\rr{4}}, {\rr{5}}{\rr{7}},{\rr{6}}8} \; - \; \yshort{1{\rr{3}}, 2{\rr{5}}, {\rr{4}}{\rr{7}},{\rr{6}}8} \; \cdots 
$$
\end{example}





\begin{definition} A tableau is called \textbf{semistandard} if it is strictly increasing along columns and weakly increasing along rows.
\end{definition}

The homogeneous coordinate ring of $\gr(k,n)$ is $\k[\gr(k,n)] = R(k,n)/\cJ$. It is the direct sum of its graded pieces:
\[
\k[\gr(k,n)] = \bigoplus_{d = 0}^\infty \k[\gr(k,n)]_d, 
\]
where each $\k[\gr(k,n)]_d$ is a finite dimensional $\k$-vector space spanned by the monomials $\{\prod_{i = 1}^d [\aa^{(i)}]: \aa^{(i)}\in \binom{[n]}{k}\text{ for all }i\}$.
The following classical theorem is originally due to Hodge \cite{Hodge} (c.f. \cite[Lemma~7.2.3]{bruns1998cohen}).

\begin{thm}\label{thm:hodge}
The monomials $\prod_{i = 1}^d [\aa^{(i)}]$ that correspond to semistandard tableaux form a basis of $\k[\gr(k,n)]_d$. 
\end{thm}

%These monomials are often referred to as the \textbf{standard monomials} for $\gr(k,n)$. We now give the precise definition of standard monomials for subvarieties of $\gr(k,n)$.

Let $\cP = (\binom{[n]}{k}, \leq)$ be the poset where
%Let``$\leq$'' be the partial order on $\binom{[n]}{k}$ such that
$$
[c_1,\dots, c_k] \leq [d_1,\dots, d_k] \text{ if and only if } c_i\leq d_i \text{ for all } i=1,\dots, k.
$$
%We call this poset $\cP = (\binom{[n]}{k}, \leq)$ the \textbf{Pl\"{u}cker poset}. 
Let $<_\omega$ be a degree revlex monomial order on the polynomial ring $R(k,n):= \k[[\aa]:\aa\in \binom{[n]}{k}]$ where the Pl\"ucker variables are ordered by some linear extension of $\cP$.
%the Pl\"{u}cker poset.

\begin{definition}\label{def:std-mon}
    For a subvariety $X\subseteq \gr(k,n)$, let $\cJ_X$ be its defining ideal under the Pl\"ucker embedding. A monomial $\mm = \prod_{i = 1}^d[\aa^{(i)}]$ is called a \textbf{standard monomial} for $X$ if $\mm\notin \inn_\omega(\cJ_X)$.
\end{definition}

For each $d\in \mathbb{N}$, the degree $d$ standard monomials $\{\mm: \deg(\mm) = d, \mm \notin \inn_\omega(\cJ_X)\}$ form a basis of $\k[X]_d$. This basis is also known as the \textbf{standard monomial basis}. 
The monomials in \Cref{thm:hodge} are standard monomials by the following:

\begin{thm}[\cite{SturmfelsWhite}]\label{thm:grgrobner}
    The straightening relations in \Cref{def:Pluckerrelations} form a Gr\"obner basis of $\cJ$ under the term order $<_\omega$. The initial ideal is 
    \[\inn_\omega (\cJ) = \langle [\aa][\bb]: \aa,\bb \text{ not comparable in }\mathcal{P} \rangle.\]
\end{thm}



\subsection{Positroid varieties}
We first recall a few sets of combinatorial objects that index positroid varieties.


Define the set of \textbf{bounded affine permutations} to be  
\begin{equation}\label{eqn:bounded}
        \mathrm{Bound}(k,n) := \{f:\ZZ\to\ZZ|f(i+n)=f(i)+n, i\leq f(i)\leq  i+n \text{ for all $i\in \ZZ$}
        \}.
\end{equation}
 We will sometimes write $f\in\bound(k,n)$ as $f = [ f(1)\ f(2) \cdots f(n) ]$. The partial order on $\bound(k,n)$ is inherited from the Bruhat order on affine permutations.

Given $V\in \gr(k,n)$, let $\widetilde{V}$  denote a choice of $k\times n$ matrix such that $\mathrm{rowspan}(\widetilde{V})=V$. We write
$\widetilde{V} = [v_1,\cdots,v_n]$ where $v_i$ is the $i$th column.
We extend the sequence $v_1\cdots v_n$ by setting $v_{i} = v_{i+n} \text{ for all }i\in [n]$,
and denote by $\widetilde{V}_{[i,j]}$ the matrix with column vectors $v_i,\cdots, v_j$.
Consider the affine permutation $f_{\widetilde{V}}:\ZZ\rightarrow \ZZ$ given by
\begin{equation}\label{eqn:f_V}
    f_{\widetilde{V}}(i) = \min\{j\geq i: v_i\in \Span(v_{i+1},\cdots, v_{j})\}.
\end{equation}
It is known that $f_V\in \mathrm{Bound}(k,n)$ and that all of $\bound(k,n)$ can arise this way. Moreover, $f_{\widetilde{V}}$ only depends on the $V:=\mathrm{rowspan}(\widetilde{V})$, so we may define $f_V$ for $V\in \gr(k,n)$.

The \textbf{open positroid variety }associated to a bounded affine permutation $f$ is
$\Pi_{f}^\circ := \{V\in \gr(k,n):f_V = f\}$
and the \textbf{positroid variety} is its Zariski closure
$\Pi_f = \overline{\Pi_f^{\circ}}$. 
In fact,  positroid varieties stratify the Grassmannian:  
$\Pi_f = \bigsqcup_{f'\geq f}\Pi_{f'}^\circ$. 

Let $\chi:\gr(k,n) \rightarrow \gr(k,n)$ be the \textbf{cyclic rotation} such that for $V\in\gr(k,n)$,
    \begin{equation}\label{eqn:chi}
    \widetilde{V} = [v_1,\cdots,v_n], \quad
    \chi(\widetilde{V}) := [v_n,v_1,\cdots,v_{n-1}],\quad \chi(V):=\mathrm{rowspan}(\chi(\widetilde{V})).
    \end{equation}
We abuse the notation and define the corresponding cyclic shift on $\mathrm{Bound}(k,n)$ as
\[\chi(f)(i) = f(i-1)+1.\]

\begin{lem}\label{lemma:chi}
    For $f\in \mathrm{Bound}(k,n)$, $\chi(\Pi_{f}) = \Pi_{\chi(f)}$.
\end{lem}
An equivalent way to define positroid varieties is through \textbf{cyclic rank matrices} as defined in \cite[Corollary 3.12]{KLS13juggling}. For any $f\in\mathrm{Bound}(k,n)$, write $f$ as the $\infty\times \infty$ matrix with $1$'s at positions $(i,f(i))$ and $0$'s everywhere else. Let $r(f)$ be the infinite periodic matrix defined for all $i\in \mathbb{Z}$, $i\le j\le i+n$,
\[r(f)_{i,j}=|[i,j]|-\#\{\text{number of 1s in }f\text{'s matrix weakly southwest of }(i,j)\}.\]
Let $V\in \gr(k,n)$ so that $f_V=f$, then $r(f)_{i,j}=\rank (v_i,\cdots, v_j)$. Furthermore,
\[\Pi_f^\circ=\{U\in \gr(k,n): \rank(\widetilde{U}_{[i,j]})= r(f)_{i,j}\text{ for all }i\in\mathbb{Z}, j\in[i,i+n]\}\]
and $\Pi_f$ is obtained by replacing ``$=$'' with ``$\le$''.



%Another important indexing set of positroid varieties that is crucial to our work is the  set of \textbf{Grassmann intervals} \cite{KLS2014projections}. In \cite{KLS2014projections}, positroid varieties are realized as projections of Richardson varieties in the complete flag variety. The proof in \Cref{section: initial-ideal} heavily relies on this perspective. However, since we suppress the details of the proofs, we omit the precise definitions as well.


For any $\alpha \in[0,n-1]$ and $m<n$, define the cyclic interval \[[\alpha+1,\alpha+m]^{\circ} = \begin{cases}
\{i: \alpha+1\leq i \leq \alpha+m\} &\text{ if }\alpha+m\le n\\
\{i: \alpha+1 \le i\le n \text{ or }1\le i\le (\alpha+m)\bmod n\} &\text{ if }\alpha+m>n
\end{cases}.
\]
We call the first kind of cyclic interval an \textbf{interval} and the second kind a 
\textbf{wrapped-around interval}. In the wrapped-around case, we sometimes write
$[\alpha+1, \alpha+m-n]^\circ:=[\alpha+1,\alpha+m]^\circ$.

For any $S\subset [n]$ and any $r\in \mathbb{N}$, define
\[\Pi_{S\leq r} = \{V\in \gr(k,n): \rank(\widetilde{V}_{S}) \leq r\},\]
where $\rank(\widetilde{V}_S)$ is the rank of the submatrix of $\widetilde{V}$ with column index $S$. 


\begin{lem-def}
If $S$ is a cyclic interval,
we say $\Pi_{S\leq r}$ is a \textbf{basic positroid variety}. This is indeed an instance of a positroid variety.
\end{lem-def}

 For $f\in\bound(k,n)$, the \textbf{essential set of $f$} is: 
\[ess(f):=\{(i,j):i\in \ZZ, j\in[i,i+n], f(i-1)>j, f^{-1}(j+1)<i, f(i)\le j, f^{-1}(j)\ge i\}.\]
%A diagrammatic description  for $ess(f)$ is given in \cite[Section 2.1]{K14interval}. 
%We have 
% \begin{equation}\label{eqn:essential-rank}
%     \Pi_f^\circ=\{U\in \gr(k,n): \rank(\widetilde{U}_{[i,j]})= r(f)_{i,j}\text{ for all }(i,j)\in ess(f)\}
% \end{equation}
% and similarly for $\Pi_f$. 
The following statement, which follows from \cite[Theorem 5.1]{KLS2014projections} is crucial for our main results.

\begin{prop}
\label{prop:basic-intersection}
    Every positroid variety is the scheme-theoretic intersection of basic positroid varieties:
    \[\Pi_f = \bigcap_{(i,j)\in ess(f)} \Pi_{[i,j]^\circ\leq r(f)_{i,j}}.\]
\end{prop}
For the purpose of this paper, We only need the existence of essential sets in the context of \Cref{prop:basic-intersection} rather than its precise description.


\begin{example}
\label{ex:positroid-decompose}
    Set $k=3$, $n=6$. Let $f$ be the bounded affine permutation \[\cdots [5,2,4,7,9,12]\cdots\] where $f(1)=5$. Then 
    $\Pi_f=\Pi_{[2]\le 0}\cap \Pi_{[2,4]\le 1}\cap \Pi_{[1,5]\le 2}.$
\end{example}

Let $w_0:\gr(k,n)\rightarrow \gr(k,n)$ be the reflection map such that for 
\[\widetilde{V} = [v_1,\dots,v_n],\quad w_0 (\widetilde{V}) = [v_n,\dots, v_1].\]
It is straightforward to see that 
$w_0(\Pi_{[i,j]\leq r}) = \Pi_{[n+1-j,n+1-i]\leq r}$, hence 
$$w_0(\Pi_f) = \bigcap_{(i,j)\in ess(f)} \Pi_{[n+1-j,n+1-i]^\circ\leq r(f)_{i,j}}.$$

\begin{lem-def}\label{lemma:w_0}
    The image $w_0(\Pi_f)$ is again a positroid variety. Define $f^*$ to be the bounded affine permutation such that $\Pi_{f^*} = w_0(\Pi_f)$. 
\end{lem-def}


% We work in the Pl\"ucker embedding of the Grassmannian. Let $[\aa]$ where $\aa\in\binom{[n]}{k}$ denote the Pl\"ucker coordinate indexed by $\aa$. Let $<_\omega$ be a degree revlex monomial order on the polynomial ring $R(k,n):= \k[[\aa]:\aa\in \binom{[n]}{k}]$ where the Pl\"ucker variables are ordered by some linear extension of the Pl\"{u}cker poset.
% The degree-$d$ standard monomials of the Grassmannian $\gr(k,n)$ under the Hodge degeneration can be identified with $B(k,n,d)$, the set of rectangular-shaped semistandard tableaux of size $k\times d$ with contents in $[n]$. 

% For any $\alpha \in[0,n-1]$ and $0<m<n$, define the \textbf{cyclic interval} 
% \[[\alpha+1,\alpha+m]^{\circ} := \begin{cases}
% \{i: \alpha+1\leq i \leq \alpha+m\} &\text{ if }\alpha+m\le n\\
% \{i: \alpha \le i\le n \text{ or }1\le i\le \alpha+m-n\} &\text{ if }\alpha+m>n
% \end{cases}.
% \]

% A \emph{basic positroid variety} $\Pi_{S\le r}\subseteq \gr(k,n)$ is defined by a single cyclic rank condition $S\le r$ where $S$ is a cyclic interval (\Cref{subsec:basic-positroid}). 














\section{Standard monomials for positroid varieties}\label{section: initial-ideal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% The goal of this section is to describe the standard monomials for any positroid variety $\Pi_f$. Let $B(k,n,d)$ denote the set of rectangular semistandard tableaux of shape $k\times d$ with entries $\le n$. Recall from \Cref{subsec:plucker} that the set $B(k,n,d)$ can be identified with standard monomials for $\gr(k,n)$ under the Hodge degeneration. We write $\mm\in B(k,n,d)$ both for the monomial and its tableau. For a positroid variety $\Pi_f$, let\[B_f(k,n,d):=\{\mm\in B(k,n,d): \mm \not\in \inn_\omega(\cJ_f) \}.\]
% Then $B_f(k,n,d)$ is the set of degree-$d$ standard monomials for $\Pi_f$, and forms a basis of $\k[\Pi_f]_d$.

Let $B(k,n,d)$ denote the set of rectangular semistandard tableaux of shape $k\times d$ with entries $\le n$. We write $\mm\in B(k,n,d)$ both for the monomial and its tableau. For a positroid variety $\Pi_f$, let $\cJ_{f}:= \cJ_{\Pi_f}$ and
\[B_f(k,n,d):=\{\mm\in B(k,n,d): \mm \not\in \inn(\cJ_f) \}.\]
be the set of degree-$d$ standard monomials for $\Pi_f$.

The main theorem of this section is an explicit combinatorial description of $B_f(k,n,d)$.

\begin{thmA}
    A monomial $\mm = \prod_{i = 1}^d [\aa^{(i)}]\in B_f(k,n,d)$ if and only if $\mm\in B(k,n,d)$ and 
    \begin{enumerate}
        \item when $\Pi_f = \Pi_{S\leq r}$ for some  interval $S$, $\mm$ does not contain a \textbf{generalized antidiagonal} for $S\leq r$ (\Cref{def:antidiag});
        \item when $\Pi_f = \Pi_{S\leq r}$ for some wrapped-around interval $S$, $\mm^\vee$ does not contain a generalized antidiagonal for $S^\vee\leq r^\vee$ (\Cref{const: complement-interval});
        \item when $\Pi_f = \bigcap_{i}\Pi_{S_i\leq r_i}$, $\mm\in \bigcap_{i} B_{f_i}(k,n,d)$ where $\Pi_{f_i}$ is the basic positroid variety $\Pi_{S_i\leq r_i}$.
    \end{enumerate}
\end{thmA}
We note that for (3), any positroid variety can be written as a finite intersection of basic poistroid varieties $\Pi_{S_i\leq r_i}$ (see \Cref{prop:basic-intersection}). 

%\subsection{The initial ideal of a basic positroid variety} 

%The goal of this section is to describe the monomials that appear in the initial ideal $\inn(\cJ_f):= \inn_\omega(\cJ_f)$ for the defining ideal $\cJ_f$ of any positroid variety $\Pi_f$ with respect to the monomial order $\omega$, the degree revlex order as described in \Cref{subsec:plucker}. 

Let $\mm = \prod_{i = 1}^d [\aa^{(i)}]$ be a standard monomial of $\gr(k,n)$ where the Pl\"ucker variables $\aa^{(1)} \leq \aa^{(2)} \leq \dots \leq \aa^{(d)}$ are sorted using the partial order in the Pl\"ucker poset.


\begin{definition}\label{def:antidiag}
Fix an interval $S = [\alpha+1,\alpha+m]$ in $[n]$ and some $r<m$. A \textbf{generalized antidiagonal} of $\mm\in B(k,n,d)$ for the rank condition $S\le r$ is a vertical strip in the tableau $\mm$ of size $r+1$ with entries in $S$ that are strictly increasing from NE to SW.
% $a^{(\sigma_{\ell})}_{\rho_1}\dots a^{(\sigma_{1})}_{\rho_{\ell}}$ such that
%     \begin{enumerate}
%         \item[(1)] $\alpha+1\leq a^{(\sigma_{\ell})}_{\rho_1}<\cdots <a^{(\sigma_{1})}_{\rho_{\ell}}\leq \alpha+m$,
%         \item[(2)] $1\leq \sigma_1 \leq \sigma_2\leq \cdots\leq \sigma_{\ell}\leq d$, and
%         \item[(3)] $1\leq \rho_1< \rho_2< \cdots< \rho_{\ell}\leq k$.
%     \end{enumerate}
\end{definition}

\begin{example}
    We illustrate the definition above with a small example. Let $n=5$, $k=3$, and consider the interval positroid variety $\Pi_{[2,4]\le 2}$. The following $\mm\in B(k,n,d)$ contains a generalized antidiagonal for the rank condition $[2,4]\leq 2$:
   
\[\ytableausetup{centertableaux, boxsize=1.1em}
\yshort{{\rr{2}},{\rr{3}},{\rr{4}}}\;, \hskip 1em
\yshort{1{\rr{2}},2{\rr{3}},{\rr{4}}5}\;,\hskip 1em 
    \yshort{1{\rr{2}},3{\rr{3}},{\rr{4}}5}\;, \hskip 1em
    \yshort{1{\rr{2}},{\rr{3}}4,{\rr{4}}5}\;, \hskip 1em
    \yshort{11{\rr{2}},2{\rr{3}}4,{\rr{4}}55}\;.
    \]
    In each monomial, the entries in the generalized antidiagonal are highlighted. By part $1$ of Theorem~A, these are not standard monomials for $\Pi_{[2,4]\leq 2}$.
\end{example}

% When we write a standard monomial $\mm = \prod_{i = 1}^d [\aa^{(i)}]$ of $\k[\gr(k,n)]$ as a semistandard tableau $T$, a generalized antidiagonal consists of $r+1$ strictly increasing entries in the interval $S$ running from northeast to southwest in the tableau with entries $(\rho_i, \sigma_{\ell-i+1})$, weakly increasing in columns and strictly increasing in rows.

To study the standard monomials for a basic positroid variety $\Pi_{S\leq r}$ where $S$ is a wrapped-around interval, we need the following contruction.

%We can relate cyclic interval rank conditions where the intervals ``wrap around'' (i.e. include columns $n$ and $1$) to those which do not using the following construction. %This is because there is a stratification-preserving isomorphism $\varphi$ of varieties between $\gr(k,n)$ and $\gr(n-k,n)$ which preserves the positroid stratification. 

%In coordinates, the isomorphism is written as $\varphi:[\mathbf{a}]\mapsto [\mathbf{a}^\vee]$. It is straightforward to check that the Pl\"ucker relations are preserved under this mapping of coordinates; in particular, if $\cJ_{S\leq r} \subset R(k,n)$ is the defining ideal for a positroid variety $\Pi_{S\leq r}$, then one may define the ideal $\cJ^\vee_{S\leq r} := \cJ_{S^\vee \leq r} \subset R(n-k,n)$, and it is exactly the defining ideal for $\k[\Pi_{S^\vee \leq r^\vee}]$ in $\gr(n-k,n)$.

\begin{construction}\label{const: complement-interval}
Let $S = [\alpha+1,\alpha+m]^\circ$ be a cyclic interval and $r<m$. 
Define the following:
\begin{itemize}
    \item $S^\vee :=[\alpha+m+1,\alpha]^\circ$.
    \item $r^\vee := n-k-m+r$
    \item For any monomial $\mm = \prod_{i = 1}^d [\mathbf{a}^{(i)}]\in R(k,n)$, define $\mm^\vee$ to be $\prod_{i = 1}^d [\mathbf{a}^{(i)\vee}]\in R(n-k,n)$, where $\mathbf{a}^{(i)\vee} := [n]\setminus \mathbf{a}^{(i)}$ for all $i\in [d]$.
    \item For any polynomial $g \in R(k,n)$, define $g^\vee \in R(n-k,n)$ to be the polynomial obtained from $g$ by applying $\vee$ to each monomial summand.
\end{itemize}
\end{construction}



\begin{prop}\label{prop:init-dual}
A monomial $\mm\in \inn_\omega(\cJ_{S\leq r}))$ if and only if $\mm^\vee \in \inn_\omega(\cJ_{S^\vee\leq r^\vee}))$.
\end{prop}






\begin{example}
Fix $\gr(4,6)$ and let $S = [2,5]$ and $r = 3$. 
Then $S^\vee = [6]\setminus S = [6,1]^\circ$ and $r^\vee = 6-4-4+3 = 1$.
Given a monomial $\mm \in \inn_\omega(\cJ_{S\leq r})\subset R(4,6)$ (with the generalized antidiagonal highlighted), we obtain $\mm^\vee \in \inn_\omega(\cJ_{S^\vee\leq r^\vee}) \subset R(2,6)$ by taking complements of each of the Pl\"{u}cker coordinates dividing $\mm$.
For instance,
\[
\yshort{{\rr{2}},{\rr{3}},{\rr{4}}, {\rr{5}}}
 \xleftrightarrow[]{\text{complement}} \yshort{1,6}
\;, \quad \yshort{111{\rr{2}},22{\rr{3}}4,3{\rr{4}}55,{\rr{5}}666} \xleftrightarrow[]{\text{complement}} \yshort{4321,6543} =\yshort{1234,3456}
\]
\end{example}

Using the Proposition below, we prove part $(3)$ of Theorem~A.

\begin{prop}
\label{prop:unique-min}
    Let $\mm:=\prod_{i=1}^d[\aa^{(i)}]\in B(k,n,d)$, where $\aa^{(1)}\le \cdots \le \aa^{(d)}$. Then there exists a unique minimal positroid variety $\Pi_f$ such that $\mm\in B_f(k,n,d)$. In other words, if $\Pi_{f'}$ is a positroid variety such that $\mm \in B_{f'}(k,n,d)$, then $\Pi_f\subseteq \Pi_{f'}$.
\end{prop}

Proof sketch for part $(3)$ of Theorem~A: Notice first that  if $\Pi_f\subseteq \Pi_{f'}$ then  $\cJ_f\supseteq \cJ_{f'}$ and thus $B_f(n,k,d)\subseteq B_{f'}(n,k,d)$. This proves $B_f(k,n,d)\subseteq\bigcap_i B_{f_i}(n,k,d)$. For $\mm \in \bigcap_i B_{f_i}(n,k,d)$, by \Cref{prop:unique-min}, there is a unique minimal $\Pi_g\subseteq \bigcap_i \Pi_{f_i} = \Pi_f$ such that $\mm\in B_g(n,k,d)$. Therefore $\mm\in B_f(n,k,d)$.


% \begin{example}[Standard monomials of a positroid variety] Let $\Pi_f$ be the positroid variety defined in \Cref{ex:positroid-decompose}. Then $\Pi_f=X_{[2]\le 0}\cap X_{[2,4]\le 1} \cap X_{[1,5]\le 2}$. Equipped with \Cref{thm:initial-ideal}, we are ready to describe the Stanley--Reisner complex, denoted $\Delta(\Pi_f)$, of the Hodge degeneration of this positroid variety directly, instead of considering it as a projection of an order complex in the Bruhat order.  The vertices of $\Delta(\Pi_f)$ are labeled by the Pl\"{u}cker coordinates $[136],[146],[156],[356],[456]$. Notice that these form a chain in the Pl\"ucker poset, but they do not form a face of $\Delta(\Pi_f)$ since ${\ytableausetup{centertableaux, smalltableaux}
% \yshort{1{\rr{3}}, {\rr{4}}5,66}}$ contains a generalized antidiagonal for $X_{[2,4]\le 1}$. The facets of $\Delta(\Pi_f)$ correspond to the monomials
% \[{\ytableausetup{centertableaux, boxsize = 1.1em}
% \yshort{1114,3455,6666}}
% \text{ \ \ and \ \ }{
% \yshort{1134,3555,6666}} \; 
% ,\]
% each of dimension 3.
% \end{example}

%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gr\"{o}bner bases for  positroid varieties}\label{section: gb-equations}
%%%%%%%%%%%%%%%%%%%%%%%%

Using straightening relations and our characterizations for standard monomials, we obtain an explicit Gr\"obner basis for positroid varieties.

\begin{thmB}%[Theorem~\Cref{thm: gb-basic-positroid} $+$ \Cref{prop:dual-gb} $+$ \Cref{thm:concat-gb}]
\label{thmB}
Let $\cJ_f$ be the defining ideal for the positroid 
variety $\Pi_f$. $\cJ_f$ has a Gr\"{o}bner basis with respect to $<_\omega$ (see bottom of p.4) consisting of straightening relations (\Cref{def:Pluckerrelations}), and
\begin{enumerate}
    \item [(1)] when $\Pi_f=\Pi_{S\le r}$ for some interval $S$, 
     $\sum_{w\in \symm_{r+1}} (-1)^{\ell(w)} w\cdot \mm$ where $\mm$ minimally contains a generalized antidiagonal for $S\leq r$ and $\symm_{r+1}$ acts on $\mm$ by permuting entries in the generalized antidiagonal;

    \item[(2)] when $\Pi_f=\Pi_{S\le r}$ for some wrapped-around interval $S$, $\sum_{w\in \symm_{r^\vee+1}} (-1)^{\ell(w)} (w\cdot \mm^\vee)^\vee$ where $\mm^\vee$ minimally contains a generalized antidiagonal for $S^\vee\leq r^\vee$; 

    \item[(3)] when $\Pi_f = \bigcap_{i}\Pi_{S_i\leq r_i}$, the union of the Gr\"obner basis for each $\Pi_{S_i\le r_i}$.
\end{enumerate}
 
\end{thmB}

% The following statement is a direct consequence of \Cref{thm:std-mon-intersect}.
% \begin{thm}
% \label{thm:concat-gb}
% %Suppose $\Pi_f=\bigcap_i\Pi_{f_i}$ where each $\Pi_{f_i}$ is a basic positroid variety. Then $\bigcup_i\mathrm{gb}(\cJ_{f_i})$ is a minimal Gr\"obner basis for $\cJ_{f}$.
% A minimal Gr\"{o}bner basis for $\cJ_f$ with respect to the Hodge degeneration is $\bigcup_i\mathrm{gb}(\cJ_{f_i})$ 
% % form a minimal Gr\"obner basis for $\cJ_{f}$ 
% where the $\Pi_{f_i}$ are the basic positroid varieties corresponding to $ess(f)$. 
% \end{thm}

% Therefore, it suffices to find an explicit description of the Gr\"{o}bner bases for basic positroid varieties. The following theorem provides exactly this.



% % \begin{thm}\label{thmB}
% % Let $S = [\alpha+1,\alpha+m]$ in $[n]$. Then the ideal $\cJ_{S\leq r}$ has a minimal Gr\"{o}bner basis with respect to $<_\omega$ with two classes of generators:
% % \begin{enumerate}
% %     \item[(1)] classical quadratic Pl\"{u}cker relations, with leading terms of the form $[\aa]\cdot [\bb]$ where $[\aa]$ and $[\bb]$ are incomparable elements of the Pl\"{u}cker poset and neither $\aa$ nor $\bb$ is contains $r+1$ elements in the interval $S$;
% %     \item[(2)] fix a monomial with a generalized antidiagonal as in \Cref{thm:initial-ideal}, and sum all of the monomials where the elements of the generalized antidiagonal have been permuted (weighted with the appropriate sign).
% % \end{enumerate}
% % Moreover, for a cyclic interval $S = [\alpha+1, \alpha+m]^\circ$, one has that $\mathrm{gb}(\cJ_{S\le r})$ is a minimal Gr\"obner basis of $\cJ_{S\le r}$ if and only if $\{g^\vee: g\in \mathrm{gb}(\cJ_{S\le r})\}$ is a minimal Gr\"obner basis for $\cJ_{S^\vee\le r^\vee}$.
% % \end{thm}

% \begin{proof}[Proof sketch] We first prove that a generalization of the homogeneous polynomials of the form (2) above are indeed in the ideal $\cJ_{S\leq r}$. To do this, proceed by double induction on the degree $d$ of the polynomial and the number of elements $\lambda_d$ of column $d$ of the leading monomial which are contained in the generalized antidiagonal. We apply \Cref{lem: unsigned-plucker-relation} to every term of such a polynomial and use the inductive hypothesis to show that the polynomial is indeed in the ideal.

% Once we have established that the polynomials described above are in the ideal, it is straightforward to check that the monomial ordering $<_\omega$ selects the monomials from \Cref{thm:initial-ideal} as the leading terms, giving the result.
% \end{proof}

% \begin{thm}\label{thm: gb-basic-positroid} 
% Let $S = [\alpha+1,\alpha+m]$ where $\alpha+m\leq n$ and consider the positroid variety $X_{S\leq r}$. 
% Let $<_\omega$ be a degree revlex monomial order with respect to some linear extension of the Pl\"{u}cker poset. 
% Then the defining ideal of $X_{S\leq r}$, denoted $\cJ_{S\leq r}$, has a minimal Gr\"{o}bner basis with two classes of generators:

% \begin{enumerate}
%     \item[(1)] degree $2$ generators coming from the classcal Pl\"{u}cker relations. The leading terms of these generators are of the form $[\aa]\cdot [\bb]$ where $[\aa]$ and $[\bb]$ are incomparable elements of the Pl\"{u}cker poset and neither $\aa$ nor $\bb$ contains $r+1$ elements of $S$;
%     \item[(2)]  generators of the form \begin{equation}\label{eq: gb-equation}
% \sum_{\sigma \in \symm_{r+1}} \sgn(\sigma) \prod_{j=1}^d [a_{j,1} \cdots a_{j,\alpha_j} c_{\sigma(p_{j-1}+1)} \cdots c_{\sigma(p_j)} b_{j,1},\cdots, b_{j ,\beta_j}]
% \end{equation}
% where the following conditions are satisfied:
% \begin{itemize}
%     \item $\cc = \{c_1 < c_2 < \cdots < c_{r+1}\} \subseteq S$,
%     \item $\lambda = (\lambda_1,\cdots, \lambda_d)$ is a composition of $r+1$ with $\lambda_i>0$ for all $i$,
%     \item $a_{j,1}<\cdots < a_{j,\alpha_j} < c_{p_{j-1}+1} < \cdots < c_{p_j} < b_{j,1} \cdots < b_{j,\beta_j}$,
%     \item $\alpha_j\geq \alpha_{j-1}+\lambda_{j-1}$ and $\beta_j = k-\lambda_{j}-\alpha_j$.
%     \end{itemize}
%     The leading terms of the generators of the form (\ref{eq: gb-equation}) contain the generalized antidiagonal $\cc$ as described in \Cref{thm:initial-ideal}.
% \end{enumerate}

% \end{thm}

% \begin{proof}
% The first class of relations is obviously in the ideal, and relations of the form (\ref{eq: gb-equation}) are in the ideal by \Cref{prop: positroid-equations}. Moreover, each of these equations has exactly one term of the form described in \Cref{thm:initial-ideal}.

% It remains to check that $<_\omega$ picks out the desired leading terms. This is clear for the usual Pl\"{u}cker relations. To see this for the relations of the form (\ref{eq: gb-equation}), observe that swapping any $c_i$ from the bottom row to a higher row will automatically force the bottom row to contain a larger number, so it will have higher rank in the Pl\"{u}cker poset $\cP$ than the original smallest variable did before swapping. Since $<_\omega$ is a revlex monomial ordering on a linear extension of $\cP$, this implies that the leading term must be of the form described in \Cref{thm:initial-ideal}.
% \end{proof}
\begin{example}
    Let $n=5$, $k=3$, $S=[2,4]$, $r=2$, and consider the basic positroid variety $X_{[2,4]\le 2}$. The generators of the Gr\"obner basis of $\cJ_{[2,4]\le 2}$ which are not Pl\"ucker relations for $\gr(3,5)$ consist of the following:

\[{\ytableausetup{centertableaux, boxsize = 1.1em}
\yshort{{\rr{2}},{\rr{3}},{\rr{4}}} \; ,\hskip 2em
\yshort{1{\rr{2}},2{\rr{3}},{\rr{4}}5}\; - \; \yshort{1{\rr{2}},2{\rr{4}},{\rr{3}}5}\; ,\hskip 1.5em 
    \yshort{1{\rr{2}},3{\rr{3}},{\rr{4}}5}\; - \; \yshort{1{\rr{3}},{\rr{2}}{\rr{4}},35} \; , \hskip 1.5em
    \yshort{1{\rr{2}},{\rr{3}}4,{\rr{4}}5} \; - \;\yshort{1{\rr{3}},{\rr{2}}4,{\rr{4}}5} \; ,
    }\]

 \[{\ytableausetup{centertableaux}
\yshort{11{\rr{2}},2{\rr{3}}4,{\rr{4}}55}\;-\;\yshort{11{\rr{2}},2{\rr{4}}4,{\rr{3}}55} \;- \;\yshort{11{\rr{3}},2{\rr{2}}4,{\rr{4}}55}
 \; . }\]
    % A generalized antidiagonal consists of $r+1$ strictly increasing  entries with values in $S$ running NE to SW in the tableau, strictly in rows and weakly in columns. The terms are ordered by $<_\omega$ in each generator. The leadterm of every generator is the first term, which  contains a generalized antidiagonal with entries 2, 3, 4. In each term the entries in the permuted generalized antidiagonal are highlighted. 
    % and consider $X_{[5,1]^\circ \le 1}$. 
    % Then $\varphi(X_{[5,1]^\circ \le 1})=X_{[2,4]\le 2}\subset \gr(3,5)$. 
    We see above that there is a single degree 3 generator of $\mathrm{gb}(\cJ_{[2,4]\le 2})$.
%     \[{\ytableausetup{centertableaux, boxsize = 1.1em}
% \yshort{11{\rr{2}},2{\rr{3}}4,{\rr{4}}55} \; - \; \yshort{112,244,355}\; - \; \yshort{113,224,455} \;
%  .}\] 
 Therefore, we also know that we have a degree 3 generator of $\mathrm{gb}(\cJ_{[5,1]^\circ\le 1})$:
 \[\ytableausetup{centertableaux, boxsize = 1.1em}\yshort{123,345} \; - \; \yshort{124,335} \; - \; \yshort{133,245} \; .\]
\end{example}

\section{Promotion on standard monomials}
\label{sec:promotion}



    Our next application concerns \emph{promotion} and \emph{evacuation} introduced by Sch\"utzenberger \cite{evacuation,promotion}. 
Evacuation appears in the theory of canonical bases where the bijection by Berenstein-Zelevinsky \cite{BZ96} between canonical bases and semistandard tableaux sends Lusztig's involution to evacuation.
Stembridge \cite{Ste94,Ste96} observed the \emph{$q=-1$ phenomenon}, meaning that the number of fixed points under an involution is the value of a polynomial $f(q)$ at $q=-1$.
Motivated by the work of Stembridge, Reiner--Stanton--White \cite{RSW04} introduced the \emph{cyclic sieving phenomenon} for the enumeration of fixed points under cyclic group actions. Rhoades \cite{rhoades2010cyclic} showed that promotion on $B(k,n,d)$ exhibits the cyclic sieving phenomenon, where the number of fixed tableaux after applying promotion $a$ times is the value of a polynomial at the roots of unity $\xi_n^a$.
More recently, these operations are of interest in the study of dynamical algebraic combinatorics (see e.g., the survey
\cite{Roby} and the references therein). 
 
 We relate promotion and evacuation to the cyclic and reflection symmetry of positroid varieties, respectively.

\begin{definition}
Let \textbf{promotion} be the map $\mathrm{prom}:B(k,n,d)\longrightarrow B(k,n,d)$ defined as follows:
\begin{itemize}
    \item[-] If $\mm\in B(k,n,d)$ does not contain $n$, then increase each entry of $\mm$ by $1$.
    \item[-] If $\mm$ contains $n$, replace each $n$ with $\bullet$ and perform the \textbf{jeu de taquin (jdt)} slide:
    \begin{align*}
        \begin{ytableau}
            a & c\\
            b & \bullet
        \end{ytableau}\longrightarrow 
        \begin{cases}
            \begin{ytableau}
            a & \bullet\\
            b & c
        \end{ytableau}& \text{ if }b\leq c \text{ or }a,b \text{ do not exist},\vspace{0.5cm}\\
        \ytableausetup{boxsize=1em}
            \begin{ytableau}
            a & c\\
            \bullet & b
        \end{ytableau}& \text{ if }b> c\text{ or }a,c \text{ do not exist}
        \end{cases}
    \end{align*}
    until no longer possible. Replace each $\bullet$ with $0$ and increase all entries by $1$. 
\end{itemize}
\end{definition}

\begin{definition}
Let \textbf{evacuation} be the map $\evac:B(k,n,d)\to B(k,n,d)$ defined by:
\begin{enumerate}
    \item replace every entry $x$ with $n+1-x$,
    \item rotate the tableau by $180^\circ$. 
\end{enumerate}
\end{definition}


Our main theorem of this section is
the following. %Recall the notation for $B(k,n,d)$ and $B_f(k,n,d)$ introduced at the start of Subsection~\ref{subsec: B(k,n,d)}.

\begin{thmC}
    Promotion (resp. evacuation) bijects the set of standard monomials of a positroid variety $\Pi_f$ and those of its cyclic shift $\chi(\Pi_f)$ (resp. its reflection $\Pi_{f^*}=w_0\cdot \Pi_f$).
\end{thmC}

Since $\mathrm{prom}$ and $\evac$ are both bijections from $B(k,n,d)$ to $B(k,n,d)$, Theorem~C is equivalent to showing $\mathrm{prom}$ (resp. $\evac$) bijects $\inn(\cJ_f)$ with $\inn(\cJ_{\chi(f)})$ (resp. $\inn(\cJ_{f^*})$). Here we give an example to see how promotion (resp. evacuation) maps monomials in $\inn(\cJ_f)$ to $\inn(\cJ_{\chi(f)})$ (resp. $\inn(\cJ_{f^*})$).

%Since promotion and evacuation are both bijections from $B(k,n,d)$ to $B(k,n,d)$, we will give an example to see how promotion (resp. evacuation) maps monomials in $\inn(\cJ_f)$ to $\inn(\cJ_{\chi(f)})$ (resp. $\inn(\cJ_{f^*})$).

\begin{example}\label{ex:promotion}
    Consider the basic positroid variety $\Pi_f = \Pi_{[2,4]\leq 1}\subset \gr(3,7)$, then $\chi(\Pi_f) = \Pi_{[3,5]\leq 2}$ and $w_0\cdot \Pi_f = \Pi_{[4,6]\leq 2}$.  For $\mm = [1,3,4]\cdot [2,5,7]\in \inn(\cJ_f)$, we have $\mathrm{prom}(\mm) = [1,2,3]\cdot [3,4,5]\in \inn(\cJ_{\chi(f)})$ and $\evac(\mm) = [1,3,6]\cdot [4,5,7]\in \inn(\cJ_{f^*})$. 
     We demonstrate the process below:  \[\mathrm{prom}:
    \ytableausetup{centertableaux,boxsize = 1.1em}
\yshort{1{\rr{2}},{\rr{3}}5,{\rr{4}}7}\rightarrow 
\yshort{1{2},3{5},4{\bullet}}\rightarrow \yshort{1{2},3{\bullet},4{5}}\rightarrow
\yshort{1{2},\bullet{3},4{5}}\rightarrow
\yshort{\bullet{2},1{3},4{5}}\rightarrow
\yshort{0{2},1{3},4{5}}\rightarrow
\yshort{1{\rr{3}},2{\rr{4}},{\rr{5}}{6}}.\]
\[\evac: \ytableausetup{centertableaux,boxsize = 1.1em}
\yshort{1{\rr{2}},{\rr{3}}5,{\rr{4}}7} \rightarrow \yshort{76,53,41} \rightarrow \yshort{1{\rr{4}},3{\rr{5}},{\rr{6}}7}\]
\end{example}



\section{Hilbert series and cyclic Demazure characters}

Motivated by the cyclic symmetry of the Grassmannian and its positroid varieties, Lam introduced \text{cyclic Demazure modules} $V_f(d\omega_k)$ as the cyclic intersections of Demazure modules. They are isomorphic to the space of sections $H^0(\Pi_f, \mathcal{L}_{d\omega_k})$ as $T$-modules.

Let $\omega_k$ be the $k$-th fundamental weight, $\mathcal{L}_{\omega_k}$ the associated line bundle on $\gr(k,n)$ and $\mathcal{L}_{d\omega_k}:= \mathcal{L}_{\omega_k}^{\otimes d}$. Let $\mathrm{res}_f:H^0(\gr(k,n),\mathcal{L}_{d\omega_k})\rightarrow H^0(\Pi_f,\mathcal{L}_{d\omega_k})$ be the restriction map.

\begin{prop}
    The map $\mathrm{res}_f$ is surjective and $H^0(\Pi_f,\mathcal{L}_{d\omega_k})\cong \k[\Pi_f]_d $ as $T$-modules.
\end{prop}

We establish a recurrence on the multigraded Hilbert series of positroid varieties with the grading induced by the standard $T$-action on the Grassmannian. This allows us to give an inductive formula for the $T$-character of cyclic Demazure modules, answering a question of Lam \cite[Problem 24]{Lam19}. 

\begin{thmD}
For $f\in \bound(k,n)$, if $d=0$, then
    $\ch(V_f(d\omega_k)) = 1$. Otherwise,
\[
        \ch (V_f(d\omega_k))=\mathbf{t}^{\topp(f)}\ch (V_f((d-1)\omega_k)+\sum_{f'\in \Lenart_0(f)\cap \bound(k,n)}(-1)^{\ell(f')-\ell(f)+1} V_{f'}(d\omega_k).
        \]
\end{thmD}
Here $\topp(f)\in\binom{[n]}{k}$ corresponds to the smallest Pl\"ucker coordinate that does not vanish on $\Pi_f$, $\mathbf{t}^{\topp(f)}:= \prod_{i\in \topp(f)}t_i$, and $\Lenart_0(f)$ is derived from Lenart's $K$-theoretic Monk's rule for Grothendieck polynomials defined below.

\begin{definition}\label{def:L_0}
Let $\Gamma_0(f)$ denote the set of saturated chains in the Bruhat order of affine permutations:
\[f\lessdot ft_{a_1,b_1}\lessdot ft_{a_1,b_1}t_{a_2,b_2} \lessdot\cdots \lessdot ft_{a_1,b_1}t_{a_2,b_2}\cdots t_{a_m,b_m},\]
where $a_i \leq 0, b_i\in [1,n]$, and the pairs $(a_i,b_i)$ satisfies either ($b_i>b_{i+1}$) or $(b_i=b_{i+1} \text{ and } a_i<a_{i+1})$. 
    Let $\Lenart_0(f)$ be the set of permutations that are endpoints of chains in $\Gamma_0(f)$.
\end{definition}





%Assign each Pl\"ucker variable $[\aa]$ where $\aa\in\binom{[n]}{k}$ degree $(d_1,\cdots,d_n)$ where $d_i=0$ if $i\not\in \aa$ and $d_i=1$ if $i\in \aa$. 
%For a bounded affine permutation $f\in \bound(k,n)$, define $\topp(f)\in \binom{[n]}{k}$ to be smallest Pl\"{u}cker coordinate with respect to $<$ such that $[\topp(f)]$ does not vanish on $\Pi_f$. In other words, $\topp(f)$ is the smallest index of a torus-fixed point of $\Pi_f$.

\begin{prop} [{\cite[Corollary 7.3]{KLS13juggling}}]
Let $C(f) := \{f'\in \bound(k,n): f\lessdot f'=ft_{a,b}, a\le 0<b\}$. Then the following formula holds scheme-theoretically:
\begin{equation}
\label{eqn:cotrans}
    \{[\topp(f)]=0\} \cap \Pi_f = \bigcup_{f'\in C(f)}\Pi_{f'}.
\end{equation}
\end{prop}

Consider the $\ZZ^n$-grading on the polynomial ring $R(k,n)$ such that for any $\aa\in\binom{[n]}{k}$, the degree of the Pl\"ucker variable $[\aa]$ is $(d_1,\cdots,d_n)$ where $d_i=0$ if $i\not\in \aa$ and $d_i=1$ if $i\in \aa$. 
For $\mm\in B_f(k,n,d)$, define $\content(\mm):=(c_1,\cdots,c_n)$ where $c_i$ counts the number of appearance of $i$ in $\mm$. Set $\mathbf{t}^{\content(\mm)}:=\prod_{i=1}^n t_i^{c_i}$. 
\vskip 0.6em

\noindent\textbf{Key Observation:} By definition, the Hilbert series 
\[\hilb(R/\cJ_f;\mathbf{t})=\sum_{d=0}^\infty \sum_{\mm \in B_f(k,n,d)}\mathbf{t}^{\content(\mm)},\]
and the degree $d$ part of $H(R/\cJ_f;\mathbf{t})$ is the character of $V_f(d\omega_k)$. Since $[\topp(f)]$ is not a zero divisor of $R/\cJ_f$, the Hilbert series of the LHS of \eqref{eqn:cotrans} is
\begin{equation}\label{eqn:Nov15}
    \hilb(R/(\cJ_f+\langle [\topp(f)]\rangle); \mathbf{t}) = (1-\mathbf{t}^{\topp(f)}) \hilb(R/\cJ_f;\mathbf{t}).
\end{equation}
For the RHS, we have

\begin{prop}\label{prop:hilb}
\[\hilb(R/(\bigcap_{f'\in C(f)}\cJ_f');\mathbf{t})=\sum_{f'\in \mathcal{L}_0(f)\cap \bound(k,n)}(-1)^{\ell(f')-\ell(f)+1}\hilb(R/\cJ_f'; \mathbf{t}),\]
\end{prop}
The proof of the above proposition relies on proving  an isomorphism of posets of varieties, generated by taking the closure of the ``intersect and decompose'' operation on the varieties appearing in the Monk's rule for positroid and matrix Schubert varieties.

%The technical part of the proof for this proposition consists of justification for why we can use Monk's rule for Grothendieck polynomials. To do this, we rely on two main ingredients: 1) matrix Schubert varieties (MSVs) and positroid varieties are Frobenius split subvarieties, and 2) 

Theorem~D then follows from taking the degree $d$ piece of the Hilbert series on both sides of \eqref{eqn:cotrans} by combining \eqref{eqn:Nov15} and \Cref{prop:hilb}.

\acknowledgements{We thank Allen Knutson, Thomas Lam, and David Speyer for inspiring conversations and helpful comments. DH would like to thank ICERM for hosting the Combinatorial Algebraic Geometry Reunion Event, where many productive conversations happened.}

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