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\title{A Type $B$ Analog of the Whitehouse Representation}

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\author[S.~Brauner]{Sarah Brauner\thanks{\href{mailto:braun622@umn.edu}{braun622@umn.edu}. S.B. is supported by the NSF Graduate Research Fellowship (DMS-0007404).}\addressmark{1}}

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\address{\addressmark{1}School of Mathematics, University of Minnesota, Twin Cities, MN}

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 \abstract{
 We give a Type $B$ analog of Whitehouse's lifts of the Eulerian representations from $S_n$ to $S_{n+1}$ by introducing a family of $B_{n}$-representations that lift to $B_{n+1}$. As in Type $A$, we interpret these representations combinatorially via a family of orthogonal idempotents in the Mantaci--Reutenauer algebra, and topologically as the  graded pieces of the cohomology of a certain $\Z_{2}$-orbit configuration space of $\R^{3}$. We show that the lifted $B_{n+1}$-representations also have a configuration space interpretation, and further parallel the Type $A$ story by giving analogs of many of its notable properties, such as connections to equivariant cohomology and the Varchenko--Gelfand ring.

}


\keywords{configuration spaces, equivariant cohomology, Eulerian idempotents, symmetric group representations, hyperoctahedral group, Mantaci--Reutenauer algebra}
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\begin{document}

\maketitle

\section{Introduction}\label{sec:intro}
Let $V$ be a representation of a finite group $H$; then $V$ is said to have a \emph{lift} to a group $G$ containing $H$ if there is a representation of $G$ that restricts to $V$. The goal of this abstract is to $(1)$ identify a family of representations of the hyperoctahedral group $B_{n}$ that decompose the regular representation $\Q[B_{n}]$ and lift to $B_{n+1}$, and $(2)$ interpret these representations combinatorially and topologically. 


This work is inspired by the well-documented Type $A$ story of a family of $S_{n}$-representations lifting to representations of $S_{n+1}$ studied by Whitehouse \cite{Whitehouse}, Early--Reiner \cite{Early-Reiner}, Mathieu \cite{mathieu1996hidden}, Getzler--Kapranov \cite{getzler1995cyclic}, Moseley--Proudfoot--Young \cite{moseley-proudfoot-young}, and others. These $S_{n}$-representations and their lifts arose from two distinct perspectives. The first is via a family of orthogonal idempotents $\{\frake_{k} \}_{0 \leq k \leq n-1}$ known as the \emph{Eulerian idempotents}. The $\frake_k$ are in \emph{Solomon's descent algebra} $\Sigma[S_{n}]$, the subalgebra of $\Q[S_{n}]$ generated by sums of permutations $\sigma=(\sigma_1,\ldots,\sigma_n)$ with the same descent set
\[ \Des(\sigma_{1}, \ldots, \sigma_{n}) := \{ i \in [n-1]: \sigma_{i}> \sigma_{i+1} \}. \]
The Eulerian idempotents have been extensively researched in the world of algebraic combinatorics, and generate the \emph{Eulerian representations} $\frake_{k}\Q[S_{n}]$, which lift to a family of $S_{n+1}$-representations called the \emph{Whitehouse representations} \cite{Whitehouse}, defined in Section~\ref{sec:whitehousereps}. 

The second viewpoint comes from the study of $\conf_{n}(\R^{3})$, the configuration space comprised of $n$ distinct ordered points in $\R^{3}$. Through this lens, one obtains a family of $S_{n}$-representations as the graded pieces of $H^{*}\conf_{n}(\R^{3})$, and lifted representations of $S_{n+1}$ by considering the cohomology of a particular quotient of the configuration space of $\SSS^{3},$ the one-point compactification of $\R^{3}$. The cohomology of $\conf_{n}(\R^{3})$ is intrinsically linked to $H^{*}\conf_{n}(\R)$, a ring with an elegant combinatorial description via \emph{Heaviside functions} due to Varchenko--Gelfand \cite{varchenkogelfand} (see Section~\ref{sec:configcohom}).  

Though not obvious, both viewpoints turn out to be equivalent and serve as a beautiful link between classical combinatorial objects and important topological ones. 

Our goal here is to construct an analog to both perspectives for Type $B$. In our analogy, Solomon's descent algebra is replaced by the \emph{Type $B$ Mantaci--Reutenauer algebra} $\Sigma'[B_{n}]$, a combinatorially defined subalgebra of $\Q[B_{n}]$ generalizing $\Sigma[S_{n}]$ and containing the Type $B$ Descent algebra $\Sigma[B_{n}]$. The role of the Eulerian idempotents will be played by a family of orthogonal idempotents $\{ \mathfrak{g}_{k} \}_{0 \leq k \leq n},$ obtained as a sum of certain orthogonal idempotents in $\Sigma'[B_{n}]$ introduced by Vazirani \cite{vazirani}. The Type $B$ analog of the space $\conf_{n}(\R^{3})$ will be a $\Z_{2}$-\emph{orbit configuration space} $\conf_{n}^{\langle \varphi \rangle}(\R^{3})$ (defined in \eqref{eq:bconfdef}) first introduced by Feichtner--Ziegler in \cite{feichtner2002orbit}, and its lift will be a quotient of the $\Z_{2}$-orbit configuration space of $\SSS^{3}$ coming from the antipodal action on $\SSS^{3}$ (see \eqref{eq:bliftdef}). In contrast to Type $A$, the strategy we adopt here is to begin with the ``lifted'' $B_{n+1}$-representations and use them to obtain representations of $B_{n}$ which should lift.

Our main result is to give a full analogy to the Type $A$ story by showing that the representations $ \mathfrak{g}_{k}\Q[B_{n}]$ describe the graded pieces of $H^{*}\conf_{n}^{\langle \varphi \rangle}(\R^{3})$, and that these representations lift to $B_{n+1}$, where they also have a cohomological interpretation. Further, we fully flesh out the connection between $\conf_{n}^{\langle \varphi \rangle}(\R^{3})$ and $\conf_{n}^{\langle \varphi \rangle}(\R)$, and give a combinatorial description for $H^{*}\conf_{n}^{\langle \varphi \rangle}(\R)$ that parallels the one by Varchenko--Gelfand. 

The remainder of the abstract proceeds as follows. Section \ref{sec:typea} describes in detail the Type $A$ motivation, including a ``wish list'' of properties for a Type $B$ analog (Section~\ref{sec:Aproperties}); Sections \ref{sec:MRalgebra} and \ref{sec:typeBtopology} introduce the Type $B$ representations and topology, respectively. Section \ref{sec:mainresults} then gives the main results, where we realize the properties on our wishlist. 
\acknowledgements{The author is very grateful to Vic Reiner for guidance and encouragement at every stage in this project, to Sheila Sundaram for insightful questions that served as the initial inspiration for this work, to Monica Vazirani for sharing her undergraduate thesis, and to Marcelo Aguiar, Patty Commins, Allen Knutson, Nick Proudfoot and Dev Sinha for enlightening discussions. }

\section{Type $A$ Motivation}\label{sec:typea}
\subsection{The Eulerian and Whitehouse representations}\label{sec:whitehousereps}
The Eulerian idempotents $\{ \frake_k \}_{0 \leq k \leq n-1}$ were originally introduced by Reutenauer in \cite{reutenauer1986theorem}, and have been extensively studied and generalized since then; see for instance \cite{saliola2009face}. They are obtained as a sum over a complete, primitive, orthogonal family of idempotents $\{ \frake_{\lambda} \}_{\lambda\vdash n}$ in $\Sigma[S_{n}]$ constructed by Garsia--Reutenauer\footnote{The definition of the $\frake_{\lambda}$ is technical and therefore omitted.} in \cite{garsia}:
\begin{equation}
    \frake_{k-1}:= \sum_{\substack{\lambda \vdash n\\ \ell(\lambda) = k}}  \frake_{\lambda}.
\end{equation}

Our focus will be on the family of $S_{n}$-representations generated by the $\frake_{k}$ and decomposing $\Q[S_{n}]$, called the \emph{Eulerian representations}, $E^{(k)}_{n}:=  \frake_{k}\Q[S_{n}]$. The Eulerian representations have connections to many beloved objects such as the free Lie algebra \cite{Hanlon}, hyperplane arrangements \cite{brauner2020eulerian} and configuration spaces (see Section~\ref{sec:configcohom}). 

For the purposes of this abstract, we are most interested in a property observed by Whitehouse in \cite{Whitehouse}: that each $E_{n}^{(k)}$ has a lift to $S_{n+1}$. View $S_{n} \leq S_{n+1}$ as the subgroup fixing the element $n+1$, let $\lambda_{n+1}$ be the $n+1$ cycle $(12\dots (n+1)) \in S_{n+1}$, and define
\[ \Lambda_{n+1} := \frac{1}{n+1} \sum_{i=0}^{n} (\lambda_{n+1})^{i} .\]
Whitehouse shows the element $f_{n+1}^{(k)} := \Lambda_{n+1} e_{n}^{(k)}$
is an idempotent in $\Q[S_{n+1}]$, 
generating a family of representations
$F_{n+1}^{(k)} := f_{n+1}^{(k)} \Q[S_{n+1}]  $
which we will call the \emph{Whitehouse representations}. She then proves that the $F_{n+1}^{(k)}$ are lifts of the $E_{n}^{(k)}$ \cite[Proposition 1.4]{Whitehouse}. 

\begin{example}[$n = 3$]  Denote by $S^\lambda$ the irreducible symmetric group representation indexed by the partition $\lambda$. Then the $S_{3}$ Eulerian representations and their $S_{4}$ lifts are
\[
\begin{array}{rlccll}
E_{3}^{(0)} &= S^{(2,1)} & & &  F_{4}^{(0)} &= S^{(2,2)} \\
    E_{3}^{(1)} &= S^{(2,1)} \oplus S^{(1,1,1)} & & & F_{4}^{(1)} &= S^{(2,1,1)}\\
    E_{3}^{(2)} &= S^{(3)}& && F_{4}^{(2)} &= S^{(4)}.
\end{array}
\]
Each $F_{4}^{(k)}$ restricts to the representation $E_{3}^{(k)}$ via the symmetric group branching rules.
\end{example}
\subsection{Configuration space cohomology}\label{sec:configcohom}
We will momentarily switch tracks here and focus on the topology of 
\[ \conf_{n}(\R^{d}):= \{ (x_{1}, \ldots, x_{n}) \in \R^{dn}: x_{i} \neq x_{j} \},\]
a space with many fascinating and far-reaching mathematical connections. When $d=2$, for example, $\conf_{n}(\R^{2})$ is the classifying space of the pure Artin braid group, and when $d=1$, $\conf_{n}(\R)$ is the complement of the Braid arrangement. The symmetric group naturally acts on $\conf_{n}(\R^{d})$ by permuting coordinates, and this action induces a representation in cohomology.\footnote{When considering representations of $H^{*}\conf_{n}(\R^{d})$, we will assume our coefficients are in $\Q$. Otherwise, we will use coefficients in $\Z$, \textit{e.g.}, for Theorems \ref{thm:vgring} and \ref{thm:cohen}.}

In the case that $d=1$, the space $\conf_{n}(\R)$ is a disjoint union of $n!$ contractible pieces. Each piece is parametrized by a relative ordering of $x_{1}, \ldots, x_{n}$ in $\R$, and $H^{*}\conf_{n}(\R)$ is concentrated in degree $0$, \textit{i.e.}\ the space of linear functionals on $\conf_{n}(\R)$. Varchenko--Gelfand give a combinatorial set of generators for $H^{0}\conf_{n}(\R)$ called 
\emph{Heaviside functions}, 
\[ u_{ij}(x_{1}, \ldots, x_{n}) := \begin{cases} 1 & x_{i} < x_{j} \\
0 & x_{i} > x_{j}
\end{cases} \]
for $i \neq j \in [n] := \{ 1, \ldots, n \}$. The space of such Heaviside functions forms a $\Z$-algebra, where the $u_{ij}$ are endowed with linear addition and component-wise multiplication:
\[
    u_{ij}\cdot u_{k\ell}(x_{1}, \ldots, x_{n}) = \begin{cases} 1 & x_{i} < x_{j} \textrm{ and } x_{k} < x_{\ell} \\
    0 & \textrm{ otherwise.}
    \end{cases}
\]
This implies certain natural relations, for example that $u_{ij}^{2} = u_{ij}$. Similarly, one can deduce that $1-u_{ij} = u_{ji}$, so that $u_{ij}\cdot u_{jk} \cdot (1-u_{ik}) = 0$, since it is impossible that $x_{i} < x_{j} < x_{k}$ but $x_{i} > x_{k}$. This is the essential idea behind Theorem \ref{thm:vgring}.

\begin{thm}[{\cite{varchenkogelfand}}]\label{thm:vgring}
The ring $H^{0}\conf_{n}(\R)$ has presentation $\Z[u_{ij}]/\I$, where $\I$ is generated by  
\[ (i) \hspace{.5em} u_{ij}^{2} = u_{ij}, \hspace{2.5em}(ii)\hspace{.5em} u_{ij} = (1-u_{ji}), \hspace{2.5em} (iii)\hspace{.5em} u_{ij}u_{jk}(1-u_{ik}) + (1-u_{ij})(1-u_{jk})u_{ik} = 0. \]
\end{thm}
Call the ring $\Z[u_{ij}]/\I$ the \emph{Varchenko--Gelfand ring.} The presentation in Theorem \ref{thm:vgring} imposes an ascending filtration on the Varchenko--Gelfand ring obtained from the natural degree grading on $\Z[u_{ij}]/\I$: the $m^{th}$ layer in the filtration is the span of monomials in the variables $u_{ij}$ having degree at most $m$. We will see that the associated graded coming from this filtration, $\gr(H^{0}\conf_{n}(\R))$, is closely related to $H^{*}\conf_n(\R^{d})$ for $d >1$.

The space $\conf_{n}(\R)$ is relevant in part because it has a ``hidden'' $S_{n+1}$-action. To recover this action, let $U(1)$ be the circle group and consider $\conf_{n+1}(U(1))$, the space of $n+1$ distinct points in $U(1)$. The group $U(1)$ acts (left) diagonally, and the quotient by this action, $\V_{n+1}^{1}:= \conf_{n+1}(U(1))/U(1)$ is $S_{n}$-equivariantly homeomorphic to $\conf_{n}(\R)$ via the map
\begin{align}
   f_{A} \colon \V_{n+1}^{1} &\xrightarrow{\cong} \conf_{n}(\R) \label{eq:homeospaces1},\\
    (p_{1}, \dots, p_{n+1}) &\mapsto (\pi(p_{n+1}^{-1} p_{1}), \dots, \pi(p_{n+1}^{-1} p_{n})), \label{eq:homeotypea}
\end{align}
where $\pi$ is the stereographic projection\footnote{The point $\infty$ here is $1 \in U(1)$, and since $p_{n+1}^{-1} p_{i} \neq 1$, the map $\pi$ is well-defined.} from $U(1)$ to $\R$. The intuition here is that $ \V_{n+1}^{1}$ has representatives $(p_{1}, \ldots, p_{n},1)$ for $p_{i} \neq p_{j} \neq 1$ and like $\conf_{n}(\R)$, is comprised of $n!$ contractible pieces. Each disjoint piece of $\V_{n+1}^{1}$ is parametrized by a relative ordering of $p_{1}, \ldots, p_{n+1}$ around the circle; these disjoint pieces ($S_{n}$-equivariantly) biject with the pieces of $\conf_{n}(\R)$. To move from $\V_{n+1}^{1}$ to $\conf_{n}(\R)$, read the ordering of $p_{1}, \ldots, p_{n}$ around $U(1)$ counter-clockwise beginning after $p_{n+1}$.

The advantage of studying $\V_{n+1}^{1}$ is that it has an explicit $S_{n+1}$-action by coordinate permutation as well as a natural $S_{n}$-action given by permuting only $p_{1}, \ldots, p_{n}$. 

When we move to cohomology, the Heaviside functions $u_{ij}$ also lift to \emph{cyclic Heaviside functions} $v_{ijk} \in  \V_{n+1}^{1}$, defined in \cite{moseley-proudfoot-young} by Moseley--Proudfoot--Young as:
\[ v_{ijk}(p_{1}, \ldots, p_{n}) := \begin{cases} 1 & p_{i} < p_{j} < p_{k} \textrm{ in counter-clockwise order on } U(1)\\
0 & \textrm{ otherwise.}
\end{cases} \]
The $v_{ijk}$ again form a $\Z$-algebra and provide an elegant combinatorial description for the ring $H^{0} \V_{n+1}^{1}$. In fact the presentation can be recovered from the presentation in Theorem \ref{thm:vgring} via the induced isomorphism $f_{A}^{*}$ sending $u_{ij}$ to $v_{ij(n+1)}$, along with the additional relation due to Early--Reiner \cite{Early-Reiner}:
\begin{equation*}
v_{ijk} - v_{ij\ell} + v_{ik \ell} - v_{jk\ell} = 0;
\end{equation*}
see also \cite{moseley-proudfoot-young}. As in the case of $H^{0}\conf_{n}(\R)$, the degree grading on $H^{0} \V_{n+1}^{1}$ from the $v_{ijk}$ imposes an ascending filtration with associated graded $\gr(H^{0} \V_{n+1}^{1})$.
\begin{example}
Consider the two representatives $\vec{q}$ and $\vec{r}$ of $\V_{3}^{1}$ and their images under $f_{A}$:
%\begin{center}

\begin{tikzpicture}[thick]
 % State q2
\draw (-5+5,0)-- (-2+5,0);
 % State q1
\node[point] (a) at (-4+5,0){};
\node[] (aa) at (-4.25 +5,.5){$\pi(p_{1})$};

% State q0
\node[point] (b) at (-3+5,0){};
\node[] (bb) at (-2.75+5,.5){$\pi(p_{2})$};

\node[] (cc) at (-3.5+5,-.75){$f_{A}(\vec{q})$};

% State q2
\node[] (A) at (.25-4,0){$p_1$};
 
% State q1
\node[] (B) at (1.75-4,0){$p_2$};
 
% State q0
\node[] (C) at (1-4,1.25){$p_3$};

\node[] (cc) at (1-4,-.75){$\vec{q}$};

\node[] (x) at (-1,0){$\longmapsto$};
\node[] (xx) at (-1,.5){$f_{A}$};

% Transition q2 to q0
\draw (A) to[bend left] node[left]{}(C) ;
 
% Transition q0 to q1
\draw (C) to[bend left] node[right]{}(B);
 
% Transition q1 to q2
\draw (B) to[bend left]node[below]{} (A);
 
\begin{scope}[xshift=-1cm]
% Initial state
\draw (5+4,0)-- (8+4,0);
 % State q1
\node[point] (a) at (6+4,0){};
\node[] (aa) at (5.75+4,.5){$\pi(p_{2})$};

% State q0
\node[point] (b) at (7+4,0){};
\node[] (bb) at (7.25+4,.5){$\pi(p_{1})$};

\node[] (cc) at (6.5+4,-.75){$f_{A}(\vec{r})$};

% State q2
\node[] (A) at (9.25-4,0){$p_2$};
 
% State q1
\node[] (B) at (10.75-4,0){$p_1$};
 
% State q0
\node[] (C) at (10-4,1.25){$p_3$};

 \node[] (cc) at (10-4,-.75){$\vec{r}$};

\node[] (x) at (8,0){$\longmapsto$};
\node[] (xx) at (8,.5){$f_{A}$};
% Transition q2 to q0
\draw (A) to[bend left] node[left]{}(C) ;
 
% Transition q0 to q1
\draw (C) to[bend left] node[right]{}(B);
 
% Transition q1 to q2
\draw (B) to[bend left]node[below]{} (A);
 
% Initial state
\end{scope}
\end{tikzpicture}
%\end{center}

\noindent
Note that $v_{123}(\vec{q}) =u_{12}(f_{A}(\vec{q})) =  1$, while $v_{123}(\vec{r}) =u_{12}(f_{A}(\vec{r})) = 0.$ On the other hand $v_{213}(\vec{q}) = u_{21}(f_{A}(\vec{q}))  = 0$ and $v_{213}(\vec{r}) = u_{21}(f_{A}(\vec{r})) =1$.
\end{example}

When $d \geq 2$, the space $\conf_{n}(\R^{d})$ is no longer comprised of contractible, disjoint pieces but nonetheless has an elegant presentation due to F. Cohen. 
\begin{thm}[{\cite{cohen}}]\label{thm:cohen}
For $d \geq 2$, the ring $ H^{*}\conf_{n}(\R^{d})$ has presentation $\Z\langle u_{ij} \rangle /\J$ for distinct $i,j,k,\ell \in [n]$, where $\J$ is generated by the relations $u_{ij}u_{k\ell} = (-1)^{d+1}u_{k\ell}u_{ij}$ and 
 \[(i)\hspace{.5em} u_{ij}^{2} = 0, \hspace{2.5em}(ii)\hspace{.5em}u_{ij} = (-1)^{d}u_{ji}, \hspace{2.5em} (iii)\hspace{.5em}u_{ij}u_{jk} + u_{jk}u_{ki} + u_{ki}u_{ij} = 0. \]
\end{thm}

\noindent The generator $u_{ij}$ lies in $H^{d-1}\conf_{n}(\R^{d})$,
which together with the relations in $\J$, implies that $H^{*}\conf_{n}(\R^{d})$ is concentrated in degrees $0, (d-1), 2(d-1), \ldots, (n-1)(d-1).$ 

\subsubsection{Property wish list for Type $B$}\label{sec:Aproperties}
We are most concerned with the case that $d=3$. In this situation, there are five notable properties of $H^{*}\conf_{n}(\R^{3})$ which will inspire our Type $B$ work.
\begin{enumerate}
    \item There is an isomorphism of $S_n$-representations\footnote{In fact \eqref{eq:typeaeulerianconfig} holds for any $d \geq 3$ and odd by replacing $H^{2k}\conf_{n}(\R^{d})$ with $H^{(d-1)k}\conf_{n}(\R^{d})$.} for $0 \leq k \leq n-1$:
\begin{equation}\label{eq:typeaeulerianconfig}
     E^{(n-1-k)}_{n} \cong_{S_{n}} H^{2k}\conf_{n}(\R^{3}).
\end{equation}
 This was first deduced by comparing a result of Sundaram--Welker for subspace arrangements \cite[Theorem~4.4(iii)]{sundaramwelker} with descriptions of the characters of $E^{(k)}_{n}$ by Hanlon \cite{Hanlon}, and was later proved in the context of Coxeter groups in \cite{brauner2020eulerian}.
\item Equation \eqref{eq:typeaeulerianconfig} ``lifts'' to an isomorphism of $S_{n+1}$ representations \cite[Theorem]{Early-Reiner}:
\begin{equation}\label{eq:typeawhitehouseconfig}
    F^{(n-1-k)}_{n+1} \cong_{S_{n+1}} H^{2k}(\V_{n+1}^{3}),
\end{equation}
where $\V_{n+1}^{3}:= \conf_{n+1}(SU_{2})/SU_{2}$. Recall that $SU_{2}$ is the group of $2 \times 2$ unitary matrices over $\C$ and is homeomorphic to $\SSS^{3}$; the quotient is by the diagonal action of $SU_{2}$ on $\conf_{n+1}(SU_{2}).$ Intuitively, \eqref{eq:typeawhitehouseconfig} comes from a $S_{n}$-equivariant homeomorphism found in
Early-Reiner \cite{Early-Reiner} and Moseley-Proudfoot-Young \cite{moseley-proudfoot-young} analogous to \eqref{eq:homeotypea}. The notation $\V_{n+1}^{3}$ (resp. $\V_{n+1}^{1}$) indicates the relationship to $\SSS^{3}$ (resp. $\SSS^1$).
\item There is a recursion relating the Eulerian and Whitehouse representations of $S_n$:
\begin{equation}
    E_{n}^{(k)} = F_{n}^{(k-1)} \oplus \left( S^{(n-1,1)} \otimes F_{n}^{(k)} \right),
\end{equation}
where $S^{(n-1,1)}$ is the reflection representation of $S_{n}$ \cite[Proposition 1]{Early-Reiner}.

\item The circle group $U(1)$ acts on $\R^{3}$ by rotation around the $x$-axis, thereby inducing an action on $\conf_{n}(\R^{3})$. The filtration induced from the $U(1)$-equivariant cohomology $H^{*}_{U(1)}\conf_{n}(\R^{3})$ implies a \emph{graded} isomorphism of $S_{n}$-modules:
\begin{equation}\label{eq:grtypea}
    \gr(H^{0}\conf_{n}(\R)) \cong_{S_{n}} H^{*}\conf_{n}(\R^{3}),
\end{equation}
where $ \gr(H^{0}\conf_{n}(\R))$ coincides with the associated graded coming from the filtration by Heaviside functions \cite{moseley2017equivariant}. 

\item Equation \eqref{eq:grtypea} also lifts to a graded $S_{n+1}$-module isomorphism \cite{moseley-proudfoot-young}:
\begin{equation}\label{eq:liftedtypeagradediso}
    \gr(H^{0}\V_{n+1}^{1}) \cong_{S_{n+1}} H^{*}(\V_{n+1}^{3}),
\end{equation}
where again \eqref{eq:liftedtypeagradediso} comes from a $U(1)$-action on $\V^{3}_{n+1}$ and subsequent computation of $H^{*}_{U(1)}\V^{3}_{n+1}$. The grading on the left-hand-side also coincides with the associated graded coming from the filtration by cyclic Heaviside functions.
\end{enumerate}
Our goal is to find a family of $B_{n}$-representations exhibiting analogs of these properties. 

\section{The Mantaci--Reutenauer algebra and idempotents}\label{sec:MRalgebra}
We will begin our Type $B$ story by introducing the family of $B_n$-representations arising in a generalization $\Sigma[S_{n}]$.
Perhaps the most obvious generalization of the Type $A$ descent algebra is the Type $B$ descent algebra, with Coxeter length used to describe $\Des(\sigma)$. However, it turns out that the corresponding Eulerian representations of $B_{n}$ (studied by the author in \cite{brauner2020eulerian} for instance) do \emph{not} lift to $B_{n+1}$! 

Instead, we will work in the \emph{Type $B$ Mantaci--Reutenauer algebra} introduced in \cite{mantaci1995generalization} and defined as follows. Consider $\sigma = (\sigma_{1}, \ldots, \sigma_{n}) \in B_{n}$ to be a signed permutation, meaning that $\sigma_{i} \in \{ -n, \ldots, -1, 1 \cdots, n \}$. The \emph{Mantaci--Reutenauer descents} of $\sigma$ is the set
\[ \mrdes(\sigma) := \left\{ i \in [n-1] : \begin{array}{l} |\sigma_{i}| > |\sigma_{i+1}| \textrm{ and } \sigma_{i} \textrm{ and } \sigma_{i+1} \textrm{ have the same sign or} \\
\sigma_{i} \textrm{ and } \sigma_{i+1} \textrm{ have opposite signs} \end{array} \right\}.
\]
Note that $\mrdes(\sigma)$ partitions $\sigma$ into $|\mrdes(\sigma)|+1$ ordered blocks between each descent. Let 
$[n]^{\pm}:= \{ 1, 2, \ldots, n, \overline{1}, \overline{2}, \ldots, \overline{n} \}.$ A \emph{signed composition} of $n$ is a sequence $(a_{1}, \ldots, a_{\ell})$ where $a_{i} \in [n]^{\pm}$ and $|a_{1}| + \cdots + |a_{\ell}| = n$. (Here $|\overline{j}| = j$.)
Denote by $\sh(\sigma)$ the signed composition of $n$ obtained from $\mrdes(\sigma)$, where each block
$\{ \sigma_{i}, \ldots, \sigma_{i+m} \}$ contributes an $m+1$ to $\sh(\sigma)$ if each $\sigma_{i}$ is positive and an $\overline{m+1}$ if $\sigma_{i}$ is negative.

\begin{example}
If $\sigma = (3,4,-1,-5,-2)$, then $\mrdes(\sigma) = \{ 2, 4 \}$, which partitions $\sigma$ into ordered blocks $(\{3,4\}, \{-1, -5 \}, \{-2\})$. Therefore $\sh(\sigma) = (2, \overline{2}, \overline{1})$.
\end{example}
The \emph{Mantaci--Reutenauer algebra} is the algebra $\Sigma'[B_{n}]$ generated by $x_{\alpha} \in \Q[B_{n}]$ where
\[ x_{\alpha}:= \sum_{\substack{\sigma \in B_{n}\\ \sh(\sigma) = \alpha}} \sigma. \]

Within $\Sigma'[B_{n}]$ is a family of complete, primitive and orthogonal idempotents\footnote{As in the case of the $\frake_{\lambda}$, the definition of these idempotents is technical and therefore omitted; see \cite{vazirani}.} $\frakg_{(\lambda^{+}, \lambda^{-})}$ introduced by Vazirani in \cite{vazirani}, where $\lambda^{+}, \lambda^{-}$ are integer partitions with $|\lambda^{+}|+|\lambda^{-}| = n.$ 

The analog of the Eulerian idempotents will come from summing over these $\mathfrak{g}_{(\lambda^{+}, \lambda^{-})}$:
\begin{equation}
    \frakg_{k}:= \sum_{\substack{(\lambda^{+}, \lambda^{-}) \\ \ell(\lambda^{+}) = k}} \frakg_{(\lambda^{+}, \lambda^{-})},
\end{equation}
and the analog of the Eulerian representations is precisely $G_{n}^{(k)}:= \frakg_{k}\Q[B_{n}] $ for $0 \leq k \leq n$.

The above analogies are quite natural in the following sense.\footnote{The author is grateful to M. Aguiar for suggesting this line of inquiry.} Let $\tau \colon B_{n} \to S_{n}$ be the projection which forgets the signs of $\sigma \in B_{n}$. In \cite{aguiar2004peak} Aguiar--Bergeron--Nyman study the properties of $\tau$ and show that it extends to a surjective algebra homomorphism  $\tau \colon \Sigma'[B_{n}] \to \Sigma[S_{n}]$. This homomorphism then relates the $\frakg_{k}$ to the $\frake_{k}$:
\begin{prop}
We have $\tau(\frakg_{0}) = 0$ and for $1 \leq k \leq n,$ $\tau(\frakg_{k}) = \frake_{k-1}$.
\end{prop}
\section{Topology in Type $B$}\label{sec:typeBtopology}
In contrast to Type $A$, in Type $B$ it is more natural to begin with the topology of the ``hidden'' action spaces analogous to $\V_{n+1}^{1}$ and $\V_{n+1}^{3}$. Recall that the antipodal map acts on $SU_{2}$ (\textit{e.g.}~$\SSS^{3}$), and $U(1)$ (\textit{e.g.}~$\SSS^1$) by $-1$. One then has two $\Z_{2}$-orbit configuration spaces
\begin{align*}
  \conf_{n+1}^{\Z_{2}}(U(1))&:= \{ (p_{1}, \ldots, p_{n+1}) \in U(1)^{n}: p_{i} \neq \pm p_{j} \},\\
 \conf_{n+1}^{\Z_{2}}(SU_{2})&:= \{ (p_{1}, \ldots, p_{n+1}) \in SU_{2}^{n}: p_{i} \neq \pm p_{j} \},
 \end{align*}
and corresponding quotients by the diagonal action of $U(1)$ and $SU_{2}$, respectively:
\begin{equation}\label{eq:bliftdef} \Y_{n+1}^{1}:= \conf_{n+1}^{\Z_{2}}(U(1))/U(1),  \hspace{5em} \Y_{n+1}^{3}:= \conf_{n+1}^{\Z_{2}}(SU_{2})/SU_{2}. \end{equation}

\subsection{Signed cyclic Heaviside functions and the $d=1$ case}
In direct analogy with Type $A$, the space $\Y_{n+1}^{1}$ is comprised of $2^{n}n!$ contractible pieces, each of which is parametrized by arrangements of $p_{1},\cdots, p_{n+1}$ \emph{and} $-p_{1}, \dots, -p_{n+1}$ on $U(1)$, where we require that each $p_{i}$ be opposite its antipode $-p_{i}$. Given a point $\vec{p} = (p_{1}, \ldots, p_{n+1}) \in \Y_{n+1}^{1}$, write $C(\vec{p}) = C(p_{1}, \ldots, p_{n+1})$ as its arrangement \emph{with} antipodes on $U(1)$ and $-p_{i}$ as $p_{\overline{i}}$. By convention $\overline{\overline{i}} = i$. 

We define \emph{signed cyclic Heaviside functions} $y_{ijk}$ for distinct $i,j,k \in [n+1]^{\pm}$ as 
\[y_{ijk}(\vec{p}):=  \begin{cases} 1 & p_{i} < p_{j} < p_{k} \textrm{ counter-clockwise in }C(\vec{p}), \\
0 & \textrm{ otherwise.}
\end{cases} \]

Once again, the $y_{ijk}$ form a $\Z$-algebra with multiplication given by
\[y_{ijk}\cdot y_{qrs}(\vec{p}):=  \begin{cases} 1 & p_{i} < p_{j} < p_{k}\textrm{ and } p_{q} < p_{r} < p_{s}  \textrm{ counter-clockwise in }C(\vec{p}), \\
0 & \textrm{ otherwise.}
\end{cases} \]

Analyzing the combinatorial properties of the $y_{ijk}$ (and employing a standard Gr{\"o}bner basis argument) allows one to determine a presentation for $H^{0}(\Y_{n+1}^{1})$.
\begin{thm}\label{thm:bpres1lift} The ring $H^{0}(\Y_{n+1}^{1})$ has presentation $\Z[y_{ijk}]/\mathcal{I'}$ for distinct $i,j,k \in [n+1]^{\pm}$, where $\mathcal{I}'$ is generated by the relations
\begin{equation*}
      (i) \hspace{.5em} y_{ijk}^{2} = y_{ijk}, \hspace{5em}   (ii) \hspace{.5em} y_{ijk} =1 -y_{jik}, \hspace{5em}   (iii) \hspace{.5em}
\hspace{.5em} y_{\overline{i \, j}\ \, k} = y_{i \, j \, \overline{ k}},
\end{equation*}
\begin{equation*}
     (iv) \hspace{.5em} y_{ijk} - y_{ij\ell} + y_{ik \ell} - y_{jk\ell} = 0,  \hspace{2.5em}  (v) \hspace{.5em}  y_{ij\ell}y_{jk \ell }(1-y_{ik\ell }) + (1-y_{ij\ell}) (1-y_{jk \ell })y_{ik \ell } = 0.
\end{equation*}
\end{thm}
Note that although the generators $y_{ijk}$ are now indexed by $[n+1]^{\pm}$, the only new relation needed for $H^{0}\Y_{n+1}^{1}$ compared to $H^{0}\V_{n+1}^{1}$ is relation $(iii)$. Like $H^{0}\V_{n+1}^{1}$, there is an ascending filtration on $H^{0}\Y_{n+1}^{1}$ by degree in the $y_{ijk}$, and the corresponding associated graded $\gr(H^{0}\Y_{n+1}^{1})$ will again play an important role in understanding $H^{*}(\Y_{n+1}^{3})$.
\newpage

 In further parallel with Section~\ref{sec:configcohom}, we would like to identify a genuine orbit configuration space of $\R$ (rather than a quotient) which is $B_{n}$-equivariantly homeomorphic to $\Y_{n+1}^{1}$. However, in this context we must be careful about how the antipodal map behaves under stereographic projection $\pi\colon \SSS^{d} \to \R^{d}$. In particular,
\[ \pi( - p_{i})  =  \frac{-\pi(p_{i})}{|\pi(p_{i})|^{2}} := \varphi(\pi(p_{i})). \]
Hence, using the same map as in \eqref{eq:homeotypea}, we obtain a $B_{n}$-equivariant homeomorphism:
\begin{equation}\label{eq:typebhomeo1}
    f_{B}\colon \Y_{n+1}^{1} \xrightarrow{\cong} \conf_{n}^{\langle \varphi \rangle}(\R \setminus \{ 0 \}),
\end{equation}
where 
\begin{equation}\label{eq:bconfdef}
 \conf_{n}^{\langle \varphi \rangle}(\R^{d} \setminus \{ 0 \}):= \{ (x_{1}, \ldots, x_{n}) \in (\R^{d} \setminus \{ 0 \})^{n}: x_{i} \neq x_{j} \neq\varphi(x_{j}) \}.\end{equation}
\begin{example} The space $\conf_{2}^{\langle \varphi \rangle}(\R\setminus \{ 0 \})$ is the complement of the (non-linear!) spaces:
\begin{center}
\begin{tikzpicture}
 \draw (-1,-1)-- (1,1);
 \draw (0,-1.5) -- (0,1.5);
 \draw (-1.5,0) -- (1.5,0);
\node[] (B) at (-1.65,.05){};
 
% State q0
\node[] (C) at (-.05,1.65){}; 
\draw (C) to[bend left] node[right]{}(B);
 
 \node[] (D) at (1.65,-.05){};
 
% State q0
\node[] (E) at (.05,-1.65){}; 
\draw (D) to[bend right] node[right]{}(E);
 

 
% \node[] (A) at (.25-4,0){$p_2$};
% \node[] (AA) at (1.75-4,.8){$p_{\overline{2}}$};

% % State q1
% \node[] (B) at (1.75-4,0){$p_1$};
% \node[] (BB) at (.25-4,.8){$p_{\overline{1}}$};

% % State q0
% \node[] (C) at (1-4,1.25){$p_3$};
% \node[] (CC) at (1-4,-.4){$p_{\overline{3}}$};


 
% % Transition q2 to q0
% \draw (C) to[bend left] node[left]{}(AA) ;
% \draw (AA) to[bend left] node[left]{}(B) ;
% \draw (B) to[bend left] node[left]{}(CC) ;
% \draw (CC) to[bend left] node[left]{}(A) ;
% \draw (A) to[bend left] node[left]{}(BB) ;
% \draw (BB) to[bend left] node[left]{}(C) ;


% % Transition q0 to q1
% %\draw (C) to[bend left] node[right]{}(B);
 
% % Transition q1 to q2
% %\draw (B) to[bend left]node[below]{} (A);
 \end{tikzpicture}
 \end{center}
\end{example}

In cohomology, \eqref{eq:typebhomeo1} induces an isomorphism of $B_{n}$-modules which identifies
\begin{equation}\label{eq:identification} y_{ij(n+1)} \longleftrightarrow \begin{cases} z_{ij} & i \neq \overline{j} \\
z_{j} & i = \overline{j}. 
\end{cases} \end{equation}
Theorem \ref{thm:bpres1lift} then determines a presentation for $H^{0} \conf_{n}^{\langle \varphi \rangle}(\R \setminus \{ 0 \})$ in terms of $z_{ij}$ and $z_{i}$ for distinct $i,j \in [n]^{\pm}$; it too has an ascending filtration coming from the degree grading in the $z_{i}$ and $z_{ij}$. The $z_{i}$ variables can be interpreted as Heaviside-like functions where $z_{i}(\vec{x}) = 1$ if $x_{i}> 0$ and 0 otherwise. Unfortunately, unlike the $u_{ij}$ in Type $A$, the $z_{ij}$ have a decidedly more complicated description, which we omit for the sake of brevity.
\subsection{The $d=3$ case}
In the case of $H^{*}(\Y_{n+1}^{3})$, there is also a simple presentation mirroring that of $H^{*}(\V_{n+1}^{3})$. 
\begin{thm}\label{thm:bpres3lift} The ring $H^{*}(\Y_{n+1}^{3})$ has presentation $\Z[y_{ijk}]/\mathcal{J}'$ for distinct $i,j,k \in [n+1]^{\pm}$, where $\mathcal{J}'$ is generated by the relations
\begin{equation*}
     (i) \hspace{.5em}  y_{ijk}^{2} = 0, \hspace{5em}  (ii) \hspace{.5em}  y_{ijk} = -y_{jik}, \hspace{5em}   (iii) \hspace{.5em}  y_{\overline{i \, j}\ \, k} = y_{i \, j \, \overline{ k}},
\end{equation*}
\begin{equation*}
      (iv) \hspace{.5em} y_{ijk} - y_{ij\ell} + y_{ik \ell} - y_{jk\ell} = 0,  \hspace{5em}   (v) \hspace{.5em}y_{ij\ell}y_{jk \ell }-y_{ik\ell }y_{ij\ell} -y_{ik \ell }y_{jk \ell } = 0.
\end{equation*}
\end{thm}
\newpage
The generators $y_{ijk}$ are of cohomological degree $2$, and so Theorem \ref{thm:bpres3lift} implies that $H^{*}(\Y_{n+1}^{3})$ is concentrated in degrees $0,2, \ldots, 2n$. 

We would like to recover from Theorem \ref{thm:bpres1lift} a presentation\footnote{This question was first studied in \cite{feichtner2002orbit}. However, the presentation given has an error (Lemma 7) which is corrected using the lifted presentation in Theorem \ref{thm:bpres3lift} and identification of generators in \eqref{eq:identification}.} for the cohomology of $\conf_{n}^{\langle \varphi \rangle}(\rthree)$. As in the $d=1$ case, there is a $B_{n}$-equivariant homeomorphism between $\Y^{3}_{n+1}$ and $\conf_{n}^{\langle \varphi \rangle}(\rthree)$ analogous to \eqref{eq:homeotypea}. This again induces a $B_{n}$-module isomorphism in cohomology identifying the generator $y_{ij(n+1)}$ with $z_{ij}$ or $z_i$ as in \eqref{eq:identification}. 

From this identification, one can readily use Theorem \ref{thm:bpres3lift} to obtain a presentation for $H^{*}\conf_{n}^{\langle \varphi \rangle}(\rthree)$ with respect to $z_{ij}$ and $z_{i}$ for $i,j \in [n]^{\pm}$.


\section{Main results: Type $B$ wishlist realized}\label{sec:mainresults}
We now present an analog of the properties described in Section~\ref{sec:Aproperties} for Type $B$.
\begin{thm}
\mbox{}
\begin{enumerate}
    \item There is an isomorphism of $B_{n}$-representations for $0 \leq k \leq n$:
    \begin{equation}\label{eq:typebreps} G_{n}^{(n-k)} \cong_{B_{n}} H^{2k} \conf_{n}^{\langle \varphi \rangle}(\rthree), \end{equation}
    and thus the total representation of $H^{*}\conf_{n}^{\langle \varphi \rangle}(\rthree)$ is $\Q[B_{n}]$.
    \item The representation in \eqref{eq:typebreps} lifts to $B_{n+1}$, where it is described by $H^{2k}\Y_{n+1}^{3}$;
    \item For $0 \leq k \leq n$, there is an isomorphism of $B_n$-representations:
    \begin{equation*}
         H^{2k}\conf_{n}^{\langle \varphi \rangle}(\rthree)  \cong_{B_n} H^{2(k-1)}(\Y_{n}) \oplus \left( V \otimes H^{2k}(\Y_{n})  \right),
    \end{equation*}
    where $V = \chi^{((n-1,1), 0)} \oplus \chi^{((n-1),(1))}$; this notation refers to the fact that irreducible representations of $B_{n}$ are indexed by partitions $(\lambda^{+}, \lambda^{-})$ where $|\lambda^{+}|+ |\lambda^{-}| = n$; see \cite{branchingrules}. 
    \item The circle group $U(1)$ acts on $\R^{3}$ by rotation around the $x$-axis, inducing an action on $\conf_{n}^{\langle \varphi \rangle}(\rthree)$. The filtration induced from the $U(1)$-equivariant cohomology implies a \emph{graded} isomorphism of $B_{n}$-modules:
\begin{equation}\label{eq:grtypeb}
    \gr(H^{0}\conf_{n}^{\langle \varphi \rangle}(\R \setminus \{ 0 \})) \cong_{B_{n}} H^{*}\conf_{n}^{\langle \varphi \rangle}(\rthree),
\end{equation}
where $\gr(H^{0}\conf_{n}^{\langle \varphi \rangle}(\R \setminus \{ 0 \}))$ coincides with the associated graded coming from the filtration by degree in the variables $z_{i}$ and $z_{ij}$ for distinct $i,j \in [n]^{\pm}$.
\item Equation \eqref{eq:grtypeb} also lifts to a graded $B_{n+1}$-module isomorphism 
\begin{equation}\label{eq:liftedtypebgradediso}
    \gr(H^{0}\Y_{n+1}^{1}) \cong_{S_{n+1}} H^{*}(\Y_{n+1}^{3}),
\end{equation}
where again \eqref{eq:liftedtypebgradediso} comes from a $U(1)$-action on $\Y^{3}_{n+1}$ and subsequent computation of $H^{*}_{U(1)}\Y^{3}_{n+1}$. Once more $\gr(H^{0}\Y_{n+1}^{1})$ coincides with the filtration by degree in the signed cyclic Heaviside functions $y_{ijk}$ for distinct $i,j,k \in [n]^{\pm}$.
\end{enumerate}
\end{thm}
\begin{proof}[Proof idea] \noindent
\begin{enumerate}
    \item The isomorphism \eqref{eq:typebreps} comes from a combination of character computations of $\frakg_{(\lambda^{+}, \lambda^{-})}\Q[B_n]$ in \cite{douglass2018decomposition}, adapting techniques in \cite[Theorem~9.1]{Berget}, and analyzing a finer (descending) filtration of the ring $H^{*} \conf_{n}^{\langle \varphi \rangle}(\rthree)$ by degree in the variable $z_{i}$ for $i \in [n]^{\pm}$.
    \item The lift follows by using the $B_{n}$-equivariant homeomorphism $\conf_{n}^{\langle \varphi \rangle}(\rthree) \cong \Y_{n+1}^{3}$.
    \item The recursion comes from studying the $B_{n}$-action on the cohomology induced by the spectral sequence
 $ SU_{2} \setminus \{ \pm p_{1}, \pm p_{2}, \ldots,  \pm p_{n}\} \longrightarrow \Y_{n+1}^3 \longrightarrow \Y_{n}^{3}.$
 \item The techniques used to prove \eqref{eq:grtypeb} are adapted from \cite[Lemma 4.2]{moseley2017equivariant}.
 \item The techniques used to prove \eqref{eq:liftedtypebgradediso} are adapted from \cite[Remark~2.9]{moseley-proudfoot-young}.
\end{enumerate} \vspace{-1em}
\end{proof}
%% 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
%\bibliographystyle{abbrv}
%\bibliography{bibliography}
\end{document}
