%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2023}
\articlenumber{37}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% OUR PACKAGES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{mathtools,enumitem}
%\usepackage[usenames,dvipsnames,svgnames,table]{xcolor}
\usepackage[dvipsnames]{xcolor}

\usepackage{valuediagrams}
\setkeys{valuediagram}{
    classic color=DarkGreen,
    quantum color=DarkRed,
    fontsize=\tiny,
    xscale=0.4,
    yscale=0.2,
    critical windows=None,
    critical window color=Yellow!50,
    critical window padding=0.1,
    node size=3pt,
    padding=0.5,
    xmin=1,
    xmax=None,
    height=None,
    thickness={very thick},
    arrow={-},
    baseline=0pt,
    tikzpicture options={},
}%


\newcommand{\Fl}{{\mathbb F}\ell}
\newcommand{\Fln}{{\mathbb F}\ell_n}
\newcommand{\CC}{{\mathbb C}}
\newcommand{\NN}{{\mathbb N}}
\newcommand{\ZZ}{{\mathbb Z}}

\newcommand{\frakS}{{\mathfrak S}}
\newcommand{\calF}{{\mathcal F}}
\newcommand{\calL}{{\mathcal L}}
\newcommand{\calM}{{\mathcal M}}
\newcommand{\qu}{{\mathbf u}}
\newcommand{\qv}{{\mathbf v}}
\newcommand{\bF}{{\mathbf F}}
\newcommand{\bT}{{\mathbf T}}
\newcommand{\bR}{{\mathbf R}}
\newcommand{\bC}{{\mathbf C}}
\newcommand{\bzero}{{\mathbf 0}}
\newcommand{\bx}{{\mathbf x}}
\newcommand{\precdot}{{\prec\!\!\!\cdot\,}}

\newcommand{\ie}{\textit{i.e.}}

\DeclareMathOperator{\het}{ht}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\nd}{end}
\DeclareMathOperator{\Lab}{{\it Lab}}
\DeclareMathOperator{\FK}{{\it FK}}
\DeclareMathOperator{\qFK}{{\it qFK}}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Span}{Span}

\DeclareMathOperator{\kactionop}{{\scriptstyle \bullet}}
\newcommand{\kaction}[1][k]{\kactionop_{#1}}

\newcommand{\FireBrick}[1]{{\color{BrickRed}#1}}
\newcommand{\defcolor}[1]{{\color{blue!80!black}#1}}
\newcommand{\demph}[1]{\defcolor{{\sl #1}}}
\newcommand{\ncycle}{\mathfrak o}
\newcommand{\hookGamma}{\mathsf{\Gamma}}

\def\R{\mathbf{R}}
\def\C{\mathbf{C}}
\def\F{\mathbf{F}}
\def\T{\mathbf{T}}
\def\v{\mathbf{v}^\bullet}
\def\u{\mathbf{u}^\bullet}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% TODO / NOTES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{xargs}
\usepackage[colorinlistoftodos,prependcaption,textsize=tiny,linecolor=red,backgroundcolor=red!25,bordercolor=red]{todonotes}
\setlength{\marginparwidth}{2cm}
\newcommandx{\franco}[2][1=]{\todo[linecolor=blue,backgroundcolor=blue!25,bordercolor=blue,#1]{#2 ---Franco}}
\newcommandx{\laura}[2][1=]{\todo[linecolor=orange,backgroundcolor=orange!25,bordercolor=orange,#1]{#2 ---Laura}}
\newcommandx{\caro}[2][1=]{\todo[linecolor=purple,backgroundcolor=purple!25,bordercolor=purple,#1]{#2 ---Caro}}
\newcommandx{\nantel}[2][1=]{\todo[linecolor=red,backgroundcolor=red!25,bordercolor=red,#1]{#2 ---Nantel}}
\newcommandx{\frank}[2][1=]{\todo[linecolor=green,backgroundcolor=green!35,bordercolor=green,#1]{#2 ---Frank}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Environments %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{openprob}[theorem]{Open Problem}

\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}

%% define your title in the usual way
\title[Quantum Murnaghan--Nakayama rule]{A Murnaghan--Nakayama rule for quantum cohomology of the flag manifold}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[Benedetti, Bergeron, Colmenarejo, Saliola, Sottile]{%
    C.~Benedetti\thanks{\href{mailto:c.benedetti@uniandes.edu.co}{c.benedetti@uniandes.edu.co}. C. Benedetti was partially supported by the grant FAPA of the Faculty of Science of U. de Los Andes.}\addressmark{1},
    N.~Bergeron\thanks{\href{mailto:bergeron@yorku.ca}{Bergeron@yorku.ca}. N.~Bergeron was partially supported by NSERC and York Research Chair.}\addressmark{2},
    L.~Colmenarejo\thanks{\href{mailto:lcolmen@ncsu.edu}{lcolmen@ncsu.edu}.}\addressmark{3},\\
    F.~Saliola\thanks{\href{mailto:saliola.franco@uqam.ca}{saliola.franco@uqam.ca}. F.~Saliola was partially supported by an NSERC Discovery Grant.}\addressmark{4}
    \and
    F.~Sottile\thanks{\href{mailto:sottile@tamu.edu}{sottile@tamu.edu}. F.~Sottile was partially supported by NSF grant DMS-2201005.}\addressmark{5}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Universidad de Los Andes, Bogot\'a, Colombia\\
\addressmark{2}York University, Toronto, Canada\\
\addressmark{3}North Carolina State University, Raleigh, USA\\
\addressmark{4}Universit\'e du Qu\'ebec \`a Montr\'eal, Montr\'eal, Canada\\
\addressmark{5}Texas A\&M University, College Station, USA}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{We give a rule for the multiplication of a Schubert class by
    a tautological class in the (small) quantum cohomology ring of the full
    flag manifold. As an intermediary step, we establish a formula for the
    multiplication of a Schubert class by a quantum Schur polynomial indexed by
a hook partition.}

%% put your French abstract here, or comment this out if you don't have one
\resume{On d\'{e}crit une r\`{e}gle pour la multiplication d'une classe de
    Schubert par une classe tautologique dans le (petit) anneau de cohomologie
    quantique de la vari\'{e}t\'{e} de drapeaux. En cours de route, on obtient
    une formule pour la multiplication d'une classe de Schubert par un
    polyn\^{o}me de Schur quantique correspondant \`{a} une \'{e}querre.}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Murnaghan--Nakayama Formula, Schubert Polynomials.}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
\addbibresource{references.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\frenchspacing

\begin{document}

\maketitle


The \emph{Murnaghan--Nakayama rule}
is a method for computing values of the irreducible characters of symmetric groups.
Under the identification of the character ring with the
ring of symmetric functions, it translates into a rule for the multiplication
of a Schur symmetric function $s_\lambda$ by a Newton power sum $p_r$.
As explained in~\cite[Sect.\ 2]{MorrisonSottile}, in the cohomology ring of the full flag manifold,
the Newton power sums
correspond to tautological classes that are components of the Chern character.
Accordingly, a formula describing multiplication by a tautological class is
also called a \emph{Murnaghan--Nakayama rule} since it corresponds to
multiplication by a Newton power sum.

Here, in \cref{C:quantumMN} we describe a rule for multiplication by a tautological class in the
small quantum cohomology ring of the full flag manifold. 
As an intermediary step, we show in 
\cref{thm:q_hook_product} a formula for multiplication by a hook quantum Schur
polynomial. This entails a detailed analysis of chains and intervals in the
quantum Bruhat order. Making use of results by N.~C.~Leung and C.~Li
\cite{LeungLi2012} and A.~Postnikov \cite{Postnikov_symmetry}, it turns out
that quantum products by hook Schur polynomials can be reduced to the
classical product from~\cite{So96}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Murnaghan--Nakayama rule for (classic) cohomology}

We begin by reviewing combinatorial models for the cohomology ring of the flag
ma-nifold. (We omit the precise definitions of the flag manifold and its
cohomology as the combinatorial models will suffice for our purposes;
the interested reader can consult~\cite{Fu97}.)
Then we will derive a formula for the product of a Schubert polynomial and
a Newton power sum, which will be instructive in deriving our
quantum analogue.

Let \defcolor{$\Fln$} denote the manifold of complete flags in $\CC^n$.
Its cohomology ring \defcolor{$H^*\Fln$} with coefficients in $\ZZ$
is a $\ZZ$-module with a distinguished basis
\defcolor{$\{\frakS_w\}_{w \in S_n}$} whose elements $\frakS_w$ are called
\demph{Schubert classes} and are indexed by elements of the symmetric group
\defcolor{$S_n$}.

Borel~\cite{Borel} gave a presentation of $H^* \Fln$ as a quotient of a polynomial ring:
specifically,
$H^* \Fln \simeq \ZZ[x_1,\dotsc,x_n] / I_n$, where
$I_n = \langle e_l(x_1,\dotsc,x_n)\mid l \in [n] \rangle$ where $e_l$ is the $l$th elementary symmetric polynomial
and $[n]=\{1,\dots,n\}$.
% so that we can write $H^*\Fln \simeq \ZZ[x] / \langle e_1(x), \ldots, e_n(x) \rangle$.

Lascoux and Sch\"utzen\-ber\-ger~\cite{LaSch82} defined a set of polynomials in
$\ZZ[x]$ called \demph{Schubert polynomials} that correspond to the Schubert
classes $\frakS_w$ under Borel's isomorphism. We write $\frakS_w(x)$ for the
Schubert polynomial associated with $\frakS_w$ on the set of variables $x: = \{x_1, \ldots, x_n\}$.
Every Schur polynomial $s_\lambda(x_1, \ldots, x_k)$ is
a Schubert polynomial, as follows.
Let $w\in S_n$ be a permutation with a unique descent at $k$.
Then the sequence $(w(k){-}k,\dotsc, w(2){-}2,w(1){-}1)$ is a partition contained in the rectangular
partition \defcolor{$R_{k,n-k}$} of $k(n{-}k)$ into $k$ parts with each part of
size $n{-}k$.
This is a bijection and
for $\lambda \subseteq R_{k, n-k}$ we write $\defcolor{v(\lambda,k)}\in S_n$ for
the corresponding permutation.
Then $\frakS_{v(\lambda, k)}(x) = s_\lambda(x_1, \ldots, x_k)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{openprob}\label{OP:general rule}
    Find a combinatorial rule expressing $\frakS_u \cdot \frakS_v$
    in terms of the basis \linebreak $\{ \frakS_w \}_{w \in S_n}$ of $H^*\Fln$.
    That is, find a combinatorial rule for the coefficients $c_{u,v}^{w}\in\ZZ$ in the product
    \begin{equation}
        \label{eq:classical-coeffs}
        \frakS_u(x) \cdot \frakS_v(x) = \sum_{w} c_{u, v}^{w} \frakS_w(x).
    \end{equation}
\end{openprob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A remarkable property of the Schubert polynomials is that
the computation in \cref{OP:general rule} is performed in the polynomial ring, not in the quotient.
The resulting coefficients $c_{u, v}^w$ in the quotient will be the same.

Although Problem~\ref{OP:general rule} remains open in general, several special cases are known.
\demph{Monk's Formula}~\cite{Monk} treats the case where $v$ is the simple transposition $(k, k+1)$:
for any $u \in S_n$, we have $\frakS_u\cdot \frakS_{(k,k+1)} = \sum \frakS_{u (i,j)}$
with the sum ranging over all transpositions $(i, j)$ satisfying $i\leq k<j \leq n$ and
$\ell(u)+1=\ell(u (i,j))$, where $\defcolor{\ell(u)}:=\#\{i<j\mid u(i)>u(j)\}$.

The terms appearing in Monk's Formula define a partial order on $S_n$ graded by $\ell(u)$.
The \demph{$k$-Bruhat order $\leq_k$} is defined by the covering relation
$\defcolor{u\lessdot_k u (i,j)}$ if $i\leq k<j \leq n$ and $\ell(u)+1=\ell(u(i,j))$; that is,
$u\lessdot_k u (i,j)$ if $\frakS_{u(i,j)}$ appears in the product $\frakS_{u}(x) \cdot \frakS_{(k,k+1)}$.
%The \demph{Bruhat order $\leq$} is the union of the $k$-Bruhat orders over all $k$.
%
%  We do not use this -F
%



Write \defcolor{$[u,w]_k$} for the interval between $u$ and $w$
in the $k$-Bruhat order.
Label each cover relation $u\lessdot_k u(i,j)$ by $u(i)$
and write $\defcolor{u \xrightarrow{\,\raisebox{0pt}[0pt][0pt]{\text{\scriptsize$u(i)$}}\,} u(i,j)}$.
The maximal element of a (saturated) chain
$\gamma = (u\xrightarrow{a_1}u_1\xrightarrow{a_2}\dotsb\xrightarrow{a_r}u_r)$
is denoted $\defcolor{\nd(\gamma)} = u_r$ and this chain has \demph{length $r$}.
If $a_1>\dotsb >a_l<a_{l+1}<\dotsb<a_{r}$, then $\gamma$ is a \demph{peakless chain of height $l$ and length $r$}.


The following case of \cref{OP:general rule} was  proved in~\cite{So96}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
    \label{pieri-formulas-version-2}
    Let $u \in S_n$,  $l\leq k$, and $m\leq n{-}k$. Then
    \begin{equation}
        \label{eq:schubert-times-hook-via-peakless-chains}
        \frakS_u(x)\cdot s_{(m,1^{l-1})}(x_1,\dotsc,x_k)\ =\ \sum_\gamma \frakS_{\nd(\gamma)}(x)
    \end{equation}
    summing over all peakless chains of height $l$ and length $m+l-1$ in the labeled $k$-Bruhat order.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To describe the multiplicities in \eqref{eq:schubert-times-hook-via-peakless-chains}, we
define the \demph{Grassmannian--Bruhat order} on $S_n$ by
    $\defcolor{\eta\preceq\zeta}$
    if
    {there are a permutation $u \in S_n$ and an integer $k$ with $u\leq_k \eta u\leq_k\zeta u$.}
This is a graded partial order on $S_n$ with minimal element the identity
permutation $e$.
Its rank function is ${\calL(\zeta)}:=\ell(\zeta u)-\ell(u)$ for any $u \in
S_n$ and $k \in \NN$ with $u\leq_k\zeta u$.
The cover relations also inherit the labelling from the $k$-Bruhat orders.
As shown in~\cite{BS-Bruhat}, neither $\preceq$, $\calL$ nor the labelling depend on the choice of $k$ or $u$.

We say that a permutation $\zeta$ is \demph{minimal} if
$\calL(\zeta) = \#\supp(\zeta) - s(\zeta)$,
where $\defcolor{\supp(\zeta)} = \{i\in[n]\mid\zeta(i)\neq i\}$
and $s(\zeta)$ is the number of nontrivial cycles in the factorization of $\zeta$ into disjoint cycles.
The \demph{height} of a minimal permutation $\zeta$ is $\defcolor{\het(\zeta)} := \#\{i\in[n]\mid i<\zeta(i)\}$.
Using results from~\cite{BS-iso}, the following characterizes minimal permutations in terms of the
Grassmannian--Bruhat order.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{L:peaklessChains}
 A permutation $\zeta$ is minimal if and only if the interval $[e,\zeta]_\preceq$ has a peakless chain.
 The number of peakless chains of height $l$ in $[e,\zeta]_\preceq$ is the binomial coefficient
 $\left(\begin{smallmatrix} s(\zeta)-1 \\ \het(\zeta)-l \end{smallmatrix}\right)$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\cref{pieri-formulas-version-2} now reads as follows.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{P:classical_hook}
    Let $u \in S_n$, $l\leq k$ and $m\leq n{-}k$.
    Then
    \begin{equation}
        \label{Eq:Schubert-times-hook}
        \frakS_u(x) \cdot s_{(m, 1^{l-1})}(x_1,\dotsc,x_k)\ =\
        \sum
        \left(\begin{smallmatrix} s(\zeta)-1 \\ \het(\zeta)-l \end{smallmatrix}\right)
        \frakS_{\zeta u}(x) \,,
    \end{equation}
    summing over all minimal permutations $\zeta \in S_n$ such that
    $u\leq_k\zeta u$ and $\calL(\zeta)=m+l-1$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

We use this to describe the product of a Schubert polynomial and a power sum symmetric polynomial.
Let $\defcolor{p_r(x_1,\dotsc,x_k)}:=x_1^r+\dotsb+x_k^r$ denote the $r$th power sum polynomial.
Expanded in terms of Schur polynomials~\cite{Macdonald}, we have\vspace{-5pt}
%
 \begin{multline}\label{Eq:PowerSumHook}
    \qquad     p_r(x_1, \ldots, x_k) = \\
    s_{(r)}(x_1, \ldots, x_k) - s_{(r-1,1)}(x_1, \ldots, x_k) + \cdots + (-1)^{r+1} s_{(1^r)}(x_1, \ldots, x_k).
    \qquad
\end{multline}
%
\begin{corollary}
    \label{C:classical-schubert-times-powersum}
    Let $u \in S_n$.
    Then
     \begin{equation*}
      \frakS_u(x) \cdot p_r(x_1,\dotsc,x_k)\ =\ \sum (-1)^{\het(\zeta)+1} \frakS_{\zeta u}(x)\,,
     \end{equation*}
    %
    summing over all minimal cycles $\zeta \in S_n$ such that
    $u\leq_k\zeta u$ and $\calL(\zeta)=r$.
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}
  Expanding the  product $\frakS_u(x) \cdot p_r(x_1,\dotsc,x_k)$  using Equations \eqref{Eq:PowerSumHook}
  and \eqref{Eq:Schubert-times-hook} shows  that
the coefficient of $\frakS_{\zeta u}$ for $u\leq_k\zeta u$ is zero unless
 $\calL(\zeta)=r$ and $\zeta$ is minimal.
 When $\zeta$ is the product of $s(\zeta)$ minimal cycles whose supports are disjoint, this coefficient is
\[
   (-1)^{\het(\zeta)+1} \sum_{l=1}^{r} (-1)^{l-\het(\zeta)}\binom{s(\zeta)-1}{\het(\zeta)-l}\ =\ \begin{cases}
            0,  & \text{ if } s(\zeta)\neq 1,\\
           (-1)^{\het(\zeta)+1},& \text{ if } s(\zeta)=1.
       \end{cases}
       \qedhere
\]
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This proof differs from that given in~\cite{Morrison,MorrisonSottile}.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Murnaghan--Nakayama rule for quantum cohomology}

The (small) quantum cohomology ring \defcolor{$qH^*\Fln$} of the flag manifold
is a deformation the cohomology ring $H^*\Fln$ whose product
encodes the three-point Gromov--Witten invariants of $\Fln$.
To simplify notation, write \defcolor{$\ZZ[q]$} for $\ZZ[q_1,\dotsc,q_{n-1}]$,
where $q_1,\dotsc,q_{n-1}$ are indeterminates.
As a $\ZZ$-module, we have \hbox{$qH^*\Fln=\ZZ[q] \otimes_\ZZ H^*\Fln$}.
It follows that the Schubert classes $\{\frakS_w\}_{w \in S_n}$ in $H^*\Fln$ form a $\ZZ[q]$-basis of $qH^*\Fln$.
We write $\defcolor{\ast}$ to distinguish this \demph{``quantum product''} from the \demph{``classical product''} in $H^*\Fln$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{openprob}\label{OP:qgeneral rule}
  Find a combinatorial rule for the coefficients $C_{u, v}^{w}\in\ZZ[q]$ in the product
    \begin{equation}
        \label{eq:quantum-coeffs}
        \frakS_u \ast \frakS_v = \sum_{w \in S_n} C_{u, v}^{w} \frakS_w.
    \end{equation}
\end{openprob}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Givental and Kim~\cite{GiventalKim} proved an analogue of Borel's
presentation, expressing $qH^*\Fln$ as a quotient of $\ZZ[q][x]$.
We write $\psi: \ZZ[q][x] \to qH^*\Fln$ for the corresponding surjection.
Fomin, Gelfand, and Postnikov~\cite{FGP} defined \demph{quantum Schubert polynomials}
$\defcolor{\frakS^q_w(x)}\in \ZZ[q][x]$ so that the quantum product of
Schubert classes satisfies $\frakS_u \ast \frakS_v = \psi(\frakS^q_u(x) \cdot \frakS^q_v(x))$.
To simplify notation, for $f(x) \in \ZZ[q][x]$, we will write
$ \frakS_u \ast f(x) $ for $\psi( \frakS^q_u(x) \cdot f(x))$.

As we already mentioned, if $\lambda \subseteq R_{k, n-k}$, then
$s_\lambda(x_1, \ldots, x_k) = \frakS_{v(\lambda, k)}(x)$.
Analogously, for $\lambda \subseteq R_{k, n-k}$, define the \demph{quantum Schur polynomials} as
\[\defcolor{s_\lambda^q(x_1,\dotsc,x_k)} := \frakS^q_{v(\lambda,k)}(x),\]
and following Equation~\eqref{Eq:PowerSumHook}, define the \demph{quantum power sum} as:
%
 \begin{multline}\label{Eq:Q-powerSum}
     \qquad   \defcolor{p^q_r(x_1,\dotsc,x_k)} := \\
        s^q_{(r)}(x_1,\dotsc,x_k) - s^q_{(r-1,1)}(x_1,\dotsc,x_k) +
                   \dotsb + (-1)^{r+1} s^q_{(1^r)}(x_1,\dotsc,x_k).
    \qquad
\end{multline}
%
We derive a formula for $\frakS_u \ast p_r^q(x)$ via intermediary
formulas for $\frakS_u \ast s_{(b, 1^{a-1})}^q(x)$, as in the proof of
\cref{C:classical-schubert-times-powersum}.\bigskip


We begin with the \demph{quantum Monk Formula} of Fomin, Gelfand, and Postnikov~\cite{FGP}:
for $u\in S_n$ and $1\leq k<n$,
    $\frakS_u \ast \frakS_{(k,k+1)} =
    \sum \frakS_{u (i,j)} +
    \sum q_{i,j} \frakS_{u (i,j)}$,
where the first sum is given by the classical Monk Formula and
the second sum is over transpositions $(i, j)$ with $i\leq k<j$ and $\ell(u)+1=\ell(u (i,j))+2(j{-}i)$,
and $\defcolor{q_{i,j}}:=q_iq_{i+1}\dotsb q_{j-1}$.
For example,
%
 \[
   \frakS_{1432} \ast \frakS_{(2,3)} = \frakS_{2431} + \frakS_{3412} + q_2\frakS_{1342} + q_{2,4}\frakS_{1234}.
 \]
%

The terms appearing in the quantum Monk Formula
define a ranked partial order on
$\defcolor{S_n[q]} := \{ q^\alpha u : u\in S_n \text{ and }
q^\alpha\text{ is a monomial in }q_1,\dotsc,q_{n-1}\}$ as follows.
The \demph{quantum $k$-Bruhat order $\leq_k^q$} on $S_n[q]$ is defined by the following cover relations:
\begin{enumerate}[noitemsep, topsep=0em]
    \item $\defcolor{u\lessdot_k^q u (i,j)}$ if $i\leq k<j$ and $\ell(u)+1=\ell(u (i,j))$,
         so that $u\lessdot_k u (i,j)$;
    \item $\defcolor{u\lessdot_k^q q_{i,j} u (i,j)}$ if $i\leq k<j$ and
           $\ell(u)+1=\ell(u (i,j))+2(j{-}i)$\,; and
    \item extend $q$-multiplicatively:  $u\leq_k^q w$  if and
        only if $q^\alpha u\leq_k^q q^\alpha w$ for $u,w\in S_n[q]$.\vspace{2pt}
\end{enumerate}
We display two  levels of the quantum 2-Bruhat order on $S_4[q]$ above $1432$.
%
 \[
    \begin{picture}(330,80)(10,0)
        \thicklines
        \put( 10, 70){$3421$}          \put( 52, 70){$q_{2} q_{3} 2134$}
        \put(110, 70){$q_{2} 2341$}       \put(160, 70){$q_{2} 1432$}
        \put(230, 70){$q_{2} 3142$}       \put(290, 70){$q_{2} q_{3} 1324$}

        \put( 65,45){\color{LimeGreen}\line(-2,1){40}}
        \put( 72,45){\color{BrickRed}\line(0,1){20}}
        \put( 80,45){\color{BrickRed}\line(2,1){40}}

        \put(150,45){\color{LimeGreen}\line(-1,1){20}}
        \put(165,45){\color{LimeGreen}\line( 1,2){10}}

        \put(240,45){\color{BrickRed}\line( 1,2){10}}

        \put(300,45){\color{LimeGreen}\line( 1,2){10}}

        \put( 60, 35){$2431$}          \put(140, 35){$q_{2} 1342$}
        \put(220, 35){$3412$}         \put(275, 35){$q_{2} q_{3} 1234$}

        \put(150,10){\color{LimeGreen}\line(-3,1){60}}
        \put(158,10){\color{BrickRed}\line( 0,1){20}}
        \put(166,10){\color{LimeGreen}\line(3,1){60}}
        \put(174,10){\color{BrickRed}\line(5,1){100}}
        \put(150,0){$1432$}
       \end{picture}
 \]
%
The rank function $\ell$ on the quantum $k$-Bruhat order is given by
$\ell(q^\alpha u) := 2 \deg q^\alpha +  \ell(u)$,
where $\deg q^\alpha$ is the usual degree of the monomial $q^\alpha$ in $\ZZ[q]$.
As far as we know, there is no analogue of the Grassmannian--Bruhat order related to the quantum $k$-Bruhat order.
Consequently, rather than speak of minimal permutations (as in the classical case),
we will define minimal intervals in the
quantum $k$-Bruhat order.
Suppose that $u,w\in S_n$ are permutations and $u\leq_k^q q^{\alpha}w$.
We say that the interval $[u,q^\alpha w]_k^q$ is \demph{minimal} if
its rank $\ell(q^{\alpha}w)-\ell(u)$ equals $\#\supp(wu^{-1}) - s(wu^{-1})$.
When $q^\alpha=1$, the interval is minimal if and only if $wu^{-1}$ is a minimal permutation.


We state our two main results.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{thm:q_hook_product}
Let $u \in S_n$\,, $l \leq k$ and $m \leq n-k$. Then
    \begin{equation*}
        \frakS_u
        \ast
        s^q_{(m,1^{l-1})} (x_1, \dots, x_k)
        = \sum
        \left(\begin{smallmatrix} s(wu^{-1})-1 \\ \het(wu^{-1})-l \end{smallmatrix}\right)
        q^\alpha
        \frakS_{w}\,,
    \end{equation*}
summing over all minimal intervals $[u,  q^{\alpha}w]_k^q$ such that $\ell(q^\alpha w)-  \ell(u)= m+l-1$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The following quantum Murnaghan--Nakayama formula was conjectured in~\cite{Morrison}.
It follows from \cref{thm:q_hook_product} in the same way that
\cref{C:classical-schubert-times-powersum} follows from \cref{P:classical_hook}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}
    \label{C:quantumMN}
    Let $u \in S_n$. Then
    \begin{equation*}
        \frakS_u \ast p_r^q(x_1,\dotsc,x_k) = \sum (-1)^{\het(w^{-1}u)+1} q^\alpha \frakS_{w},
    \end{equation*}
    summing over all minimal intervals $[u, q^{\alpha}w]_k^q$ of rank $r$ such that $w^{-1}u$ is a
    single cycle.
\end{corollary}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For example, in $qH^*\Fl_8$, if $u=38254671$, then  the product $\frakS_{u}\ast p^q_4(x_1,\dotsc,x_4)$ is
\begin{eqnarray*}
 &
   \frakS_{(2,5,6,3,4)u}
    + \frakS_{(2,3,5,6,4)u}
    - \frakS_{(2,5,7,6,4)u}
    - \frakS_{(3,5,7,6,4)u}
    -q_4 \frakS_{(2,4,7,6,5)u}
    -q_4 \frakS_{(3,4,7,6,5)u}
  &\\ &
   +q_4 \frakS_{(2,3,4,6,5)u}
   +q_{2,5} \frakS_{(2,4,6,5,8)u}
   +q_{2,5} \frakS_{(2,5,6,4,8)u}
   -q_{2,5} \frakS_{(2,3,4,5,8)u}
  &\\ &
    +q_{4,8} \frakS_{(1,7,6,5,4)u}
   -q_{2,8} \frakS_{(1,7,6,5,8)u}
    +q_{2,8} \frakS_{(1,8,2,3,4)u}
   +q_4q_{2,8} \frakS_{(1,8,2,4,5)u}\,.
   &
\end{eqnarray*}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Sketch of Proof and Remarks}
We prove \cref{thm:q_hook_product} in two parts.
In the first part we show that if $[u, q^\alpha w]_k^q$ is a minimal interval, then there is a hook partition
$\lambda$ with $|\lambda|=\ell(q^\alpha w)-\ell(u)$ such that $C^{q^\alpha w}_{v(\lambda,k),u}$ is nonzero.
This uses a study of chains in the interval $[u,q^\alpha w]_k^q$ in terms of certain left operators modeled on those for
the ordinary $k$-Bruhat order as developed in~\cite{BS-Monoid}.
This is sketched in \cref{SS:LO}.

Our second part of the proof uses a Corollary to the proof of Theorem~1.2 in~\cite{LeungLi2012}, the result of Leung and Li
that ``quantum equals classical''.
We discuss this in \cref{SS:CC}.
A consequence is that if $C^{q^\alpha w}_{v(\lambda,k),u}\neq 0$ for some partition $\lambda$,  then there are $y,z\in S_n$ with $y\leq_k z$,
$\ell(z)-\ell(y)=\ell(q^\alpha w)-\ell(u)=|\lambda|$, and $zy^{-1}=wu^{-1}$  such that for all partitions $\mu$ with
$|\mu|=|\lambda|$,  $C^{q^\alpha w}_{v(\mu,k),u}=q^\alpha c^z_{v(\mu,k),y}$.
If $\lambda$ is a hook partition, then \cref{P:classical_hook} implies that the interval $[u, q^\alpha w]_k^q$ is minimal.
Then \cref{thm:q_hook_product} follows from these results and \cref{P:classical_hook}.


We point out Theorem 15 of~\cite{MPP} also claims that quantum equals classical for hook partitions, and from
which one could in principle deduce our \cref{thm:q_hook_product}.
Unfortunately, its proof invokes a lemma (Lemma 13 in~\cite{MPP}) whose hypotheses do not apply to the case (hook
partitions) that is invoked.
We sketch this in \cref{SS:R}.
This necessitates the alternative proof we provide.

%The journal version of this extended abstract will contain complete proofs of these assertions.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Left Operators}\label{SS:LO}

We now sketch a proof of the first part in the following result.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem}\label{Th:nonZero}
  Let $u,w\in S_n$ and $q^\alpha w\in S_n[q]$.
  Then $[u,q^\alpha w]^q_k$ is a minimal quantum interval if and only if
  there is a hook partition $\lambda$ with $|\lambda|=\ell(q^\alpha w)-\ell(u)$ such that
  $C^{q^\alpha w}_{v(\lambda,k), u}\neq 0$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%We prove this by studying chains in the interval $[u,q^\alpha w]^q_k$ using a theory of (left) operators $\qv_{a,b}$ on
%$S_n[q]$ that model chains in the quantum $k$-Bruhat order, and whose relations give other chains in this order.

A cover $y\lessdot^q_k q^\alpha z$ in the quantum $k$-Bruhat order gives a transposition $(a,b)=zy^{-1}$.
Therefore,
a chain in $[u,q^\alpha w]^q_k$ provides a factorization of $\zeta:=wu^{-1}$ into transpositions
%
 \begin{equation}\label{Eq:factorization}
   \zeta\ =\ (a_r,b_r)\; \dotsb\; (a_2,b_2)\; (a_1,b_1)\,,
 \end{equation}
%
which in turn defines a (multi) graph with vertex set $[n]$ whose edges are $(a_i,b_i)$ for each transposition above.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{L:minimalIsForest}
  Suppose that $[u,q^\alpha w]^q_k$ is minimal and~\eqref{Eq:factorization} is a factorization of $\zeta=w u^{-1}$
  corresponding to a chain in $[u,q^\alpha w]^q_k$.
  Let $F$ be the corresponding multigraph.
  Then
  %
  \begin{enumerate}[noitemsep, topsep=3pt]
    \item $F$ is a forest: it is acyclic without multiple edges and its connected components are trees.
    \item The vertices in connected components of $F$ form a noncrossing partition of $[n]$.
    \item $\zeta$ is a product of disjoint cycles, one for each tree in $F$.
  \end{enumerate}
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{example}\label{Ex:minimalForest}
  We have the following chain in the quantum 3-Bruhat order:
%
 \begin{equation*}
     \begin{aligned}
         635214 = u
         & ~~ \lessdot_3^q ~~
         q_{1,5} \, (1\, 6) u
         ~~ \lessdot_3^q ~~ 
         q_{1,5} \, (1\, 2) (1\, 6) u = q_{1,5} \, 235164
         \\
         & ~~ \lessdot_3^q ~~
         q_{1,5} \, (3\,4)(1\,2)(1\,6)u
         ~~ \lessdot_3^q ~~
         \begin{aligned}[t]
             q_{1,5} q_{3} \, (1\,5)(3\,4)(1\,2)(1\,6)u = q_{1}q_{2}q_{3}^2q_{4} \, 241563.
         \end{aligned}
     \end{aligned}
 \end{equation*}
%
 The interval $[635214, q_{1}q_{2}q_{3}^2q_{4}\, 241563]^q_3$ has rank four, as
 \[
     \ell(q_{1}q_{2}q_{3}^2q_{4} \, 241563)=
     10+5=15 \quad\mbox{and}\quad \ell(635214)=11.
 \]
 Set  $\zeta=(1,5)(3,4)(1,2)(1,6)=(3,4)(1,6,2,5)$.
 Its support has six elements and $\zeta$ consists of two cycles, and thus this interval is minimal.
 The supports of the two cycles in $\zeta$ are  non-crossing.
 We draw its corresponding forest.
 \[
 \begin{picture}(170,38)(-7,-10)
   \put(0,0){\includegraphics{forest}}
   \put( 0,-10){\small$1$} \put( 30,-10){\small$2$} \put( 60,-10){\small$3$}
   \put(90,-10){\small$4$} \put(120,-10){\small$5$} \put(150,-10){\small$6$}
 \end{picture}
 \]
\end{example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Postnikov~\cite{Postnikov_symmetry} observed that the quantum Bruhat order (union over $k$ of all quantum $k$-Bruhat orders)
has a cyclic symmetry.
We describe this for the quantum $k$-Bruhat order.
Write $\defcolor{\ncycle}$ for the $n$-cycle $(1,2,\dotsc,n)$.
If $i>j$, define $q_{i,j}:=q^{-1}_{j,i}$, a Laurent monomial.
For $u,w\in S_n$, define $\defcolor{q(u,w)}:= q_{w^{-1}(n),u^{-1}(n)}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}\label{L:cyclic_symmetry}
 Let $u,w\in S_n$.
 We have $u<_k^q q^\alpha w$ if and only if $\ncycle u <_k^q q(u,w) q^\alpha \ncycle w$.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For $a<b$ in $[n]$, define an operator \defcolor{$\qv_{a,b}$} on $S_n[q]\cup\{\bzero\}$ by
$\qv_{a,b}\kactionop\bzero=\bzero$, and for $u\in S_n$,
\[
   \qv_{a,b}\kactionop u\ :=\  \begin{cases}
            (a,b)u & \text{ if } u\lessdot_k (a,b)u,\\
           q_{i,j}(b,a)u & \text{ if } u\lessdot^q_k q_{i,j}(b,a)u \mbox{ with }u^{-1}(b)=i\leq k < u^{-1}(a)=j,\\
            \bzero& \text{ otherwise\,,}
       \end{cases}
\]
and extend this action multiplicatively.
We will say that $\qv_{a,b}$ is a \demph{classical operator} if the first case holds, and that it is a
\demph{quantum operator} when the second case holds.
%
% F: I am finssing this distinction between quantum and classical
%
%

A composition  of operators $\qv=\qv_{a_r,b_r}\dotsb \qv_{a_1,b_1}$ is zero if for all $u\in S_n$, $\qv\kactionop u=\bzero$
and otherwise it is nonzero.
If $\qv'=\qv_{c_r,d_r}\dotsb \qv_{c_1,d_1}$ is another composition of operators, then it is \demph{equivalent to $\qv$} if
for all $u\in S_n$, $\qv\kactionop u=\qv'\kactionop u$.
When $\qv$ is nonzero, there are $u,w\in S_n$ and $q^\alpha w\in S_n[q]$ such that $\qv\kactionop u=q^\alpha w$.
In this case, $u\leq_k^q q^\alpha w$, $\qv$ corresponds to a chain in $[u,q^\alpha w]^q_k$, and
$\qv$ corresponds to a factorization~\eqref{Eq:factorization} of $\zeta=wu^{-1}$.
Then the chains in  $[u,q^\alpha w]^q_k$ correspond to compositions $\qv'$ of operators equivalent to $\qv$.

By \cref{L:cyclic_symmetry}, \demph{cyclic shift} (multiplication by $\ncycle$) is an isomorphism of
intervals \linebreak $[u,q^\alpha w]^q_k$, and it acts (via conjugation) on chains in intervals and on the operators $\qv_{a,b}$, preserving
relations.
If $b\neq n$, then $\ncycle\qv_{a,b}\ncycle^{-1}=\qv_{a+1,b+1}$ and this preserves classical/quan\-tum operators.
However, if $b=n$, then
$\ncycle\qv_{a,n}\ncycle^{-1}=\qv_{1,a+1}$, and it interchanges classical operators with quantum operators.
Key to our proof of the forward implication of \cref{Th:nonZero} is showing that a chain in a minimal interval may be cyclically
shifted (conjugated by some power of $\ncycle$) to
become a chain of classical operators.


The relations among classical operators were determined in~\cite{BS-Monoid},  and they are all of degree two or
three.
In particular, given two classical operators $\qv_{a,b}$ and $\qv_{c,d}$:
\begin{itemize}[noitemsep, topsep=0pt]
    \item
        they commute if $(a,b)(c,d)$ is non-crossing, in the sense of \cref{L:minimalIsForest}(2);
    \item
        their product is nonzero and they do not commute if $b=c$ or $a=d$; and
    \item
        their product is zero if the partition $\{a,b\}\sqcup\{c,d\}$ is crossing or if $a=c$ or $b=d$.
\end{itemize}
There is an interesting relation of degree three that we do not describe.
If we also consider quantum operators, we do not have a complete set of relations,
but do know those of degrees two and three, for they are induced from the classical relations by the cyclic
symmetry of \cref{L:cyclic_symmetry}.

A composition $\qv=\qv_{a_r,b_r}\dotsb\qv_{a_1,b_1}$ is a \demph{connected (classical) row} if
$b_1=a_2$, $b_2=a_3$, $\ldots$, $b_{r-1}=a_r$.
It is a \demph{connected (classical) column} if $a_1=b_2,\dotsc,a_{r-1}=b_r$.
The graph with vertices $\{a_1,b_1,\dotsc,a_r,b_r\}$ and edges $(a_ib_i)$ is a path in either case.
A \demph{classical row} is a composition of connected rows in which every pair has noncrossing support (a noncrossing
composition), and the same for a \demph{classical column}.
A composition is a \demph{row} (\demph{column}) if may be cyclically shifted to a classical row (column).

Given these definitions, we may express Postnikov's quantum Pieri formula~\cite{Postnikov} as
\[
\frakS_u \ast s^q_{(m)}(x_1,\dotsc,x_k)=\sum q^\alpha \frakS_v,
\]
summing over all $q^\alpha v=\bR\kactionop u$, where $\bR$ is a row of length $m$.
For quantum multiplication by $s^q_{1^m}(x_1,\dotsc,x_k)$, replace row by column.
A consequence of the Pieri formula and \cref{L:cyclic_symmetry} is that if $u\leq_k^q q^\alpha w$ and $\lambda$ is a
partition with $|\lambda|=\ell(q^\alpha w)-\ell(u)$, then
%
 \begin{equation}\label{Eq:cyclicShiftCoefficients}
   q(u,w) C^{q^\alpha w}_{v(\lambda,k),u} = C^{ q(u,w) q^\alpha \ncycle w}_{v(\lambda,k),\ncycle u}\,.
 \end{equation}
%


A composition $\bT = \qv_{a_t,b_t}\dotsb \qv_{a_1b_1}$ is a \demph{tree} if the graph with vertices
$\{a_t,b_t,\dotsc,a_1,b_1\}$ given by the corresponding factorization is a tree.
By \cref{L:minimalIsForest},  if
$[u,q^\alpha  w]^q_k$ is a minimal quantum interval, then any chain has corresponding composition of operators that is
a \demph{noncrossing forest}: a composition of trees in which every pair has noncrossing support.
We state our main technical result.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}
    \label{main-lemma}
    Suppose that $\qv$ is a nonzero noncrossing forest.
    Then  $\qv$ is equivalent to the product of a row $\bR$ and a column $\bC$, and the product $\bR\cdot\bC$ may be
    cyclically shifted to the product of a classical row and a classical column.
\end{lemma}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof of the forward implication of \cref{Th:nonZero}]
  Suppose that $[u,q^\alpha w]^q_k$ is a minimal interval.
  By \cref{L:minimalIsForest}, there is a noncrossing forest
  $\bF$ of left operators of length $\ell(q^\alpha w)-\ell(u)$ with $q^\alpha w=\bF\kactionop u$.
  By \cref{main-lemma}, there are a row $\bR$ and a column $\bC$ of lengths $m$ and $l$, respectively,
  with $m+l=\ell(q^\alpha w)-\ell(u)$ and
  $q^\alpha w=(\bR\cdot\bC)\kactionop u$.
  By \cref{main-lemma} again, $\bR\cdot\bC$ may be cyclically shifted to  the product of a classical row and a
  classical column.
  If $r$ is that shift, then $\frakS_{\ncycle^r w}$ appears in
  %
  \begin{multline*}
   \qquad   s_{(m)}(x_1,\dotsc,x_k)\cdot s_{1^l}(x_1,\dotsc,x_k)\cdot\frakS_{\ncycle^r u}= \\
  (s_{(m,1^l)}(x_1,\dotsc,x_k) + s_{(m+1,1^{l-1})}(x_1,\dotsc,x_k))\cdot\frakS_{\ncycle^r u}, \qquad
    \end{multline*}
  %
  which is a product in $H^*\Fln$.
  By~\eqref{Eq:cyclicShiftCoefficients}, at least one of
  $C^{q^\alpha w}_{v((m,1^l),k),u}$ or $C^{q^\alpha w}_{v((m+1,1^{l-1}),k),u}$ is nonzero.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Quantum equals Classical}\label{SS:CC}

Leung and Li~\cite{LeungLi2012} gave a different identity among the quantum coefficients
than~\eqref{Eq:cyclicShiftCoefficients}.
In it, the exponent of $q^\alpha$ is reduced.
For $i\in[n-1]$ let \defcolor{$e_i$} be the exponent of $q_{i}$, so that $q^{e_i}=q_{i}$.
We state the results of Leung and Li, in the form that we use.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{P:LL_I}
  Suppose that $u,w\in S_n$, $q^\alpha w\in S_n[q]$, and that $\lambda$ is a partition such that
  $C^{q^\alpha w}_{v(\lambda,k),u}\neq 0$.
  Then there is an $i\in[n{-}1]$ such that one of the following two conditions hold:
  %
  \begin{enumerate}[noitemsep, topsep=0em]
   \item We have $i\neq k$ and $1=2\alpha_i-\alpha_{i-1}-\alpha_{i+1}$,
     $\ell(w(i,i{+}1))=\ell(w)-1$, $\ell(u(i,i{+}1))=\ell(u)+1$, and
     $q^{\alpha-d_i}w(i,i{+}1)\geq^q_k u(i,i{+}1)$.
     %
     %
     %

    \item We have $i=k$ and $2=2\alpha_k-\alpha_{k-1}-\alpha_{k+1}$,
     $\ell(w(k,k{+}1))=\ell(w)-1$,  $\ell(u(k,k{+}1))=\ell(u)+1$, and
     $q^{\alpha-e_k}w(k,k{+}1)\geq^q_k u(k,k{+}1)$.

\end{enumerate}
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}\label{P:LL_II}
  Suppose that  $u,w\in S_n$, $q^\alpha w\in S_n[q]$, and $i\in[n{-}1]$, and either condition of \cref{P:LL_I}
  holds.
  Then for any partition $\lambda$,
  $C^{q^\alpha w}_{v(\lambda,k),u}=q_iC^{q^{\alpha-e_i} w(i,i+1)}_{v(\lambda,k),u(i,i+1)}$.
\end{proposition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In~\cite{LeungLi2012} these are formulated in a weaker form; the stronger result of \cref{P:LL_II} is
not stated.
The journal version of this abstract will contain a complete explanation.

Observe that in the statement of \cref{P:LL_II}, $w u^{-1}=(w(i,i+1))(u(i,i+1))^{-1}$.
An induction on $\deg(q^\alpha)$ using \cref{P:LL_I,P:LL_II} shows that if
$C^{q^\alpha w}_{v(\lambda,k),u}\neq 0$, then there exist $y,z\in S_n$ with $y\leq_k z$,
$|\lambda|=\ell(z)-\ell(y)=\ell(q^\alpha w)-\ell(u)$,
$zy^{-1}=wu^{-1}$, and $q^\alpha c^z_{v(\lambda,k),y}=C^{q^\alpha w}_{v(\lambda,k),u}\neq 0$.
If $\lambda$ is a hook partition, then this implies that $zy^{-1}$ is a minimal permutation
and thus $[u,q^\alpha w]_k^q$ is minimal, which is
the reverse implication in  \cref{Th:nonZero}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proof}[Proof of \cref{thm:q_hook_product}]
  By \cref{Th:nonZero},  $[u,q^\alpha w]^q_k$ is a minimal interval if and only if there is a
  hook partition $\lambda=(m,1^{l-1})$ with $C^{q^\alpha w}_{v(\lambda),u}\neq 0$.

  It remains to compute the coefficient $C^{q^\alpha w}_{v(\lambda,k),u}$.
  By the induction  sketched above, and using the same notation, this coefficient equals $q^\alpha c^z_{v(\lambda,k),y}$. Now by
  \cref{P:classical_hook}, this equals
  $q^\alpha\left(\begin{smallmatrix}s(wu^{-1})-1\\\het(wu^{-1})-l\end{smallmatrix}\right)$,
  as $w u^{-1}=zy^{-1}$.
\end{proof}

%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Remarks}\label{SS:R}

\cref{P:classical_hook} may be deduced from a stronger result of M\'esz\'aros, Panova, and
Postnikov~\cite[Thm.\ 8]{MPP}.
There, they establish a formula for a `lift' of the Schur polynomial
$s_{(m,1^{l-1})}(x_1,\dotsc,x_k)$ to the Fomin--Kirillov~\cite{FominKirillov}
algebra ${\mathcal E}_n$, which is a subalgebra (generated by certain Dunkl elements $\theta_i$) of the quotient of a free
associative algebra with generators $x_{i,j}$ for $1\leq i<j\leq n$ by certain relations, which include $x_{i,j}^2=0$.
Their paper contains another result~\cite[Thm.\ 15]{MPP} which is a quantum analog of~\cite[Thm.\ 8]{MPP}, giving a formula
for a `quantum lift' $s^q_{(m,1^{l-1})}(x_1,\dotsc,x_k)$ in the quantum Fomin--Kirillov algebra
$\mathcal{E}^q_n$~\cite{Postnikov}.
This formula would imply \cref{thm:q_hook_product}.
Unfortunately, as we now sketch, there is a gap in the proof of~\cite[Thm.\ 15]{MPP} (we have communicated this to the
authors).

In~\cite{MPP}, the result~\cite[Thm.\ 15]{MPP} is deduced from the proof of~\cite[Thm.\ 8]{MPP} using a technical
result~\cite[Lem.\ 13]{MPP} which roughly asserts the following:
If an identity involving positive terms holds in ${\mathcal E}_n$, and if no relations  $x_{i,j}^2=0$ were used to deduce
the identity, then the same identity holds in ${\mathcal E}^q_n$.
We are convinced this lemma is correct, but it unfortunately does not apply in the case
invoked to prove~\cite[Thm.\ 15]{MPP}.

In addition to using relations (but not  $x_{i,j}^2=0$), the proof of~\cite[Thm.\ 8]{MPP} uses in a fundamental
way~\cite[Cor.\ 10]{MPP}, that
$e_a(\theta_1,\dotsc,\theta_a)h_b(\theta_1,\dotsc,\theta_a)=0$ in $\mathcal{E}_{a+b}$ (in their notation).
However, this identity can only be proven directly in  $\mathcal{E}_{a+b}$ using the relations $x_{i,j}^2=0$.
For example, when $a=b=1$, as $\theta_1=x_{1,2}$, it becomes $0=e_1(x_{1,2})h_1(x_{1,2})=x_{1,2}^2$.\bigskip

We can prove a stronger version of \cref{Th:nonZero} which is nearly a quantum analog of
\cref{pieri-formulas-version-2}.
Its arguments involve a significantly more detailed study of the left operators and their relations than our
proof of \cref{Th:nonZero}.

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


\acknowledgements{This collaboration was initiated in the {\it Algebraic Combinatorics}
  seminar at the Fields Institute in 2017.}

%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way

\printbibliography


\end{document}
