%% if you are submitting an initial manuscript then you should have submission as an option here
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%% otherwise options taken by the article class will be accepted
\documentclass[final]{FPSAC2022}
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\usepackage{tikz-cd, xcolor, mathtools}

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\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}
\newtheorem{lem}{Lemma}
\newtheorem{prop}{Proposition}
\theoremstyle{definition}
\newtheorem{defn}{Definition}

\newcommand{\symm}{\mathfrak{S}}
\newcommand{\NC}{\mathrm{NC}}
\newcommand{\Cat}{\mathrm{Cat}}
\newcommand{\Nar}{\mathrm{Nar}}
\newcommand{\Frob}{\mathrm{Frob}}
\newcommand{\sign}{\mathrm{sign}}

\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}


\usepackage{lipsum}

%% define your title in the usual way
\title{Set Partitions, Fermions, and Skein Relations}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[J.~Kim and B.~Rhoades]{Jesse Kim\addressmark{1}\thanks{\href{mailto:jvkim@math.ucsd.edu}{jvkim@math.ucsd.edu}} \and Brendon Rhoades\addressmark{1}\thanks{\href{mailto:bprhoades@math.ucsd.edu}{bprhoades@math.ucsd.edu}. Partially supported by NSF grant
DMS-1953781.}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California, San Diego}

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

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%\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{The second author defined an action of the symmetric group
$\symm_n$ on the vector space spanned by noncrossing partitions of $\{1, \dots , n \}$ by introducing
new skein relations which resolve local crossings in set partitions.
On the other hand, the second author and Jongwon Kim defined and studied
 the {\em fermionic diagonal coinvariant
ring} $FDR_n$ which has a definition analogous to the Garsia-Haiman diagonal coinvariant ring $DR_n$,
but with fermionic (anticommuting) variables. We prove that set partition skein relations arises naturally 
in the context of $FDR_n$. This clarifies and sharpens results on the skein action and gives an 
$\symm_n$-equivariant way to resolve an arbitrary set partition into a linear combination of 
noncrossing partitions.}



%% put your keywords here, or comment this out if you don't have them yet
\keywords{noncrossing set partition, exterior algebra, skein relation}

%% 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, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
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\usepackage[backend=bibtex]{biblatex}
\addbibresource{sample.bib}
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\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above

\section{Introduction}
\label{Introduction}

This extended abstract relates two representations of the symmetric group $\symm_n$ --- one combinatorial
and one algebraic.
We describe the combinatorial module first, and then turn to the algebraic one.

A set partition $\pi$ of $[n] := \{1, \dots, n \}$ is {\em noncrossing} if for all 
 $1 \leq a < b < c < d \leq n$  such that $a \sim c$ and $b \sim d$ in $\pi$, we have
$a \sim b \sim c \sim d$.
We let $\NC(n)$ denote the family of noncrossing set partitions of $[n]$ and 
$\NC(n,k) \subseteq \NC(n)$ denote the subfamily of noncrossing set partitions with $k$ blocks.
It is well-known that these families are counted by the Catalan and Narayana numbers
\begin{equation}
|\NC(n)| = \Cat(n) = \frac{1}{n+1}{2n \choose n}, \quad \quad |\NC(n,k)| = \Nar(n,k) = 
\frac{1}{n} {n \choose k}{n \choose k-1}.
\end{equation}





   

  
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.7]


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    \end{tikzpicture}
    
      \caption{The three skein relations defining the action of $\symm_n$ on $\CC[\NC(n)]$. The red vertices
      are adjacent 
      and the shaded blocks have at least three elements. The  3-term relation obtained
      by reflecting the middle relation across the $y$-axis is not shown.}
     \label{fig:skein}
    
 \end{figure}


Let $\Pi(n)$ be the  family of {\bf all} set partitions of $[n]$ and $\Pi(n,k)$
be the set partitions of $[n]$ with $k$ blocks.
The sets $\Pi(n)$ and $\Pi(n,k)$ carry a permutation action $\pi \mapsto w(\pi)$
of the symmetric group $\symm_n$. However, this action does not preserve the noncrossing
property; the subsets $\NC(n) \subseteq \Pi(n)$ and $\NC(n,k) \subseteq \Pi(n,k)$ are not 
$\symm_n$-closed.
Nevertheless, the second author introduced \cite{RhoadesSkein} new ``set partition skein 
relations'' to define an  $\symm_n$-action on the linearized 
versions $\CC[\NC(n)]$ and $\CC[\NC(n,k)]$ of these sets.



For $1 \leq i \leq n-1$, let $s_i = (i,i+1) \in \symm_n$ be the adjacent transposition.
Given $\pi \in \NC(n)$, the {\em skein action} of $s_i$ on $\pi$ is given by\footnote{This
version of the skein action is slightly modified from that in \cite{RhoadesSkein}. The 
skein relations are the same, but the sign convention when $s_i(\pi)$ is noncrossing is different.}
\begin{equation}
\label{skein-si-action}
s_i \cdot \pi := \begin{cases}
- s_i(\pi) & \text{if $s_i(\pi)$ is noncrossing} \\
\sigma(s_i(\pi)) & \text{otherwise}
\end{cases}
\end{equation}
where $\sigma(s_i(\pi)) \in \CC[\NC(n)]$ resolves the crossing at $i, i+1$ using the skein relations 
 in Figure~\ref{fig:skein}.  More precisely, if $B_i$ and $B_{i+1}$ are the blocks of $s_i(\pi)$
containing $i$ and $i+1$  we have
\begin{equation}
\label{sigma-equation}
\sigma(s_i(\pi)) := \begin{cases}
\pi_1 + \pi_2 & \text{if $|B_i| = |B_{i+1}| = 2$,} \\
\pi_1 + \pi_2 - \pi_3 & \text{if $|B_i| > 2$ and $|B_{i+1}| = 2$,} \\
\pi_1 + \pi_2  \quad \quad \,  \,  - \pi_4 & \text{if $|B_i| = 2$ and $|B_{i+1}| > 2$,} \\
\pi_1 + \pi_2 - \pi_3 - \pi_4 & \text{if $|B_i|, \, |B_{i+1}| > 2$,}
\end{cases}
\end{equation}
where the set partitions $\pi_1 = \pi$ and $\pi_2, \pi_3, \pi_4$ are obtained from $s_i(\pi)$ as follows:
\begin{itemize}
\item $\pi_2$ replaces $B_i$ and $B_{i+1}$ with $(B_i \cup B_{i+1}) - \{i,i+1\}$ and
$\{i,i+1\}$,
\item $\pi_3$ replaces $B_i$ and $B_{i+1}$ with $B_i - \{i\}$ and $B_{i+1} \cup \{i\}$, and
\item $\pi_4$ replaces $B_i$ and $B_{i+1}$ with $B_i \cup \{i+1\}$ and $B_{i+1} - \{i+1\}$.
\end{itemize}
Since  skein relations preserve the number of blocks
$\CC[\NC(n,k)] \subseteq \CC[\NC(n)]$ is a submodule for this action and we have
$\CC[\NC(n)] = \bigoplus_{k = 1}^n \CC[\NC(n,k)]$.


The top skein relation in Figure~\ref{fig:skein} is the famous transformation
\begin{center}

\begin{tikzpicture}[scale=0.3]


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	\node at (2,0) {$\mapsto$};
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    \end{center}
    which appears in many  contexts including Schubert calculus, centralizer algebras,
     invariant theory, cluster algebras,
    and knot theory.  In contrast, the lower two skein relations do not seem to have been defined before
    the 2017 paper \cite{RhoadesSkein}.
    
    The skein action was defined to give representation-theoretic proofs of cyclic sieving results
    of Reiner--Stanton--White and Pechenik \cite{Pechenik, RSW}.
    While its basic properties were established in 
    \cite{RhoadesSkein}, its purely combinatorial
    definition made for very involved proofs -- even verifying that the local action of $s_i$
     satisfies the Coxeter relations used a number of `miraculous' 16-term identities.
    It was also unclear whether the skein relations of Figure~\ref{fig:skein} were tied to other 
    areas of mathematics.
    
    Our algebraic module is as follows. Let 
    $\Theta_n = (\theta_1, \dots, \theta_n)$ and $\Xi_n = (\xi_1, \dots, \xi_n)$ be two lists of 
    $n$ variables and let $\wedge \{ \Theta_n, \Xi_n \}$ be the rank $2n$ exterior algebra over these 
    variables.
    This ring carries a diagonal action of $\symm_n$, viz.
    \begin{equation*}
    w \cdot \theta_i := \theta_{w(i)} \quad \quad w \cdot \xi_i := \xi_{w(i)} \quad \quad (w \in \symm_n, \, \, 
    1 \leq i \leq n)
    \end{equation*}
    Adopting the language of physics, we refer to the anticommuting variables $\theta_i$ and 
    $\xi_i$ as {\em fermionic} and general elements $f \in \wedge \{ \Theta_n, \Xi_n \}$ as 
    {\em fermions}.
    
    
    The second author and Jongwon Kim \cite{KRFermion} 
    defined the {\em fermionic diagonal coinvariant ring}
    to be the quotient
    \begin{equation}
    FDR_n := \wedge \{ \Theta_n, \Xi_n \} / I
    \end{equation}
    where $I \subseteq \wedge \{ \Theta_n, \Xi_n \}$ is the ideal generated by $\symm_n$-invariants
    with vanishing constant term.
    The ring $FDR_n$ is a bigraded $\symm_n$-module, with one grading coming from the 
    $\theta$-variables and the other from the $\xi$-variables.
    The ring $FDR_n$ is analogous to the Garsia-Haiman 
    diagonal coinvariant ring $DR_n$ (see \cite{Haiman})
     but uses anticommuting rather than commuting variables.
    In recent years, various authors 
    \cite{BergeronBF, BRT, DIW, OZ, RWSuper, RWVan, Swanson, SW, ZabrockiDelta, ZabrockiCoinv}
    have considered versions of $DR_n$ involving mixtures
    of commuting and anticommuting variables.
    
    We recall some  $\symm_n$-module terminology.
    For a partition $\lambda \vdash n$, let $S^{\lambda}$ be the corresponding $\symm_n$-irreducible.
    The {\em Frobenius image} of an $\symm_n$-module 
    $V = \bigoplus_{\lambda \vdash n} c_{\lambda} S^{\lambda}$ is 
    the symmetric function $\Frob \, V = \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda}$,
    where $s_{\lambda}$ is the Schur function of shape $\lambda$.
    
    The ring $FDR_n$ was shown \cite{KRFermion} 
    to have total dimension ${2n-1 \choose n}$, proving a conjecture
    of Zabrocki \cite{ZabrockiCoinv}.
    The bigraded piece $(FDR_n)_{i,j}$ was shown \cite{KRFermion} to vanish unless $i+j < n$.
    When $i + j < n$ we have    
    \begin{equation}
    \label{kronecker-difference}
    \Frob  \, (FDR_n)_{i,j} = s_{(i,1^{n-i})} * s_{(j,1^{n-j})} - s_{(i+1,1^{n-i-1})} * s_{(j+1,1^{n-j-1})}
    \end{equation}
    where $*$ denotes Kronecker product of Schur functions.
   

    
    The combinatorics of $FDR_n$ went largely unexplored in \cite{KRFermion}. A hint at its combinatorial
    interest is that in the `extremal bidegrees' $i + j = n-1$ it has dimension
    \begin{equation}
    \label{fdr-catalan-narayana}
    \dim (FDR_n)_{n-k,k-1} = \Nar(n,k) \quad \text{so that} \quad \sum_{k = 1}^n 
    \dim (FDR_n)_{n-k,k-1} = \Cat(n)
    \end{equation}
    which is a consequence of Equation~\eqref{kronecker-difference}. Our contributions are as follows.
    \begin{itemize}
    \item
    We enhance \eqref{fdr-catalan-narayana} by establishing
     $\symm_n$-module isomorphisms
    \begin{equation}
    \label{sn-iso}
    (FDR_n)_{n-k,k-1} \cong \CC[\NC(n,k)] \quad \text{so that} \quad
     \bigoplus_{k = 1}^n (FDR_n)_{n-k,k-1} \cong \CC[\NC(n)]
    \end{equation}
    between the extreme components of $FDR_n$ and the skein modules. 
   We prove these isomorphisms by attaching 
    fermions $F_{\pi}, f_{\pi} \in \wedge \{ \Theta_n, \Xi_n \}$ to any set partition
    $\pi \in \Pi(n)$ which satisfy the 
    skein relations in Figure~\ref{fig:skein}. 
    \item The fermions $F_{\pi}$ give a natural method of resolving crossings in set partitions which 
    extends the usual crossing resolution in chord diagrams. We describe this resolution combinatorially.
    \end{itemize}
    
    The skein relations of Figure~\ref{fig:skein} have appeared in invariant theory.
    Given two integers $\ell \leq n$ which are both $\geq 2$,
    Patrias, Pechenik, and Striker \cite{PPS} studied the projective variety $X$ of two-step flags
    $V_{\bullet} = (0 = V_0 \subseteq V_1 \subseteq V_2 \subseteq \CC^n)$
    of subspaces of $\CC^n$ with $\dim V_1 = 2$ and $\dim V_2 = \ell$.
    They found certain natural elements $g_{\pi} \in \CC[X]$ of the homogeneous coordinate ring
    of $X$ indexed by set partitions which satisfy the relations of Figure~\ref{fig:skein}.
    More general flag varieties $G/P$ give a natural setting for possible generalizations of 
    the relations in Figure~\ref{fig:skein}.
    We are hopeful that our skein relations will see further application to 
    algebra, geometry, and topology.
    



        

        
        





\section{Fermions for Set Partitions}
\label{Fermions}

We recall a notion of `differentiation' 
in exterior algebras.
Let $\Omega_m = (\omega_1, \dots, \omega_m)$ be a list of fermionic variables and let
$\wedge \{ \Omega_m \}$ be the exterior algebra over 
these variables.  For $1 \leq i \leq m$
we define 
\begin{equation}
\omega_i \odot (\omega_{j_1} \cdots \omega_{j_r}) := \begin{cases}
(-1)^{s-1} \omega_{j_1} \cdots \widehat{\omega_{j_s}} \cdots \omega_{j_r} & \text{if $i = j_s$}, \\
0 & \text{if $i \notin \{j_1, \dots, j_r \}$},
\end{cases}
\end{equation}
whenever $1 \leq j_1, \dots, j_r \leq n$ are distinct indices.
Linear extension gives an action $\odot$ of $\wedge \{ \Omega_m \}$ on itself
called  {\em contraction}.  Using our alphabet 
$(\theta_1, \dots, \theta_n, \xi_1, \dots, \xi_n)$ of variables, we have an action
\begin{equation}
\odot \colon 
\wedge \{ \Theta_n, \Xi_n \} \otimes \wedge \{ \Theta_n, \Xi_n \} \longrightarrow
\wedge \{ \Theta_n, \Xi_n \}
\end{equation}
of $\wedge \{ \Theta_n, \Xi_n \}$ on itself.


Certain derivations of $\wedge \{ \Theta_n, \Xi_n \}$ will be key to our constructions.
For any nonempty subset $B \subseteq [n]$, define the {\em block operator}
$\rho_B \colon \wedge \{ \Theta_n, \Xi_n \} \rightarrow \wedge \{ \Theta_n, \Xi_n \}$ by 
\begin{equation}
\rho_B(f) := \begin{cases}
\xi_i \cdot (\theta_i \odot f) & \text{$B = \{i\}$ is a singleton,} \\
\sum_{\substack{i, j \in B \\ i \neq j}} \xi_j \cdot (\theta_i \odot f) & \text{otherwise,}
\end{cases}
\end{equation}
for $f \in \wedge \{ \Theta_n, \Xi_n \}$.
The operator $\rho_B$ raises $\xi$-degree and lowers $\theta$-degree by one.
It bears formal similarity to the {\em polarization operators} 
$y_1 \partial^j_{x_1} + \cdots + y_n \partial^j_{x_n}$
(see, \textit{e.g.},~\cite{RWVan}) in the theory of harmonic spaces.


\begin{lem}
\label{lem:block-commute}
For any nonempty subsets $A$ and $B$ of $[n]$, we have $\rho_A \circ \rho_B = \rho_B \circ \rho_A$
as operators on $\wedge \{ \Theta_n, \Xi_n \}$.
\end{lem}


Our aim is to attach fermions to set partitions.
The following construction is valid by Lemma~\ref{lem:block-commute}.

\begin{defn}
\label{fermion-definition}
Let $\pi = \{ B_1 \mid \cdots \mid B_k \} \in \Pi(n)$ be a set partition.  We define two fermions 
$F_{\pi}, f_{\pi} \in \wedge \{ \Theta_n, \Xi_n \}$ by 
\begin{equation}
F_{\pi} := (\rho_{B_1} \circ \cdots \circ \rho_{B_k})(\theta_1 \cdots \theta_n) \quad \quad \text{and} \quad \quad
f_{\pi} := (\xi_1 + \cdots + \xi_n) \odot F_{\pi}
\end{equation}
\end{defn}

As an example of these objects, for $\pi = \{ 1, \, 3 \mid 2 \}$ we have
\begin{align*}
F_{\{1, \, 3 \, \mid \, 2 \}} &= \rho_{\{1, \, 3\}} \circ \rho_{\{2\}}(\theta_1 \theta_2 \theta_3) =
\rho_{\{1,3\}} (- \xi_2 \cdot \theta_1 \theta_3) = 
\xi_3 \xi_2 \theta_3 - \xi_1 \xi_2 \theta_1,  \\
f_{\{1, \, 3 \, \mid \, 2 \}} &=
(\xi_1 + \xi_2 + \xi_3) \odot (\xi_3 \xi_2 \theta_3 - \xi_1 \xi_2 \theta_1) = 
\xi_2 \theta_3 - \xi_3 \theta_3 - \xi_2 \theta_1 + \xi_1 \theta_1.
\end{align*}

If $\pi \in \Pi(n,k)$ has $k$ blocks, the fermion $F_{\pi}$ has bidegree $(n-k,k)$ and the fermion 
$f_{\pi}$ has bidegree $(n-k,k-1)$. The $F_{\pi}$ are cleaner to work with, but the 
$f_{\pi}$ are useful for the study of $FDR_n$.

How do the $F_{\pi}$ and $f_{\pi}$ interact with the combinatorics of set partitions?
Recall the natural permutation action $w(\pi)$ of $\symm_n$ on $\Pi(n)$. Denote by
$\star$  the sign twist of this action on $\CC[\Pi(n)]$. Explicitly, we have
\begin{equation}
w \star \pi := \sign(w) \, w(\pi) \quad \quad (w \in \symm_n, \, \pi \in \Pi(n)).
\end{equation}
%The following result is a consequence of the identity
%\begin{equation}
%w \cdot (f \odot g) = (w \cdot f) \odot (w \cdot g)
%\end{equation}
%which is valid for any $w \in \symm_n$ and any $f, g \in \wedge \{ \Theta_n, \Xi_n \}$.



\begin{prop}
\label{prop-equivariant}
The assignments $\pi \mapsto F_{\pi}$ and $\pi \mapsto f_{\pi}$ both induce 
$\symm_n$-module homomorphisms
\begin{equation*}
\CC[\Pi(n)] \longrightarrow \wedge \{ \Theta_n, \Xi_n \},
\end{equation*}
where $\CC[\Pi(n)]$ is endowed with the $\star$-action.
\end{prop}

The map in Proposition~\ref{prop-equivariant} is neither surjective (for degree reasons)
or injective (by our next result).  Let $\pi \in \NC(n)$ be a noncrossing partition and let $1 \leq i \leq n$
be such that $s_i(\pi)$ is not noncrossing.  Let $B_i$ and $B_{i+1}$ be the blocks of $s_i(\pi)$ containing
$i$ and $i+1$.
For brevity, 
 define a fermion
$F_{\sigma(s_i(\pi))} \in \wedge \{ \Theta_n, \Xi_n \}$  by
\begin{equation}
F_{\sigma(s_i(\pi))} := \begin{cases}
F_{\pi_1} + F_{\pi_2} & \text{if $|B_i| = |B_{i+1}| = 2$,} \\
F_{\pi_1} + F_{\pi_2} - F_{\pi_3} & \text{if $|B_i| > 2$ and $|B_{i+1}| = 2$,} \\
F_{\pi_1} + F_{\pi_2}  \quad \quad \, \,  \,  \,  - F_{\pi_4} & \text{if $|B_i| = 2$ and $|B_{i+1}| > 2$,} \\
F_{\pi_1} + F_{\pi_2} - F_{\pi_3} - F_{\pi_4} & \text{if $|B_i|, \, |B_{i+1}| > 2$,}
\end{cases}
\end{equation}
where $\pi_1, \dots, \pi_4 \in \NC(n)$ are as in \eqref{sigma-equation}.
We also define $f_{\sigma(s_i(\pi))} \in \wedge \{ \Theta_n, \Xi_n \}$ by
\begin{equation}
f_{\sigma(s_i(\pi))}  := 
(\xi_1 + \cdots + \xi_n) \odot F_{\sigma(s_i(\pi))}.
\end{equation}
The following result states that the fermions $F_{\pi}$ and $f_{\pi}$ satisfy the skein relations.
For complete versions of the proofs in this abstract, see \cite{KRSkein}.



\begin{thm}
\label{thm-skein} 
Let $\pi \in \NC(n)$ be a noncrossing partition and $1 \leq i \leq n-1$. 
Let $B_i$ and $B_{i+1}$ be the blocks of $s_i(\pi)$ containing $i$ and
$i+1$, respectively. Then
\begin{equation}
F_{s_i(\pi)} + F_{\sigma(s_i(\pi))} = 0 \quad \text{ and } \quad
f_{s_i(\pi)} + f_{\sigma(s_i(\pi))} = 0.
\end{equation}
in $\wedge \{ \Theta_n, \Xi_n \}$ where we interpret $\sigma(s_i(\pi)) = \pi$ if $s_i(\pi)$ is noncrossing.
\end{thm}


\begin{proof}[Proof sketch]  Assume $s_i(\pi)$ is not noncrossing.
Introduce a variant $\psi_B$ of the block operator $\rho_B$ on $\wedge \{ \Theta_n, \Xi_n \}$ by 
\begin{equation}
\psi_B(f) :=
\sum_{\substack{i, j \in B \\ i \neq j}} \xi_j \cdot (\theta_j \odot f) 
\end{equation}
for $f \in \wedge \{ \Theta_n, \Xi_n \}$. In particular, we have $\psi_B = 0$ when $B$ is a singleton.
Like the $\rho$-operators, the $\psi$-operators commute.
The $\psi$-operators remove the branching in the definition of $F_{\sigma(s_i(\pi))}$.
Writing $A := B_i - \{i\}$ and $B := B_{i+1} - \{i+1\}$, one checks the operator identity
\begin{multline}
\label{skein-operator-goal}
\psi_{A \sqcup \{i+1\}} \circ \psi_{B \sqcup \{i\}} + 
\psi_{A \sqcup \{i\}} \circ \psi_{B \sqcup \{i+1\}} + 
\psi_{A \sqcup B} \circ \psi_{\{i, i+1\}} \\ - 
\psi_{A } \circ \psi_{B \sqcup \{i, i+1\}} -
\psi_{A \sqcup \{i, i+1\}} \circ \psi_{B} = 0.
\end{multline}
Applying both sides of Equation~\eqref{skein-operator-goal} to $\theta_1 \cdots \theta_n$
(together with the block operators $\rho_C$ for blocks $C \in s_i(\pi)$ other than $B_i, B_{i+1}$)
yields $F_{s_i(\pi)} + F_{\sigma(s_i(\pi))} = 0$.  The further application of 
$(\xi_1 + \cdots + \xi_n) \odot (-)$ gives
$f_{s_i(\pi)} + f_{\sigma(s_i(\pi))} = 0$.
\end{proof}


Singleton blocks play a special role in the theory of skein actions. To this end, we define 
subsets $\Pi(n,k,m) \subseteq \Pi(n,k)$ and $\NC(n,k,m) \subseteq \NC(n,k)$ by
\begin{align*}
\Pi(n,k,m) &:= \{ \pi \in \Pi(n,k) \,:\, \text{$\pi$ has $m$ singletons} \}, \\
\NC(n,k,m) &:= \{ \pi \in \NC(n,k) \,:\, \text{$\pi$ has $m$ singletons} \}, 
\end{align*}
Our families of set partitions give rise to six subspaces of $\wedge \{ \Theta_n, \Xi_n \}$ as follows:
\begin{align*}
V(n) & := \mathrm{span}\{ F_{\pi} \,:\, \pi \in \NC(n) \}, &
W(n) & := \mathrm{span}\{ f_{\pi} \,:\, \pi \in \NC(n) \}, \\
V(n,k) & := \mathrm{span}\{ F_{\pi} \,:\, \pi \in \NC(n,k) \}, &
W(n,k) & := \mathrm{span}\{ f_{\pi} \,:\, \pi \in \NC(n,k) \}, \\
V(n,k,m) & := \mathrm{span}\{ F_{\pi} \,:\, \pi \in \NC(n,k,m) \}, &
W(n,k,m) & := \mathrm{span}\{ f_{\pi} \,:\, \pi \in \NC(n,k,m) \},
\end{align*}
Theorem~\ref{thm-skein} guarantees that these subspaces are closed under the action of $\symm_n$.
The next result states that they are isomorphic to the skein modules and implies that their 
defining spanning sets are in fact bases.



\begin{thm}
\label{thm-isomorphism}
Let $m \leq k \leq n$.
The action of $s_i$ on $\CC[\NC(n)]$ defined in Equation~\eqref{skein-si-action}
satisfies the Coxeter relations, and so extends to an action of $\symm_n$ on $\CC[\NC(n)]$ for 
which $\CC[\NC(n,k)]$ and $\CC[\NC(n,k,m)]$ are submodules. 

Furthermore,
the assignments $F_{\pi} \leftrightarrow f_{\pi} \leftrightarrow \pi$ induce $\symm_n$-module
isomorphisms
\begin{gather*}
V(n) \cong W(n) \cong \CC[\NC(n)],  \quad  \quad
V(n,k) \cong W(n,k) \cong \CC[\NC(n,k)],
\\
V(n,k,m) \cong W(n,k,m) \cong \CC[\NC(n,k,m)].
\end{gather*}
for any $m \leq k \leq n$.
The common Frobenius images of these modules are 
\begin{gather*}
\Frob \, \CC[\NC(n)] = \sum_{k = 1}^n \Frob \, \CC[\NC(n,k)], \quad
\Frob \, \CC[\NC(n,k)] = \sum_{m = 1}^k \Frob \, \CC[\NC(n,k,m)],
\\
\Frob \, \CC[\NC(n,k,m)] = s_{(k-m,k-m,1^{n-2k+m})} \cdot s_{(1^m)}.
\end{gather*}
\end{thm}

\begin{proof}[Proof sketch]
By Theorem~\ref{thm-skein}, the map $\pi \mapsto F_{\pi}$ 
gives $\symm_n$-epimorphisms from the skein modules to the $V$-modules.
One then proves, using parabolic symmetrizers and antisymmetrizers in 
$\CC[\symm_n]$, that $V(n,k,0) \cong S^{(k,k,1^{n-2k})}$ is the $\symm_n$-irreducible
of {\em flag shape} $(k, k, 1^{n-2k}) \vdash n$.
Next, one checks directly that 
\begin{equation}
V(n,k,m) \cong \mathrm{Ind}_{\symm_{n-m} \times \symm_m}^{\symm_n} V(n-m,k-m,0)
\otimes \sign_{\symm_m}
\end{equation}
as $\symm_n$-modules. This proves 
$\dim V(n,k,m) = | \NC(n,k,m) |$ so that the epimorphism
$V(n,k,m) \twoheadrightarrow \CC[\NC(n,k,m)]$ is an isomorphism.
The other isomorphisms are proven in a similar way.
\end{proof}

As mentioned earlier, the proof in \cite{RhoadesSkein} that the skein action of $s_i$ on
$\CC[\NC(n)]$ satisfied the Coxeter relations was lengthy and calculation intensive.
By contrast, fermions made the proof of Theorem~\ref{thm-isomorphism}
much cleaner. The simplicity of the block operators $\rho_B$ give conceptual reason
for `why' the skein action should exist.







\section{Resolving Set Partition Crossings}
\label{Resolving}

A {\em matching} of size $n$ is a set partition of $[n]$ in which every block has size 1 or 2.
Any matching may be transformed into a linear combination of noncrossing matchings by 
repeated use of the local transformation
\begin{center}

\begin{tikzpicture}[scale=0.3]


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	\node at (2,0) {$\mapsto$};
	\node at (6.3,0) {$+$};
    \node at (9.9,-.4) {$.$};
    

    
    
    \end{tikzpicture}
    
    \end{center}
    For example, resolving crossings in the `asterisk of order 4' yields
          \begin{center}

\begin{tikzpicture}[scale=0.4]


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    \end{tikzpicture}
    
    \end{center}
    Among other things, this resolution combinatorializes the representation theory of the 
    Temperley-Lieb algebra $\mathrm{TL}_n$, the Kazhdan-Lusztig cellular basis of the 
    $\symm_n$-irreducible of 2-row rectangular shape, $\mathfrak{sl}_2$-web bases, and
    the coordinate ring of the Grassmannian  of $2$-planes in $n$-space
    \cite{KungRota, RhoadesPoly, RT}.
    Thanks to skein actions, we can extend this resolution from matchings to arbitrary set partitions.


\begin{defn}
\label{p-map-definition}
Let $\pi \in \Pi(n)$ be a set partition. We define $p(\pi) \in \CC[\NC(n)]$ by
\begin{equation*}
p(\pi) = \sum_{\tau \in \NC(n)} c_{\pi,\tau} \cdot \tau,
\end{equation*}
where the $c_{\pi,\tau}$ are the unique coefficients so that
$F_{\pi} = \sum_{\tau \in \NC(n)} c_{\pi,\tau}  \cdot F_{\tau}$.
\end{defn}


When $\pi$ is a matching, $p(\pi)$ agrees with the resolution described above up to a global sign.
For a non-matching example, if $\pi = \{ 1, \, 2, \, 6 \mid 3, \, 4, \, 8 \mid 5, \, 7 \} \in \Pi(8)$, applying 
$p$ yields
  \begin{center}
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                {
                \coordinate (p\i) at (-45*\i + 113:\r);
                \filldraw(p\i) circle (\pointradius pt);
                \node[] at (-45*\i + 113:1.5) {\footnotesize $\i$};
                }
        }
            
    \def\r{1}           % pentagon radius
    \def\pointradius{3} % points radius
    
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                \fill[black!10] (p\a) -- (p\b) -- (p\f) -- (p\a);
             \fill[black!10] (p\c) -- (p\d) -- (p\h) -- (p\c);
                  \draw[thin] (p\c) -- (p\d) -- (p\h) -- (p\c);
                 \draw[thin] (p\a) -- (p\b) -- (p\f) -- (p\a);
              \draw[thick] (p\e) -- (p\g);
            \squa
        \end{scope}
        }
        
        \node at (2.5,0) {$\mapsto$};
         \node at (7,0) {$-$};
           \node at (11.2,0) {$+$};
           \node at (15.4,0) {$-$};
             \node at (19.7,0) {$+$};
             
                 \node at (4.8,-4.2) {$-$};
                 
                   \node at (9.1,-4.2) {$+$};
           \node at (13.35,-4.2) {$-$};             
      \node at (17.55,-4.2) {$+$};   
              
      \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=140+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\a) -- (p\b) -- (p\h) -- (p\a);
             \fill[black!10] (p\c) -- (p\d) -- (p\g) -- (p\c);
                  \draw[thin] (p\c) -- (p\d) -- (p\g) -- (p\c);
                 \draw[thin] (p\a) -- (p\b) -- (p\h) -- (p\a);
              \draw[thick] (p\e) -- (p\f);
            \squa
        \end{scope}
        }
        
    \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=200+100*\shft,yshift=-120]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\c) -- (p\d) -- (p\g) -- (p\h) -- (p\c);
                 \draw[thin] (p\c) -- (p\d) -- (p\g) -- (p\h) -- (p\c);
              \draw[thick] (p\a) -- (p\b);
                \draw[thick] (p\e) -- (p\f);
            \squa
        \end{scope}
        }
   
        \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=260+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\a) -- (p\b) -- (p\g) -- (p\h) -- (p\a);
                 \draw[thin] (p\a) -- (p\b) -- (p\g) -- (p\h) -- (p\a);
              \draw[thick] (p\c) -- (p\d);
                \draw[thick] (p\e) -- (p\f);
            \squa
        \end{scope}
        }
        
       \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=320+100*\shft,yshift=-120]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\a);
                 \draw[thin] (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\a);
              \draw[thick] (p\e) -- (p\h);
                \draw[thick] (p\f) -- (p\g);
            \squa
        \end{scope}
        }
        
        
        
        
        
         \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=380+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\a) -- (p\b) -- (p\h) -- (p\a);
             \fill[black!10] (p\c) -- (p\d) -- (p\e) -- (p\c);
                  \draw[thin] (p\c) -- (p\d) -- (p\e) -- (p\c);
                 \draw[thin] (p\a) -- (p\b) -- (p\h) -- (p\a);
              \draw[thick] (p\f) -- (p\g);
            \squa
        \end{scope}
        }
        
       \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=440+100*\shft,yshift=-120]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\c) -- (p\d) -- (p\e) -- (p\h) -- (p\c);
                 \draw[thin] (p\c) -- (p\d) -- (p\e) -- (p\h) -- (p\c);
              \draw[thick] (p\a) -- (p\b);
                \draw[thick] (p\f) -- (p\g);
            \squa
        \end{scope}
        }
        
        
       \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=500+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\a) -- (p\b) -- (p\e) -- (p\h) -- (p\a);
                 \draw[thin] (p\a) -- (p\b) -- (p\e) -- (p\h) -- (p\a);
              \draw[thick] (p\c) -- (p\d);
                \draw[thick] (p\f) -- (p\g);
            \squa
        \end{scope}
        }
        
        
        
        
        \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=620+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\c) -- (p\d) -- (p\h) -- (p\c);
             \fill[black!10] (p\e) -- (p\f) -- (p\g) -- (p\e);
                  \draw[thin] (p\c) -- (p\d) -- (p\h) -- (p\c);
                 \draw[thin] (p\e) -- (p\f) -- (p\g) -- (p\e);
              \draw[thick] (p\a) -- (p\b);
            \squa
        \end{scope}
        }
        
               \foreach \a/\b/\c/\d/\e/\f/\g/\h [count=\shft from 0] in   
       {1/2/3/4/5/6/7/8}
        {
        \begin{scope}[xshift=560+100*\shft,yshift=-120]
            \squa
            \foreach \i in {1,...,8}             
                \fill[black!10] (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\a);
                 \draw[thin] (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\a);
              \draw[thick] (p\e) -- (p\f);
                \draw[thick] (p\g) -- (p\h);
            \squa
        \end{scope}
        }
   
   
   
   \end{tikzpicture}
   \end{center}
   The map $p$ fixes noncrossing set partitions and
   is equivariant with respect to $\symm_n$-actions.


\begin{thm}
\label{thm-projection}
The linear map $p \colon \CC[\Pi(n)] \twoheadrightarrow \CC[\NC(n)]$ is an $\symm_n$-equivariant
projection where $\CC[\Pi(n)]$ carries the $\star$-action and $\CC[\NC(n)]$ carries the 
skein action.
\end{thm}

\begin{proof}
This follows from Theorem~\ref{thm-skein} and Theorem~\ref{thm-isomorphism}.
\end{proof}

\begin{cor}
\label{cor:local-symmetries}
Let $w \in \symm_n$ and $\pi \in \NC(n)$ be such that $w(\pi)$ is noncrossing. In the skein 
action on $\CC[\NC(n)]$ we have $w \cdot \pi = \sign(w) w(\pi)$.
\end{cor}

\begin{proof}
By Theorem~\ref{thm-projection}  we calculate
\begin{equation}
\sign(w) w(\pi) = \sign(w) p(w(\pi)) = p(w \star \pi) = w \cdot p(\pi) = w \cdot \pi,
\end{equation}
where we used that $w(\pi)$ and $\pi$ are noncrossing, hence fixed by $p$.
\end{proof}

To see Corollary~\ref{cor:local-symmetries} in action, consider applying the long cycle
$c = (1,2,3,4,5,6) \in \symm_6$ to $\pi = \{ 1, \, 5, \,6 \mid 2 \, 4 \mid 3 \} \in \NC(6)$.
Using the skein action, we calculate the action of $c = s_1 s_2 s_3 s_4 s_5$, cancelling
terms along the way:
 \begin{center}
   \begin{tikzpicture}[scale=0.4]


    \newcommand{\squa}
        {
        \foreach \i in {1,...,6}
                {
                \coordinate (p\i) at (-60*\i + 120:\r);
                \filldraw(p\i) circle (\pointradius pt);
                \node[] at (-60*\i + 120:1.5) {\footnotesize $\i$};
                }
        }
            
    \def\r{1}           % pentagon radius
    \def\pointradius{3} % points radius
    
      \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\d);
            \squa
        \end{scope}
        
        \node at (3,0) {$\xmapsto{ \, \, s_5 \, \,  }$};
         \node at (4.5,0) {$-$};
        
        \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=190+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\d);
            \squa
        \end{scope}
        }
        
         \node at (9.5,0) {$\xmapsto{ \, \, s_4 \, \,  }$};
          \node at (11,0) {$-$};
           \node at (15.8,0) {$-$};
            \node at (20.8,0) {$+$};
         
        \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=380+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\d);
            \squa
        \end{scope}
        }
        
        
        \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=520+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\e) -- (p\d);
            \squa
        \end{scope}
        }
        
       \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=660+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\b) -- (p\d) -- (p\e) -- (p\b) -- cycle;
		 \fill[black!10] (p\b) -- (p\d) -- (p\e) -- (p\b) -- cycle;
                \draw[thick] (p\a) -- (p\f);
            \squa
        \end{scope}
        }
          \node at (26,0) {$\xmapsto{ \, \, s_3 \, \,  }$};
        
        }
    
    \end{tikzpicture}
 %   \end{center}  
 %  \begin{center}
   \begin{tikzpicture}[scale=0.4]
   
     \newcommand{\squa}
        {
        \foreach \i in {1,...,6}
                {
                \coordinate (p\i) at (-60*\i + 120:\r);
                \filldraw(p\i) circle (\pointradius pt);
                \node[] at (-60*\i + 120:1.5) {\footnotesize $\i$};
                }
        }
            
    \def\r{1}           % pentagon radius
    \def\pointradius{3} % points radius



          
      
        \node at (4.90,0) {$+$};
        
        \node at (9.1,0) {$-$};
        \node at (13.9,0) {$\xmapsto{ \, \, s_2 \, \,  }$};
         \node at (15.2,0) {$-$};
         
       \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=80+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\f) -- (p\e) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\f) -- (p\e) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\c);
            \squa
        \end{scope}}
        
             \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=200+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\e) -- (p\c);
            \squa
        \end{scope}
        }
        
       \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=320+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\b) -- (p\c) -- (p\e) -- (p\b) -- cycle;
		 \fill[black!10] (p\b) -- (p\c) -- (p\e) -- (p\b) -- cycle;
                \draw[thick] (p\a) -- (p\f);
            \squa
        \end{scope}
        }
        
       \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=490+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\c);
            \squa
        \end{scope}
        
         \node at (19.3,0) {$+$};
          \node at (23.5,0) {$+$};
          \node at (27.8,0) {$-$};
           \node at (32,0) {$+$};
           
          
         
                \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=610+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\e) -- (p\c);
            \squa
        \end{scope}}
        
        \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=730+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\c);
            \squa
        \end{scope}}
        
           \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=850+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\b) -- (p\c) -- (p\e) -- (p\b) -- cycle;
		 \fill[black!10] (p\b) -- (p\c) -- (p\e) -- (p\b) -- cycle;
                \draw[thick] (p\a) -- (p\f);
            \squa
        \end{scope}}
        
        
                   \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=970+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\b) -- (p\c) -- (p\e) -- (p\b) -- cycle;
		 \fill[black!10] (p\b) -- (p\c) -- (p\e) -- (p\b) -- cycle;
                \draw[thick] (p\a) -- (p\f);
            \squa
        \end{scope}}
        
         \draw[ultra thick, red] (33.2,2) -- (35.2,-2);
         
          \draw[ultra thick, red] (29,2) -- (31,-2);
         
        }
       
   
        
        
    
    \end{tikzpicture}
%    \end{center}
%  \begin{center}
   \begin{tikzpicture}[scale=0.4]
   
     \newcommand{\squa}
        {
        \foreach \i in {1,...,6}
                {
                \coordinate (p\i) at (-60*\i + 120:\r);
                \filldraw(p\i) circle (\pointradius pt);
                \node[] at (-60*\i + 120:1.5) {\footnotesize $\i$};
                }
        }
            
    \def\r{1}           % pentagon radius
    \def\pointradius{3} % points radius


    \node at (14.5,0) {$\xmapsto{ \, \, s_1 \, \, }$};
        \node at (16,0) {$-$};
          
             
       \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=520+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\c);
            \squa
        \end{scope}
        
         \node at (20.5,0) {$-$};
          \node at (25,0) {$+$};
          \node at (29.5,0) {$-$};
           \node at (33.8,0) {$+$};
            \node at (38,0) {$+$};
              \node at (42.3,0) {$-$};
           
          
         
   \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=645+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\c) -- (p\e) -- (p\f) -- (p\c) -- cycle;
		 \fill[black!10] (p\c) -- (p\e) -- (p\f) -- (p\c) -- cycle;
                \draw[thick] (p\a) -- (p\b);
            \squa
        \end{scope}}
        
        \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=775+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\c) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\b) -- (p\c) -- (p\a) -- cycle;
                \draw[thick] (p\f) -- (p\e);
            \squa
        \end{scope}}
        
           \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=900+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\c) -- (p\e);
            \squa
        \end{scope}}
        
        
                   \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=1020+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\f) -- (p\e) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\f) -- (p\e) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\c);
            \squa
        \end{scope}}
        
     \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=1140+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\c) -- (p\e) -- (p\f) -- (p\c) -- cycle;
		 \fill[black!10] (p\c) -- (p\e) -- (p\f) -- (p\c) -- cycle;
                \draw[thick] (p\a) -- (p\b);
            \squa
        \end{scope}}
        
        \foreach \a/\b/\c/\d/\e/\f [count=\shft from 0] in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=1270+100*\shft,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\c) -- (p\a) -- cycle;
		 \fill[black!10] (p\a) -- (p\b) -- (p\c) -- (p\a) -- cycle;
                \draw[thick] (p\e) -- (p\f);
            \squa
        \end{scope}}
        
        
         \draw[ultra thick, red] (39.2,2) -- (41.2,-2);
         
          \draw[ultra thick, red] (21.8,2) -- (23.8,-2);
          
         \draw[ultra thick, red] (29,2) -- (25,-2);
         
          \draw[ultra thick, red] (46.5,2) -- (42.5,-2);
          
         \draw[ultra thick, red] (36.5,2) -- (35.5,-2);
         
          \draw[ultra thick, red] (19,2) -- (18,-2);
         
        
         
        }
   
        
        
    
    \end{tikzpicture}
    \end{center}
     As Corollary~\ref{cor:local-symmetries} says, the end result is
        $\sign(c) \cdot c(\pi) = - c(\pi) = - \{ 1, \, 2, \, 6 \mid 3, \, 5 \mid 4 \}$.
        Corollary~\ref{cor:local-symmetries} implies a cyclic sieving result
        of Reiner, Stanton, and White \cite{RSW}. The {\em $q$-Narayana number} is 
        \begin{equation}
        \Nar_q(n,k) := \frac{q^{(n-k)(n-k+1)}}{[n]_q} {n \brack k}_q {n \brack k-1}_q,
        \end{equation}
        where we use the standard $q$-analog notation
        \begin{equation*}
        [n] := 1 + q + \cdots + q^{n-1},  \quad [n]!_q := [n]_q [n-1]_q \cdots [1]_q, \quad \text{and} \quad 
        {n \brack k}_q := \frac{[n]!_q}{[k]!_q \cdot [n-k]!_q}.
        \end{equation*}




\begin{cor}
\label{cor:cyclic-sieving}
The triple $(\NC(n,k), \ZZ_n, \Nar_q(n,k))$ exhibits the cyclic sieving phenomenon where 
$\ZZ_n$ acts on $\NC(n,k)$ by rotation.
\end{cor}

\begin{proof}[Proof sketch]
Corollary~\ref{cor:local-symmetries} implies that the skein action of
$c = (1, \dots, n)$  on $\CC[\NC(n,k)]$ is the scalar $\sign(c) = (-1)^{n-1}$ times rotation.
 Theorem~\ref{thm-isomorphism}  gives the $\symm_n$-isomorphism
type of $\CC[\NC(n,k)]$. Now apply Springer's Theorem on regular elements.
\end{proof}

Given $\pi \in \Pi(n)$, calculating $p(\pi) \in \CC[\NC(n)]$ using Definition~\ref{p-map-definition}
involves expanding $F_{\pi}$ as a linear combination of $\{ F_{\tau} \,:\, \tau \in \NC(n) \}$.
There is a purely combinatorial way to compute $p(\pi)$ as follows.


If $\pi \in \Pi(n)$ is not noncrossing, 
let $A, B \in \pi$ be two blocks whose convex hulls cross on the circle
labeled clockwise with $1, 2, \dots, n$.  The union $A \cup B$ may be expressed as a cyclic sequence
$(C_1, C_2, \dots, C_{2m})$ of nonempty sets where each $C_{2i}$ is a cyclically contiguous
subset of $A$ and each $C_{2i+1}$ is a cyclically contiguous subset of $B$.
For example, if $\pi = \{ A \mid B \} \in \Pi(16)$ is the two-block set partition with 
\[
A = \{ 1, \, 2, \, 4,  \, 8, \, 9, \, 10, \, 12, \, 13, \, 14, \, 15, \, 16 \} \text{ and ]}
    B = \{ 3, \, 5, \, 6, \, 7, \, 11 \}
\]
then $m = 3$, and we may take 
\[
(C_1, \dots, C_6) = ( \{ 1, \, 2, \, 12, \, 13, \, 14, \, 15, \, 16 \}, \{3\}, \{4\},  \{5, \, 6, \, 7 \}, \{8, \, 9, \, 10 \}, \{11\} ).
\]

\begin{thm}
\label{combinatorial-resolution}
With $\pi, A, B,$ and $(C_1, C_2, \dots, C_{2m})$ as above we have
\begin{equation*}
F_{\pi} = \sum_{1 \leq i \leq j \leq 2m} \epsilon(i,j) \cdot F_{\pi(i,j)}
\end{equation*}
where $\pi(i,j)$ is obtained from $\pi$ by replacing $A$ and $B$ with 
$C := C_i \cup C_{i+1} \cup \cdots \cup C_j$ and $D := (A \cup B) - C$ and the coefficient
$\epsilon(i,j)$ is $0$ if either of $C$ or $D$ are singletons and $(-1)^{j-i}$ otherwise.
\end{thm}

Theorem~\ref{combinatorial-resolution} resolves the local crossing between the blocks $A, B \in \pi$.
In the example above, this resolution looks like
    \begin{center}
    \begin{tikzpicture}[scale = 0.4]
        \newcommand{\squa}
        {
        \foreach \i in {1,...,6}
                {
                \coordinate (p\i) at (-60*\i + 120:\r);
                \filldraw(p\i)[color = red] circle (\pointradius pt);
                }
                \node[] at (-60*1 + 120:1.6) {\scriptsize $C_1$};
                \filldraw[color = black] (-60*1 + 120:1)  circle (\pointradius pt);
                \node[] at (-60*2 + 120:1.6) {\scriptsize $C_2$};
                \filldraw[color = black] (-60*2 + 120:1)  circle (\pointradius pt);
                \node[] at (-60*3 + 120:1.6) {\scriptsize $C_3$};
                \filldraw[color = black] (-60*3 + 120:1)  circle (\pointradius pt);
                \node[] at (-60*4 + 120:1.6) {\scriptsize $C_4$};
                \filldraw[color = black] (-60*4 + 120:1)  circle (\pointradius pt);
                \node[] at (-60*5 + 120:1.6) {\scriptsize $C_5$};
                \filldraw[color = black] (-60*5 + 120:1)  circle (\pointradius pt);
                \node[] at (-60*6 + 120:1.6) {\scriptsize $C_6$};
                \filldraw[color = black] (-60*6 + 120:1)  circle (\pointradius pt);
        }
            
    \def\r{1}           % pentagon radius
    \def\pointradius{3} % points radius
    
      \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=0,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\c) -- (p\e) -- (p\a);
		 \fill[black!10] (p\a) -- (p\c) -- (p\e) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\d) -- (p\f) -- (p\b);
                \fill[black!10] (p\b) -- (p\d) -- (p\f) -- (p\b) -- cycle;
                \draw[] (-0.51,-0.3) -- (-0.51,0.3) -- (0,0.6) -- (0.51,0.3) -- (0.51,-0.3) -- (0,-0.6) -- (-0.51,-0.3) -- cycle;
               % \fill[black!40] (-0.51,-0.3) -- (-0.51,0.3) -- (0,0.6) -- (0.51,0.3) -- (0.51,-0.3) -- (0,-0.6) -- (-0.51,-0.3) -- cycle;
            \squa
        \end{scope}}
        
        \node at (2.8,0) {$=$};
        \node at (8.4,0) {$+$};
        \node at (14.0,0) {$+$};
        \node at (5.7,-4.2) {$-$};
        \node at (11.3,-4.2) {$-$};
        \node at (16.9,-4.2) {$-$};
        \node at (22.5,-4.2) {$-$};
        \node at (28.1,-4.2) {$-$};
        \node at (33.7,-4.2) {$-$};
        \node at (8.4,-8.4) {$+$};
        \node at (14.1,-8.4) {$+$};
        \node at (19.75,-8.4) {$+$};
        
              \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=160,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\b) -- (p\c) -- (p\d) -- (p\e) -- (p\f) --  (p\b);
                \fill[black!10] (p\b) --  (p\c) -- (p\d) -- (p\e) -- (p\f) -- (p\b) -- cycle;
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=320,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\c) -- (p\e) -- (p\f) -- (p\a);
                \fill[black!10] (p\a) --  (p\b) -- (p\c) -- (p\e) -- (p\f) -- (p\a) -- cycle;
            \squa
        \end{scope}}
        
        
        
      \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=480,yshift=0]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\f) -- (p\a);
                \fill[black!10] (p\a) --  (p\b) -- (p\c) -- (p\d) -- (p\f) -- (p\a) -- cycle;
            \squa
        \end{scope}}
   
        

        
               \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=240,yshift=-120]
            \squa
            \foreach \i in {1,...,6}
                \draw[thick]  (p\c) -- (p\d) -- (p\e) -- (p\f) -- (p\c) -- cycle;
                \fill[black!10]  (p\c) -- (p\d) -- (p\e) -- (p\f) -- (p\c) -- cycle;
                \draw[thick] (p\a) -- (p\b);
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=400,yshift=-120]
            \squa
            \foreach \i in {1,...,6}
                \draw[thick]  (p\a) -- (p\d) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \fill[black!10]  (p\a) -- (p\d) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\b) -- (p\c);
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=560,yshift=-120]
            \squa
            \foreach \i in {1,...,6}
                \draw[thick]  (p\a) -- (p\b) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \fill[black!10]  (p\a) -- (p\b) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\c) -- (p\d);
            \squa
        \end{scope}}

        \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=720,yshift=-120]
            \squa
            \foreach \i in {1,...,6}
                \draw[thick]  (p\a) -- (p\b) -- (p\c) -- (p\f) -- (p\a) -- cycle;
                \fill[black!10]  (p\a) -- (p\b) -- (p\c) -- (p\f) -- (p\a) -- cycle;
                \draw[thick] (p\d) -- (p\e);
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=880,yshift=-120]
            \squa
            \foreach \i in {1,...,6}
                \draw[thick]  (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\a) -- cycle;
                \fill[black!10]  (p\a) -- (p\b) -- (p\c) -- (p\d) -- (p\a) -- cycle;
                \draw[thick] (p\e) -- (p\f);
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=1040,yshift=-120]
            \squa
            \foreach \i in {1,...,6}
                \draw[thick]  (p\b) -- (p\c) -- (p\d) -- (p\e) -- (p\b) -- cycle;
                \fill[black!10]  (p\b) -- (p\c) -- (p\d) -- (p\e) -- (p\b) -- cycle;
                \draw[thick] (p\a) -- (p\f);
            \squa
        \end{scope}}
        
        
      \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=320,yshift=-240]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\a) -- (p\b) -- (p\c) -- (p\a) -- cycle;
                  \fill[black!10] (p\a) -- (p\b) -- (p\c) -- (p\a) -- cycle;
                  \draw[thick] (p\d) -- (p\e) -- (p\f) -- (p\d) -- cycle;
                  \fill[black!10] (p\d) -- (p\e) -- (p\f) -- (p\d) -- cycle;                                   
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=480,yshift=-240]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\b) -- (p\c) -- (p\d) -- (p\b) -- cycle;
                  \fill[black!10] (p\b) -- (p\c) -- (p\d) -- (p\b) -- cycle;
                  \draw[thick] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;
                  \fill[black!10] (p\a) -- (p\e) -- (p\f) -- (p\a) -- cycle;                                   
            \squa
        \end{scope}}
        
        
       \foreach \a/\b/\c/\d/\e/\f in   
       {1/2/3/4/5/6}
        {
        \begin{scope}[xshift=640,yshift=-240]
            \squa
            \foreach \i in {1,...,6}
                  \draw[thick] (p\c) -- (p\d) -- (p\e) -- (p\c) -- cycle;
                  \fill[black!10] (p\c) -- (p\d) -- (p\e) -- (p\c) -- cycle;
                  \draw[thick] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;
                  \fill[black!10] (p\a) -- (p\b) -- (p\f) -- (p\a) -- cycle;                                   
            \squa
        \end{scope}}
        

    
    \end{tikzpicture}
    \end{center} 
    where we have compressed the sets $C_i$ to vertices and terms with an isolated
    $C_2, C_3,$ and $C_6$ do not appear since these sets are singletons. 
    For arbitrary $\pi \in \Pi(n)$, repeated applications
    of this rule yield the  resolution $p(\pi)$ of the crossings in $\pi$.






\section{Fermionic Diagonal Coinvariants and the Skein Action}
\label{Fermionic}

Thus far, we have considered the fermions $F_{\pi}$ and $f_{\pi}$ as members of the exterior
algebra $\wedge \{ \Theta_n, \Xi_n \}$.
The next theorem establishes that $\{ f_{\pi} \,:\, \pi \in \NC(n) \}$ descends to a basis of 
the `extreme bidegree part' of its quotient $FDR_n$.


\begin{thm}
\label{thm-frobenius}
Let $k \leq n$.
The set $\{ f_{\pi} \,:\, \pi \in \NC(n,k) \}$ descends to 
a basis of $(FDR_n)_{n-k,k-1}$ and 
the composite map
\begin{equation*}
W(n,k) \hookrightarrow \wedge \{ \Theta_n, \Xi_n \}_{n-k,k-1} \twoheadrightarrow (FDR_n)_{n-k,k-1}
\end{equation*}
is an isomorphism of $\symm_n$-modules.
\end{thm}

\begin{proof}[Proof sketch]
We need only prove that that $\{ f_{\pi} \,:\, \pi \in \NC(n,k) \}$ descends to a basis.
Since $\dim \, (FDR_n)_{n-k,k-1} = \Nar(n,k)$, it suffices to show linear independence.
In \cite{KRFermion, OZ} it is proven that the defining ideal $I$ of $FDR_n$ is generated by three elements:
\[
\theta_1 + \cdots + \theta_n, \quad \xi_1 + \cdots + \xi_n, \quad \text{ and } \quad \theta_1 \xi_1 + \cdots + \theta_n \xi_n.
\]
One deduces that $\{ f_{\pi} \,:\,  \pi \in \NC(n,k) \}$ is orthogonal
to $I \cap \wedge \{ \Theta_n, \Xi_n \}_{n-k,k-1}$ under the inner product 
$\langle -, - \rangle$ on $\wedge \{ \Theta_n, \Xi_n \}_{n-k,k-1}$ which declares 
the `fermionic monomial basis' 
$\{ \theta_{i_1} \cdots \theta_{i_{n-k}} \xi_{j_1} \cdots \xi_{j_{k-1}} \,:\, 1 \leq i _1 < \cdots < i_{n-k} \leq n, \, 
1 \leq j_1 < \cdots < j_{k-1} \leq n \}$
to be orthonormal. 
The linear independence of 
$\{ f_{\pi} \,:\,  \pi \in \NC(n,k) \}$ in $(FDR_n)_{n-k,k-1}$ follows from 
Theorem~\ref{thm-isomorphism}.
\end{proof}

Theorem~\ref{thm-frobenius} gives a combinatorial basis for 
the portion of $FDR_n = \bigoplus_{i+j < n} (FDR_n)_{i,j}$ in extreme bidegrees $i+j = n-1$.
In a forthcoming paper, the first author will extend our results to  all bidegrees
 $i + j < n$.







\acknowledgements{The authors are grateful to Alessandro Iraci, Jongwon Kim, 
Rebecca Patrias,  Oliver Pechenik, and Jessica Striker for helpful conversations.}

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\end{document}
