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\title[Orbit harmonics and cyclic sieving]{Orbit harmonics and cyclic sieving: a survey}
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\author{Jaeseong Oh\thanks{\href{mailto:jaeseong_oh@snu.ac.kr}{jaeseong\_oh@snu.ac.kr}. Jaeseong Oh was partially supported by the Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the 
Ministry of Education (2020R1A6A3A13076804).}}

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\address{Department of Mathematical Sciences, Seoul National University}

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\abstract{Orbit harmonics is a tool in combinatorial representation theory which promotes the
(ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient. The cyclic sieving phenomenon is a notion in enumerative combinatorics which encapsulates the fixed-point structure of the action of a finite cyclic group $C$ on a finite set $X$ in terms of root-of-unity evaluations of an auxiliary polynomial $X(q)$. In this survey, we present a variety of sieving results obtained by applying orbit harmonics.
}

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\keywords{orbit harmonics, cyclic sieving, point locus}

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\begin{document}

\maketitle
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\section{Introduction}

We survey recent works (\cite{OR20}, \cite{HJO20+}, \cite{Oh20+}) on the application of orbit harmonics to cyclic sieving phenomena. The orbit harmonics comes from the idea to model a finite set $X$ geometrically as a finite point locus in a complex space. The relevant algebra has roots in (at least) the work of Kostant \cite{Kos63}. The idea of providing CSPs using orbit harmonics has been implicitly used in \cite{ARR15}, \cite{Dou18}, and \cite{Rho10}. Rhoades and the author gave this idea in an explicit and unified way in \cite{OR20}. The author applied this idea to the complex reflection group $G(r,1,n)$ to obtain sieving results concerning twisted rotations on colored words \cite{HJO20+}. Extending the work in \cite{OR20} to diagonal orbit harmonics, the author has provided a `tri-CSP' for matrices \cite{Oh20+} by using the work of Garsia and Haiman \cite{GH96}.

Let $X$ be a finite set with an action of a finite cyclic group $C = \langle c \rangle$
and $\omega = \exp(2 \pi i / |C|)$.
Let $X(q) \in \ZZ_{\geq 0}[q]$ be a polynomial with nonnegative integer coefficients. The triple
$(X, C, X(q))$ {\em exhibits the cyclic sieving phenomenon} \cite{RSW04} if for all $r \geq 0$ we have
\begin{equation}\label{CSP definition}
|X^{c^r}| = | \{ x \in X \,:\, c^r \cdot x = x \} | = X(\omega^r) = [ X(q) ]_{q = \omega^r}.
\end{equation}

More generally, if we have a finite set $X$ with a action of product $C_1\times\cdots\times C_m$ of $m$ cyclic groups, and a polynomial $X(q_1,\dots,q_m)$ of $m$ variables satisfy similar condition in Equation \eqref{CSP definition}, then we say the triple $(X,C_1\times\cdots\times C_m, X(q_1,\dots,q_m))$ \emph{exhibits the $m$-ary cyclic sieving phenomenon} (biCSP for $m=2$ and triCSP for $m=3$). 

Although ostensibly in the domain of enumerative combinatorics, the most desired proofs of CSPs are algebraic. In this survey, we present a systematic way to prove sieving results using orbit harmonics and provide various instances of CSPs.

The remainder of this survey is organized as follows. In {\bf Section~\ref{sec:backgrounds}} we give background. In {\bf Section~\ref{sec:Theorems of Springer and Morita--Nakajima}}, we describe how orbit harmonics gives a new perspective 
on the classical results of Springer and Morita-Nakajima.
In {\bf Section~\ref{sec:CSP generator}} we provide a sieving generating theorem (Theorem~\ref{sieving-generator}) and apply this to various combinatorial loci to prove instances of CSPs. In {\bf Section~\ref{sec:Orbit harmonics in diagonal setting}}, we consider diagonal orbit harmonics and obtain a new sieving result regarding $(q,t)-$Kostka polynomials and enumeration of matrices with certain symmetries.

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

\subsection{Symmetric Functions}
We denote by $\Lambda = \bigoplus_{d \geq 0} \Lambda_d$ the graded ring of symmetric functions 
in an infinite variable set $\xx = (x_1, x_2, \dots )$ over the ground field $\CC(q,t)$.
Here $\Lambda_d$ consists of symmetric functions of homogeneous degree $d$.
Two important elements of $\Lambda_d$ are the {\em homogeneous} and {\em elementary}
symmetric functions
\begin{equation*}
h_d(\xx) := \sum_{i_1 \leq \cdots \leq i_d} x_{i_1} \cdots x_{i_d} \quad \quad \text{and} \quad \quad
e_d(\xx) := \sum_{i_1 < \cdots < i_d} x_{i_1} \cdots x_{i_d}.
\end{equation*}
By restricting $h_d(\xx)$ and $e_d(\xx)$ to a finite variable set $\xx_n = \{x_1, \dots, x_n \}$,
we obtain the homogeneous and elementary
symmetric polynomials
$h_d(\xx_n)$ and $e_d(\xx_n)$.








Bases of $\Lambda_n$ are indexed by partitions of $n$. For a partition $\lambda \vdash n$, we let
\begin{equation*}
h_{\lambda}(\xx), \quad e_{\lambda}(\xx), \quad s_{\lambda}(\xx), \quad 
 \widetilde{Q}_{\lambda}(\xx;q) \quad \text{and} \quad \widetilde{H}_{\lambda}(\xx;q,t)
\end{equation*}
denote the associated {\em homogeneous symmetric function}, {\em elementary symmetric function, Schur
function,} {\em Hall--Littlewood symmetric function,} and \emph{Macdonald symmetric function}. For any partition
$\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots )$ of $n$ the $h$-, $e$- and Schur functions are defined by 
\begin{equation*}
h_{\lambda}(\xx) := \prod_{i\ge 1}h_{\lambda_i}(\xx), \quad
e_{\lambda}(\xx) := \prod_{i\ge 1}e_{\lambda_i}(\xx) \quad \text{and} \quad
s_{\lambda}(\xx) := \sum_{T\in \operatorname{SSYT}(\lambda)} \xx^T,
\end{equation*}
where $\operatorname{SSYT}(\lambda)$ denotes the set of semistandard tableaux of shape $\lambda$. The (modified) Macdonald polynomial $\widetilde{H}_\lambda(\xx;q,t)$ are the basis of $\Lambda_n$ defined by so called \emph{triangularity} and \emph{normalization} axioms (see \cite{HHL04} for example). The (modified) Hall--Littlewood symmetric function $\widetilde{Q}_{\lambda}(\xx;q)$ is given by a specialization $\widetilde{Q}_\lambda(\xx;q)=\widetilde{H}_\lambda(\xx;0,q)$.

\subsection{Representation theory of \texorpdfstring{ $\symm_n$}{Lg}}

Irreducible representations of the symmetric group $\symm_n$ are in one-to-one correspondence
with partitions $\lambda \vdash n$. We let $S^{\lambda}$ denote the
irreducible module corresponding to $\lambda$. If $V$ is any finite-dimensional $\symm_n$-module, there are unique 
multiplicities $c_{\lambda}$ so that $V \cong \bigoplus_{\lambda \vdash n} c_{\lambda} S^{\lambda}$.
The {\em Frobenius image} of $V$ is the symmetric function
\begin{equation*}
\Frob(V) := \sum_{\lambda \vdash n} c_{\lambda} s_{\lambda}(\xx). 
\end{equation*}

If $V$ is graded (or bigraded) $\mathfrak{S}_n$-module as $V=\bigoplus_{d\ge0} V_d$ (or $V=\bigoplus_{d,e\ge0} V_{d,e}$) the graded Frobenius image is the symmetric function over $\CC(q,t)$ given by
\begin{equation*}
\grFrob(V;q) :=\sum_{d\ge0}\Frob(V_d)q^d \quad
    (\text{or }\grFrob(V;q,t):=\sum_{d,e\ge0} \Frob(V_{d,e})q^d t^e).
\end{equation*}

\subsection{Complex reflection groups} 

A finite subgroup $W \subseteq \GL_n(\CC)$ is a {\em reflection group} if it is generated by reflections. A complex reflection group $W$ acts on the polynomial ring $\CC[\xx_n] := \CC[x_1, \dots, x_n]$ by linear substitutions. Let $\langle \CC[\xx_n]^W_+ \rangle \subseteq \CC[\xx_n]$ be the ideal generated by the subspace of $W$-invariants with vanishing constant term.
The {\em coinvariant ring} attached to $W$ is the quotient $R_W := \CC[\xx_n]/\langle \CC[\xx_n]^W_+ \rangle.$ The ring $R_W$ also has a graded $W$-module structure.

For any irreducible $W$-module $U$, the {\em fake degree polynomial} $f^U(q)$ is the graded multiplicity
of $U$ in the coinvariant ring.  That is, we define
\begin{equation*}
f^U(q) := \sum_{d \geq 0} m_{U,d} q^d
\end{equation*}
where $m_{U,d}$ is the multiplicity of $U$ in the degree $d$ piece $(R_W)_d$ of $R_W$.

An element $c \in W$ is {\em regular} if it possesses an eigenvector $v \in \CC^n$ which has full $W$-orbit. Such an eigenvector $v$ is called a {\em regular eigenvector}. For example when $W = \symm_n$, an element in $W$ is a regular if and only if it is a power of an $n$-cycle or an $(n-1)$-cycle.

\subsection{Orbit harmonics}\label{subsec: orbit harmonics}

Let $X \subseteq \CC^n$ be a finite set of points which is closed under the action of $W \times C$ where $W \subseteq \GL_n(\CC)$ is a (finite) complex reflection group, and $C$ is a finite cyclic group acting on $\CC^n$ by root-of-unity scaling.
Let $\II(X) := \{ f \in \CC[\xx_n] \,:\, f(v) = 0 \, \text{ for all $v \in X$} \}$ be the ideal of polynomials in $\CC[\xx_n]$ which vanish on $X$.
Since  $X$ is finite, Lagrange Interpolation affords a $\CC$-algebra isomorphism
\begin{equation}
\label{first-orbit-iso}
\CC[X] \cong \CC[\xx_n]/\II(X)
\end{equation}
where $\CC[X]$ is the algebra of all functions $X \rightarrow \CC$.
Since $X$ is $W \times C$-stable, \eqref{first-orbit-iso} is also 
an isomorphism of ungraded $W \times C$-modules.
 
 For any nonzero polynomial $f \in \CC[\xx_n]$, let $\tau(f)$ be the highest degree component of $f$.
 That is, if $f = f_d + \cdots + f_1 + f_0$ with $f_i$ homogeneous of degree $i$ and $f_d \neq 0$, we set
 $\tau(f) := f_d$.  Given our locus $X$ with ideal $\II(X)$ as above, we define a homogeneous 
 ideal $\TT(X)$ by
 \begin{equation*}
 \TT(X) := \langle \tau(f) \,:\, f \in \II(X), \, \, f \neq 0 \rangle \subseteq \CC[\xx_n].
 \end{equation*}
 The ideal $\TT(X)$ is the {\em associated graded} ideal of $\II(X)$ and is often denoted
 $\mathrm{gr \,} \II(X)$.  From the construction, $\TT(X)$ is homogeneous and stable under $W \times C$.
 The isomorphism \eqref{first-orbit-iso} extends to an isomorphism of $W \times C$-modules
 \begin{equation}
\label{second-orbit-iso}
\CC[X] \cong \CC[\xx_n]/\II(X) \cong \CC[\xx_n]/\TT(X)
\end{equation}
where $\CC[\xx_n]/\TT(X)$ has the additional structure of a graded $W \times C$-module on which
$C$ acts by scaling in each fixed degree. 

The procedure $X \leadsto \CC[\xx_n]/\TT(X)$ which promotes the (ungraded) locus $X$ to the graded module $\CC[\xx_n]/\TT(X)$ is known as {\em orbit harmonics}. The terminology comes from the seminal work of Garsia and Haiman \cite{GH08} in which they studied point sets that were `orbits' of reflection group actions where there is a correspondence between coinvariant and `harmonic' space. The general procedure of $X \leadsto \CC[\xx_n]/\TT(X)$ is known as \emph{deformation to the normal cone} and this also can be applied to prove sieving results as noted in Vic Reiner's talk \cite{Rei18}. Nevertheless, we prefer to call this procedure orbit harmonics because we choose finite set $X$ to be orbits of reflection group actions.
\section{Theorems of Springer and Morita--Nakajima}\label{sec:Theorems of Springer and Morita--Nakajima}
Before applying orbit harmonics to prove sieving results, 
we state representation-theoretic results
of Springer \cite{Spr74} and Morita-Nakajima \cite{MN06},
which will be useful in our combinatorial work.
We explain how orbit harmonics may be used to prove these results.
\subsection{Springer's theorem on regular elements}\label{springer-subsection}
Let $W \subseteq \GL_n(\CC)$ be a complex reflection group and let $c \in W$ be a regular element
with regular eigenvector $v \in \CC^n$ whose eigenvalue is $\omega \in \CC^{\times}$.
Let $C = \langle c \rangle$ be the cyclic subgroup of $W$ generated by $c$.  We regard the coinvariant ring
$R_W$ as a graded $W \times C$-module, where $W$ acts by linear substitutions and the generator
$c \in C$ sends each variable $x_i$ to $\omega x_i$.

\begin{theorem}[Springer \cite{Spr74}]
\label{springer-theorem}
Consider the action of $W \times C$ on $W$ given by $(u, c^r) \cdot w := uwc^{-r}$.
Then $\CC[W]$ is isomorphic to $R_W$ as an ungraded $W \times C$-module.
\end{theorem}

\begin{proof}
We describe an argument of Kostant \cite{Kos63} using orbit harmonics. We let $C$ act on $\CC^n$ by the rule $c \circ v' := \omega^{-1} v'$ for all $v' \in \CC^n$. The corresponding action of $C$ on $\CC[\xx_n]$ by linear substitutions sends $x_i$ to $\omega x_i$ 
for all $i$. We may regard $\CC^n$ as a $W \times C$-module in this way.

Define the {\em Springer locus} to be the $W$-orbit of the regular eigenvector $v$ of $c$:
\begin{equation*}
W \cdot v := \{ w \cdot v \,:\, w \in W \} \subseteq \CC^n
\end{equation*}
The locus
$W \cdot v$ is closed under the action of $W$ and $\circ$-action of $C$. We may regard $W\cdot v$ as a $W \times C$-set. 
The regularity of $v$ shows that
the map $w \mapsto w \cdot v$ furnishes a 
$W \times C$-equivariant bijection 
\begin{equation*}
\label{ungraded-springer-isomorphism}
W \xrightarrow{ \, \sim \, } W \cdot v
\end{equation*}
where the action of $W \times C$ on $W$ is as in Theorem~\ref{springer-theorem}.

Chevalley \cite{Chevalley} proved that there exist 
algebraically independent $W$-invariant polynomials
$f_1, \dots, f_n$ of homogeneous positive degree such that 
$\CC[\xx_n]^W = \CC[f_1, \dots, f_n]$. Furthermore, we have isomorphisms of ungraded $W$-modules
\begin{equation*}
R_W = \CC[\xx_n]/\langle f_1, \dots, f_n \rangle \cong \CC[W].
\end{equation*}
The $W$-invariance of $f_i$ implies that $f_i-f_i(v)\in \II(W\cdot v)$ and taking the top degree component gives $f_i \in \TT(W\cdot v)$. On the other hand, 
\begin{equation*}
\dim ( \CC[\xx_n] / \langle f_1, \dots, f_n \rangle ) = \dim \CC[W] = |W| = |W \cdot v| =
\dim ( \CC[\xx_n] / \TT(W \cdot v) ),
\end{equation*}
so we have $\langle f_1, \dots, f_n \rangle = \TT(W \cdot v).$

Finally, orbit harmonics furnishes isomorphisms of ungraded $W \times C$-modules
\begin{equation*}
R_W = 
\CC[\xx_n]/\langle f_1, \dots, f_n \rangle =
\CC[\xx_n]/\TT(W \cdot v) \cong
\CC[W \cdot v] \cong \CC[W]
\end{equation*}
where the last isomorphism used
the $W \times C$-equivariant bijection \eqref{ungraded-springer-isomorphism}.
\end{proof}


\subsection{A theorem of Morita--Nakajima via orbit harmonics}\label{MN-subsection}

In this subsection we consider the case of the symmetric group $W = \symm_n$.
We fix a 
weak composition $\mu = (\mu_1, \dots, \mu_k)$ of $n$ with $k$
parts where $\mu_i = \mu_{i+a}$ for all $i$ with subscripts interpreted modulo $k$.
Let $c$ be a fixed generator of the cyclic group $\ZZ_{k/a}$.
Morita and Nakajima proved \cite{MN06} a variant
of Springer's theorem as follows. 

Let $W_{\mu}$ be the family words $w_1 \dots w_n$ over the alphabet $[k]$
in which the letter $i$ appears $\mu_i$ times. 
The set $W_{\mu}$ carries an action of 
$\symm_n \times \ZZ_{k/a}$ where $\symm_n$ acts by 
subscript permutation and  $\ZZ_{k/a}$ acts by
$c: w_1 \cdots w_n \mapsto (w_1 + a) \cdots (w_n + a)$ where letter values are interpreted modulo $k$.
Extending by linearity, the space $\CC[W_{\mu}]$ is a $\symm_n \times \ZZ_{k/a}$-module.

Let $I_{\mu} \subseteq \CC[\xx_n]$ be the {\em Tanisaki ideal} attached to the 
composition $\mu$ and let $R_{\mu} := \CC[\xx_n]/I_{\mu}$ be the corresponding 
{\em Tanisaki quotient ring}.  The ring $R_{\mu}$ is a graded $\symm_n$-module which has several descriptions (\cite{Tanisaki}, \cite{GP92}). One way to describe this module is via orbit harmonics as follows.

Let $\alpha_1, \dots, \alpha_k \in \CC$ be distinct complex numbers. We have $I_{\mu} = \TT(W_{\mu})$ as ideals in $\CC[\xx_n]$ by considering the set $W_{\mu} \subseteq \CC^n$ as a point locus under the following identification
\begin{equation*}
w_1 \dots w_n \leftrightarrow (\alpha_{w_1}, \dots, \alpha_{w_n}).
\end{equation*}
The orbit harmonics interpretation of $R_{\mu}$ was used by Garsia and Procesi \cite{GP92} to derive 
$$\grFrob(R_\mu;q)=\widetilde{Q}_\mu(\xx;q).$$

Let $\omega := \exp(2 a \pi i / k)$ be a primitive  $(k/a)^{th}$ root-of-unity. We extend the graded
$\symm_n$-action on $R_{\mu}$ to a graded $\symm_n \times \ZZ_{k/a}$-action by letting 
the distinguished generator
$c \in \ZZ_{k/a}$ scale by $\omega^d$ in homogeneous degree $d$.

\begin{theorem}
(Morita-Nakajima \cite[Theorem 13]{MN06})
\label{mn-theorem}
We have an isomorphism of ungraded $\symm_n \times \ZZ_{k/a}$-modules
\begin{equation*}
\CC[W_{\mu}] \cong R_{\mu}.
\end{equation*}
\end{theorem}
The proof in \cite{MN06} involves
tricky symmetric function manipulations involving the Hall--Littlewood polynomials
$\widetilde{Q}_{\mu}(\xx;q)$ when $q$ is a root of unity, and relies on further intricate symmetric function identities due to Lascoux-Leclerc-Thibon \cite{LLT}.
Orbit harmonics gives an easier and more conceptual proof.


\section{Cyclic sieving generating theorem}\label{sec:CSP generator}

The `generating theorem' for sieving results in \cite{OR20} is as follows. The heart of the proof comes from the isomphism~\eqref{second-orbit-iso} and Springer's theorem (Theorem~\ref{springer-theorem}).

\begin{theorem}
\label{sieving-generator}
Let $W \subseteq \GL_n(\CC)$ be a complex reflection group, $C' = \langle c' \rangle$ be the subgroup of $W$ generated by a regular element $c'$, and 
$\omega := \exp(2 \pi i / k)$.
Let $C = \langle c \rangle$ be a cyclic group of order $k$ and consider the action
of $W \times C$ on $\CC^n$ where $c$ scales by $\omega$ and $W$ acts by left multiplication. Let $X \subseteq \CC^n$ be a finite point set that is closed under the action of $W \times C$.

\begin{enumerate}
\item  Suppose that the isomorphism type of the degree $d\ge0$ piece of 
$\CC[\xx_n]/\TT(X)$ is given by
\begin{equation*}
( \CC[\xx_n]/\TT(X) )_d \cong \bigoplus_{U \in \mathrm{Irr}(W)} U^{\oplus m_{U,d}}.
\end{equation*}
The triple $(X, C \times C', X(q,t))$ exhibits the bicyclic sieving phenomenon where 
\begin{equation*}
X(q,t) = \sum_{U \in \mathrm{Irr}(W)} \sum_{d \geq 0} m_{U,d} q^d f^{U^*}(t).
\end{equation*}
\item Let $G \subseteq W$ be a subgroup. The set $X/G$ of $G$-orbits in $X$ carries a natural
$C$-action and the triple $(X/G, C, X(q))$ exhibits the cyclic sieving phenomenon where
\begin{equation*}
X(q) = \Hilb( (\CC[\xx_n]/\TT(X))^G; q).
\end{equation*}
\end{enumerate}
\end{theorem}

\begin{remark}
Note that if we have $$\grFrob(\CC[\xx_n]/\TT(X);q)=\sum_{\lambda\vdash n}c_{\lambda}(q)s_{\lambda},$$ then the Hilbert series of $G$-invariant space for $W=\mathfrak{S}_n$, for $G=C_n$ a cyclic group generated by a long cycle $(1,2,\dots,n)$, and for $G=H_{n/2}$ a hyperoctahedral group, respectively are given by
\begin{align*}
\Hilb((\CC[\xx_n]/\TT(X)^{\mathfrak{S}_n};q) &=
c_{(n)}(q), \\
\Hilb((\CC[\xx_n]/\TT(X))^{C_n};q) &= \sum_{\substack{\lambda\vdash n}}c_{\lambda}(q)a_{\lambda,0}, \\
\Hilb((\CC[\xx_n]/\TT(X))^{H_{n/2}};q) &= \sum_{\substack{\lambda\vdash n \\ \lambda: even}} c_{\lambda}(q),
\end{align*}
where $a_{\lambda,0}$ is the number of standard tableaux of shape $\lambda$ with major index $0$ modulo $n$.

\end{remark}


\subsection{The functional loci}\label{subsec:the functional loci}

The loci considered in this section correspond to arbitrary, injective, and surjective functions between finite sets inspired by Rota's {\em Twelvefold Way}.

\begin{defn}
\label{functional-locus-definition}
Given integers $n$ and $k$, set $\omega := \exp(2 \pi i / k)$.
We define the following three point sets in $\CC^n$ closed under the action of $\symm_n \times \ZZ_k$, where $\ZZ_k$ scales by 
$\omega$:
\begin{align*}
X_{n,k} &:= \{ (a_1, \dots, a_n) \,:\, a_i \in \{ \omega, \omega^2, \dots, \omega^k \} \} \\
Y_{n,k} &:= \{ (a_1, \dots, a_n) \in X_{n,k} \,:\, \text{$a_1, \dots, a_n$ are distinct} \} \\
Z_{n,k} &:= \{ (a_1, \dots, a_n) \in X_{n,k} \,:\, \{a_1, \dots, a_n\} = \{\omega, \omega^2, \dots \omega^k \} \}
\end{align*}
\end{defn}
The table below summarizes combinatorial objects for sieving results obtained by exploiting Theorem~\ref{sieving-generator} of each functional locus $X_{n,k}$, $Y_{n,k}$, and $Z_{n,k}$.

\begin{table}[H]\small
\centering
    \begin{tabular}{cccc}\toprule
     & $X_{n,k}$ & $Y_{n,k}$ & $Z_{n,k}$ \\\midrule
     
     $G=\mathfrak{S}_n$ & $WComp(n,k)$ & $[n] \choose k$ & $Comp(n,k)$ \\\midrule
     
     $G=C_n$ & necklaces & necklaces with distinct letters & necklaces with all letters used \\\midrule
     
     $G=H_{\frac{n}{2}}$ & graphs & graphs with degree $\le 1$ & graph without an isolated vertex\\\bottomrule
\end{tabular}
    \label{CSP table}
\end{table} 

\subsection{Other combinatorial loci}\label{subsec:other combinatorial loci}

\subsubsection{Springer locus}
In this subsection we return to the setting of an arbitrary complex reflection group $W \subseteq \GL_n(\CC)$
acting on $\CC^n$.  
We fix a regular element $c \in W$ with regular eigenvector $v \in \CC^n$ and corresponding regular 
eigenvalue $\omega \in \CC$, so that $c \cdot v = \omega v$. We also let $C := \langle c \rangle$ be the subgroup
of $W$ generated by $c$.


Let $W$-orbit $W \cdot v = \{ w \cdot v \,:\, w \in W \} \subseteq \CC^n$ be the Springer locus.
Subsection~\ref{springer-subsection} shows that the Springer locus is closed under the action
of the group $W \times C$, where $W$ acts by its natural action on $\CC^n$ and $C$ acts by the rule
$c: v' \mapsto \omega v'$ for all $v' \in \CC^n$.
(Note that this is different from the $\circ$-action $c \circ v' := \omega^{-1} v'$ of $C$ considered
in Subsection~\ref{springer-subsection}.) By the discussion in Subsection~\ref{springer-subsection} and Theorem~\ref{sieving-generator}, we have the following.

\begin{theorem}
\label{springer-bicsp}
Let $c, c' \in W$ be regular elements and let $C = \langle c \rangle, C' = \langle c' \rangle$ be the cyclic 
subgroups which they generate. The product of cyclic groups $C \times C'$ acts on $W$ by the rule
$(c, c') \cdot w := c' w c$. The triple $(W, C \times C', W(q,t))$ exhibits the bicyclic sieving phenomenon where
\begin{equation*}
W(q,t) := \sum_{U} f^U(q) f^{U^*}(t)
\end{equation*}
and the sum is over all (isomorphism classes of) irreducible $W$-modules $U$.
\end{theorem}

Theorem~\ref{springer-bicsp} is a result of Barcelo, Reiner, and Stanton
\cite[Thm. 1.4]{BRS08}. 
In \cite{BRS08} the polynomial $W(q,t)$ is referred to as a {\em bimahonian distribution}.
\subsubsection{The Tanisaki Locus}
Throughout this subsection, we fix 
a weak composition $\mu = (\mu_1, \dots, \mu_k)$ of $n$ into $k$ parts which satisfies
$\mu_i = \mu_{i+a}$ for all $i$, where indices are interpreted modulo $k$. 
If $\omega := \exp(2 \pi i/k)$, define the {\em Tanisaki locus} $X_{\mu} \subseteq \CC^n$ by
\begin{equation*}
X_{\mu} := \{ (\alpha_1, \dots, \alpha_n) \,:\, \alpha_j = \omega^i \text{ for precisely $\mu_i$ values of $j$} \}.
\end{equation*}
As discussed in Subsection~\ref{MN-subsection}, Garsia-Procesi \cite{GP92} proved that 
$\TT(X_{\mu})$ is the Tanisaki ideal and 
$\grFrob(\CC[\xx_n]/\TT(X_{\mu}); q) = \widetilde{Q}_{\mu}(\xx;q)$.
We have the following bicyclic sieving result.

\begin{theorem}
\label{word-bicyclic}
Let $W_{\mu}$ be the set of length $n$ words $w_1 \dots w_n$ of content $\mu$. The set $W_{\mu}$ carries an action of 
$\ZZ_n \times \ZZ_{k/a}$, where $\ZZ_n$ acts by word rotation and $\ZZ_{k/a}$ acts by adding $a$ to each letter modulo $k$. The triple $(W_{\mu}, \ZZ_n \times \ZZ_{k/a}, X(q,t))$ exhibits the 
bicyclic sieving phenomenon, where
\begin{equation*}
X_{\mu}(q,t) = \sum_{\lambda \vdash n} 
\widetilde{K}_{\lambda,\mathrm{sort}(\mu)}(q) f^{\lambda}(t)
\end{equation*}
and $\mathrm{sort}(\mu)$ is the partition obtained by sorting the parts of $\mu$ in weakly decreasing order.
\end{theorem}

%When $\mu = (1^n)$ and $a = 1$, the set $W_{\mu}$ may be identified with the family of $n \times n$ 
%permutation matrices and the action of $\ZZ_n \times \ZZ_n$ is by row and column rotation.
%In this case, Theorem~\ref{word-bicyclic} is a result of Barcelo, Reiner, and Stanton \cite{BRS}.
Theorem~\ref{word-bicyclic} was proven in the unpublished work of Reiner and White.
A proof of Theorem~\ref{word-bicyclic} in full generality using Theorem~\ref{mn-theorem} is in \cite{Rho10}. 

\subsubsection{Colored locus}\label{sec:Orbit harmonics for general complex reflection groups}
The complex reflection group $G(r,1,n)\le \GL_n(\CC)$ is a group of $n\times n$ monomial matrices whose nonzero entries are $\zeta^i$ for some $i$, where $\zeta:=e^{\frac{2\pi i}{r}}\in \mathbb{C}$. Irreducible representations of $G(r,1,n)$ are in one-to-one correspondence with $r$-tuple $\lambda^\bullet=(\lambda^{(0)},\dots,\lambda^{(r-1)})$ of partitions with total size $|\lambda^{(0)}|+\cdots+|\lambda^{(r-1)}|=n$. We denote the irreducible representation corresponding to $\lambda^\bullet$ by $S^{\lambda^\bullet}$. Let $\Lambda^{G(r,1,n)}:=\Lambda^{\otimes r}$ be $r^{th}$ tensor of the symmetric function ring $\Lambda$ (with variables $x^{(i)}$'s). For any finite-dimensional $G(r,1,n)$-module $V$, there extists unique 
multiplicity $c_{\lambda^\bullet}$ for each $\lambda^\bullet$ so that $V \cong \bigoplus_{\lambda^\bullet \vdash n} c_{\lambda^\bullet} S^{\lambda^\bullet}$.
The {\em Frobenius image} of $V$ is defined by
\begin{equation*}
\Frob^{G(r,1,n)}(V) := \sum_{\lambda^\bullet \vdash n} c_{\lambda^\bullet} s_{\lambda^\bullet}(\xx) \in \Lambda^{G(r,1,n)},
\end{equation*}
where $s_{\lambda^\bullet}(\xx)=s_{\lambda^{(0)}}(\xx^{(0)})\cdots s_{\lambda^{(r-1)}}(\xx^{(r-1)}).$ We define $\grFrob^{G(r,1,n)}$ in a usual way.

Suppose that a finite set $X\subseteq \CC^n$ is invariant under the action of $\symm_n\times C_k$, where $\symm_n$ acts on $X$ by permuting coordinates and a generator $c \in C_k$ acts on $X$ by $k^{th}$ root of unity scaling. Then a \emph{r-colored version} of $X$, 
$$\operatorname{Col_r}(X):=\{(\zeta^{c_1}x_1^{\frac{1}{r}},\dots,\zeta^{c_n}x_n^{\frac{1}{r}}\} : (x_1,\dots,x_n)\in X, c_1,\dots,c_n\in\{0,1,\dots,r-1\}\}$$
is invariant under action of $G(r,1,n)\times C_{kr}$, where $G(r,1,n)$ acts by left multiplication and $C_{kr}$ acts by scaling a $kr^{th}$ root of unity. Then we have the following equivalence as ungraded $G(r,1,n)\times C_{kr}$-modules
$$\mathbb{C}[\operatorname{Col_r}(X)]\cong \mathbb{C}[x_1,\dots,x_n]/\operatorname{Col_r}(T(X)),$$
where $\operatorname{Col_r}(T(X))$ is the image of $T(X)$ under $r^{th}$ power ring homomorphism given by $x_k\mapsto x_k^r$. If we can calculate graded Frobenius image of $\CC[\xx_n]/\operatorname{Col_r}(T(X))$, then applying Theorem~\ref{sieving-generator} gives a sieving result involving `colored words'.

For example, let $X=\{(1,1,\dots,1)\}\subseteq \CC^n$ be a set with a single element (with $\symm_n\times C_1$-action). Then $\operatorname{Col_2}(X)=\{(a_1,\dots,a_n)|a_i\in\{-1,1\}\}$ can be regarded as the set $\operatorname{BW_n}$ of binary words of length $n$. By a graded Frobenius character formula (Proposition 9 \cite{OS19}) and usual fact for a plethystic substitution, we have
\begin{align*}
    \operatorname{grFrob}^{G(2,1,n)}(\CC[\xx_n]/\operatorname{Col_2}(T(X));q)&=\operatorname{grFrob}(\mathbb{C}[\xx_n]/T(X);q^2)[\xx^{(0)}+q\xx^{(1)}]\\
    &=s_{(n)}[\xx^{(0)}+q\xx^{(1)}]=\sum_{k=0}^{n} q^k s_{(n-k)}(\xx^{(0)})s_{(k)}(\xx^{(1)}).
\end{align*}

By Theorem~\ref{sieving-generator} and a formula for fake degree polynomial $f^{\lambda^\bullet}(q)$ for $G(r,1,n)$ in \cite{BRS08}, we conclude that the polynomial
\[\operatorname{BW_n}(q,t):=\sum_{k=0}^{n}q^k t^k {n \brack k}_{t^2}\]
provides a biCSP for binary words where one cyclic group acts by twisted rotation $(a_1,\dots,a_n)\mapsto (-a_n,a_1,\dots,a_{n-1})$ and the other cyclic group $\mathbb{Z}_2$ acts by $(a_1,\dots,a_n)\mapsto (-a_1,\dots,-a_n)$. These two actions reflect the action of a regular element of $G(r,1,n)$ and the action of scaling a root of unity on $\operatorname{Col_2}(X)$. This gives a desired representation theoretic proof of the sieving result for twisted rotation on binary words in \cite{AU19}. More examples concerning sieving results coming from orbits of $G(r,1,n)$ will be provided in \cite{HJO20+}.

\section{Diagonal orbit harmonics and cyclic sieving}\label{sec:Orbit harmonics in diagonal setting}
In this section, we consider diagonal orbit harmonics. To be precise, suppose a finite set $X\subseteq \CC^{2n}$ is invariant under the action of $\symm_n \times C \times C'$, where $\symm_n$ acts diagonally, i.e. $$\sigma\cdot (x_1,\dots,x_n,y_1,\dots,y_n)=(x_{\sigma(1)},\dots,x_{\sigma(n)},y_{\sigma(1)},\dots,y_{\sigma(n)}),$$ and the cyclic group $C$ acts on $x$ coordinates by scaling a root of unity and the cyclic group $C'$ acts on $y$ coordinates by scaling a root of unity. Let $\II(X)$ be the vanishing ideal in $\CC[\xx_n,\yy_n]$ and $\TT(X)$ be the homogeneous ideal obtained by applying top degree homogeneous part in both $x$ and $y$ variables,
\begin{equation*}
    \TT(X) := \langle \tau_y (\tau_x (f)) : f \in \II(X), f\neq 0 \rangle.
\end{equation*}
Then we have the following equivalence as $W\times C \times C'$-modules.
\begin{equation}\label{isomorphism diagonal orbit harmonics}
    \CC[X]\cong\CC[\xx_n,\yy_n]/\TT(X)
\end{equation}
A sieving generating theorem for diagonal orbit harmonics can be obtained by using a similar argument in the proof of Theorem~\ref{sieving-generator} \cite[Thm. 3.1]{Oh20+}. We provide an application of this idea.

To each partition $\mu$ of $n$, one associates a bigraded $\symm_n$-module called the \emph{Garsia--Haiman module} as follows. An \emph{injective tableau} $T$ of shape $\mu\vdash n$ is a filling of cells of $\mu$ by integers $1,2,\dots,n$. For each injective tableau $T$, we assign a point $p_T\in\CC^{2n}$ by letting $i^{th}$ and $(n+i)^{th}$ coordinates of $p_T$ record the position of $i$ in $T$:
\begin{equation*}
    p_T=(\omega^{x_T(1)},\dots,\omega^{x_T(n)},\zeta^{y_T(1)},\dots,\zeta^{y_T(n)}),
\end{equation*}
where $x_T(i)$ and $y_T(i)$ are $x$ and $y$ coordinates of $i$ in $T$ (in a French notation), $\omega$ is $\ell(\mu)^{th}$ root of unity and $\zeta$ is $\mu_1^{th}$ root of unity. The point locus $X_\mu$ possesses a natural $\symm_n$ action that acts diagonally on $X_\mu$. Combining results in \cite{GH96} and \cite{Hai01}, we can conclude that the graded Frobenius image of $\CC[\xx_n,\yy_n]/\TT(X_\mu)$ is the Macdonald polynomial $\widetilde{H}_\mu(\xx;q,t)$.

To provide an instance of CSP, consider a rectangular partition $\mu=(a^b)$. The set $X_\mu \subseteq \CC^{2n}$ has not only $\symm_n$ action but also $C\times C'$ action, where a cyclic group $C$ of order $a$ acts by scaling $\omega$ to the first $n$ coordinates and a cyclic group $C'$ of order $b$ acts scaling $\zeta$ to the last $n$ coordinates. Note that the set $X_{(a^b)}$ can be identified with the set of $b\times a$ matrices where each of $1, 2, \dots, ab$ is used once as entry. Moreover, under this correspondence, $\symm_n\times C\times C'$ action on $X_{(a^b)}$ corresponds to the permutation action on entries, rotation action on columns, and rotation action on rows. Applying isomorphism \eqref{isomorphism diagonal orbit harmonics} gives
\begin{equation*}
    \CC[\xx_n,\yy_n]/\TT(X_{(a^b)})\cong \CC[X_{(a^b)}]
\end{equation*}
as $\symm_n \times C \times C'$ modules and by applying Springer's theorem, we have the following.

\begin{theorem}\label{Thm: Macdonald CSP 1}
Let $X_{(a^b)}$ be the set of $b\times a$ matrices of content of entries $(1^{ab})$. The product of cyclic groups $\ZZ_a\times\ZZ_b\times\ZZ_{ab}$ acts on $X$ by column rotation, row rotation and adding 1 modulo $ab$ to each entry. Then the triple $(X_{(a^b)},\ZZ_a\times\ZZ_b\times\ZZ_{ab},X_{(a^b)}(q,t,z))$ exhibits triCSP, where $$X_{(a^b)}(q,t,z)=\sum_{\lambda\vdash ab} \tilde{K}_{\lambda,(a^b)}(q,t)f^{\lambda}(z).$$
\end{theorem}

If we apply a similar argument in \cite{Rho10}, one can obtain tri-cyclic sieving phenomena of $a\times b$ matrices of content $\nu$ with $\nu$ has a cyclic symmetry (See \cite{Oh20+} for details).


\acknowledgements{The author is grateful to Brendon Rhoades for many helpful conversations and encouragement in writing this survey. We also thank referees for careful reading and helpful comments.}

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