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\title[Configuration spaces and peak representations]{Configuration spaces and peak representations}

\author{Marcelo Aguiar\addressmark{1}, Sarah Brauner\thanks{\href{sarahbrauner@gmail.com}{sarahbrauner@gmail.com}. Brauner is supported by the NSF MSPRF DMS 2303060.}\addressmark{2}, \and Vic Reiner\thanks{Reiner is partially supported by NSF grant DMS 2053288.}\addressmark{3}}

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\address{\addressmark{1} Department of Mathematics, Cornell University \\ \addressmark{2}  Département de mathématiques, Universit\'e du Qu\'ebec \`a Montréal \\ \addressmark{3}Department of Mathematics, University of Minnesota - Twin Cities}

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\abstract{{\it Eulerian} idempotents of types $A$ and $B$ generate representations with topological interpretations, as the cohomology of configuration spaces of types $A$ and $B$. We provide an analogous cohomological interpretation for the representations generated by idempotents in the {\it peak algebra}, called the {\it peak representations}. We describe the peak representations as sums of {\it Thrall's higher Lie characters}, give Hilbert series and branching rule recursions for them, and discuss connections to Jordan algebras.}

\keywords{Peak algebra, configuration spaces, Solomon's descent algebra, higher lie characters, hyperplane arrangements, Varchenko-Gelfand ring, Type $A$, Type $B$}


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\begin{document}

\maketitle
\section{Introduction}
\label{intro-section}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
This abstract concerns the cohomology $H^* X=H^*(X,\kk)$ with coefficients in a field $\kk$ for three different topological configuration spaces $X=X_n, Y_n, Z_n$ having large symmetry groups $W$.  For each, the (ungraded) cohomology carries the regular representation of $W$, that is,
$
H^* X \cong \kk W.
$
Our goal is to study and exploit the following surprising fact: for $\kk$ of characteristic zero, the decomposition into $H^i X$ matches a combinatorial direct sum decomposition for certain complete families $\{ E_i \}$ 
of {\it orthogonal idempotents} in $\kk W$:
\begin{equation} \label{eq:generalcorrespondance}
H^* X =\bigoplus_i H^i X  \quad \quad \cong  \quad \quad \displaystyle \bigoplus_i (\kk W) E_i =\displaystyle \kk W.
\end{equation}

The first two spaces $X_n, Y_n$ are well-studied: $X_n$ is the {\it ordered configuration space} of $n$ points in $\R^3$ while $Y_n$ is the {\it $\Z_2$-orbit configuration spaces} for the $\Z_2$-action via $\xx \mapsto -\xx$: %on the same $\R^3$:
\begin{align*}
\TypeAConf{n}&:=\Conf_n\R^3=\{\xx \in (\R^3)^n: x_i \neq x_j \text{ for }1 \leq i < j \leq n\},\\
\TypeBConf{n}&:=\Conf^{\Z_2}_n\R^3=\{\xx \in (\R^3 )^n: x_i \neq \pm x_j \text{ for }1 \leq i <j \leq n, \text{ and }x_i \neq 0 \text{ for }1 \leq i \leq n\}
\end{align*}

Note that $X_n$ has an action of the {\it symmetric group} $W=\symm{n}$ permuting the coordinates of $\xx$, while $Y_n$ carries an action of the {\it hyperoctahedral group} $W=\symmB{n}$ by permuting and negating coordinates. Both spaces have cohomology concentrated only in even degrees and total cohomology carrying the regular representation $\kk W$ for $W = \symm{n}, \symmB{n}$. \newpage

The idempotent decompositions of $\kk \symm{n}$ and $\kk \symmB{n}$ will come from the {\it type $A$ and $B$ Eulerian idempotents}  
$\{ \TypeAEulerianIdempotent{n}{k} \}_{k=0,1,\ldots,n-1}$ in $\kk \symm{n}$ and
$\{ \TypeBEulerianIdempotent{n}{k} \}_{k=0,1,\ldots,n}$ in $\kk \symmB{n}$, defined in work of Reutenauer \cite{reutenauer}, Gerstenhaber--Schack \cite{gerstenhaberschack},  and F. Bergeron and N. Bergeron \cite{bergeronbergeron}. 

The Eulerian idempotents lie within the subalgebras of the group algebras $\kk W$ known as {\it Solomon's descent algebra} $\Sol{W}$, meaning that
when expressed as $\sum_{w \in W} c_w w$, their coefficients $c_w$ depend only upon the Coxeter group {\it descent set} of $w$. Work of Hanlon \cite{hanlon}, Sundaram-Welker \cite{sundaramwelker} and Brauner \cite{brauner2022eulerian} gives a correspondence between these objects: 
\begin{align}
\label{type-A-Eulerian-idempotents-give-cohomology-reps}
H^{2k} X_n&\cong \left(\kk \symm{n} \right) \TypeAEulerianIdempotent{n}{n-1-k} \text{ for }k=0,1,\ldots,n-1,\\
\label{type-B-Eulerian-idempotents-give-cohomology-reps}
H^{2k} Y_n&\cong \left(\kk \symmB{n}\right) \TypeBEulerianIdempotent{n}{n-k} \,\,\,\, \text{ for }k=0,1,\ldots,n.
\end{align}

In this abstract, we use \eqref{type-A-Eulerian-idempotents-give-cohomology-reps} and \eqref{type-B-Eulerian-idempotents-give-cohomology-reps} as the starting point to give a third correspondence of the form \eqref{eq:generalcorrespondance} for the space $Z_n:=Y_n/\Z_2^n \cong \Conf_n(\RP^2 \times \opensegment)$, where $\Z_2^n$ is the normal subgroup of $\symmB{n}$ consisting of 
sign changes; thus $\symm{n} \cong \symmB{n} / \Z_2^n$ acts on $Z_n$.

The idempotents $\{ \PeakIdempotent{n}{k} \}$ in this new correspondence lie inside the \emph{peak algebra} $\PeakAlgebra{n}$, which is the further subalgebra of $\Sol{\symm{n}}$ inside $\kk\symm{n}$ whose elements $\sum_{w \in W} c_w w$ have coefficients $c_w$ depending only upon the {\it peak set} of $w=(w_0:=0,w_1,\ldots,w_n)$
$$
\peak(w):= \{i : 1\leq i \leq n-1\text{ and } w_{i-1} < w_i > w_{i+1} \} .
$$
%with convention $w_0:=0$.
Our main contribution is to relate the \emph{peak representations} $\left( \kk \symm{n} \right) \PeakIdempotent{n}{n-k}$ to the cohomology ring $H^*Z_n$, and to explicitly describe these families of representations in terms of Thrall's famed {\it higher Lie characters} $\lie_{\lambda}$ for $\lambda$ an integer partition of $n$. 


\begin{theorem}
\label{decomposition-of-peak-idempotent-reps-theorem}
Let $\kk$ be a field of characteristic zero.
\begin{itemize}
\item[(i)] The peak idempotent $\PeakIdempotent{n}{k}$ in $\kk \symm{n}$ vanishes unless $k \equiv n \bmod 2$.
\item[(ii)] The cohomology 
$H^i\PeakConf{n}=H^i(\PeakConf{n},\kk)$ vanishes unless $i \equiv 0 \bmod 4$.
\item[(iii)] As a $\symm{n}$-representation, the total
cohomology carries the regular representation:
$$
H^* \PeakConf{n} \cong \kk \symm{n}.
$$
\item[(iv)] For  $0 \leq k \leq n$ with $k$ even, one has $\symm{n}$-representation isomorphisms
$$
\left( \kk \symm{n} \right) \PeakIdempotent{n}{n-k} \cong
H^{2k} \PeakConf{n} 
\cong \bigoplus_{\substack{\lambda \vdash n:\\ \odd(\lambda)=n-k}} \Lie{\lambda},
$$    
where $\odd(\lambda)$ is the number of odd parts of $\lambda$.
\end{itemize}
\end{theorem}
In fact, we refine Theorem \ref{decomposition-of-peak-idempotent-reps-theorem}  (see Theorems \ref{thm:peakcohomology_typeacohomology} and \ref{thm:peakcohomology_interpretation}) by introducing several (compatible) decompositions of $H^*Z_n$ and a  family of primitive idempotents in $\PeakAlgebra{n}$. 

Although $\PeakAlgebra{n}$ is a well-known subalgebra of $\sol(\symm{n})$, it is in general difficult to directly relate the two algebras. Our work offers a step in this direction. The novelty of our approach is to avoid computations in the algebras themselves, and instead develop and utilize concrete combinatorial descriptions of the rings $H^*X_n$, $H^*Y_n$, and $H^*Z_n$.
\newpage 

The remainder of the abstract proceeds as follows. Section \ref{section:background} gives necessary background on the Type $A$ and $B$ stories. We then develop properties of $H^*Y_n$ in Section \ref{section:moreonY}, which will be instrumental in proving our main results on the peak representations in Section \ref{section:mainresults}. In Section \ref{section:hilbertandjordan} we provide generating function formulae and branching rule recursions for the peak representations, and relate this story to the free Jordan algebra.

\section{Background}\label{section:background}
We review here in more detail the spaces $X_n, Y_n$, their cohomology rings, and their relationship to the Eulerian idempotents and Lie characters $\lie_{\lambda}$ discussed in Section \ref{intro-section}.

\subsection{The (associated graded) Varchenko-Gelfand ring}\label{sec:VG}
The cohomology rings $\TypeACohomology{n}:= H^*X_n$ and $\TypeBCohomology{n}:= H^*Y_n$ are closely related to the {\it reflection hyperplane arrangements} $\A_W \subset V = \R^n$ associated to the groups $W=\symm{n}, \symmB{n}$:
 \[ \A_{\symm{n}}=\{x_i = x_j\}_{1 \leq i < j \leq n} \quad \quad \A_{\symmB{n}}=\{x_i=0\}_{1 \leq i \leq n} \sqcup  \{x_i = \pm x_j\}_{1 \leq i<j \leq n}. \]
In particular, Moseley \cite{moseley} proved there are algebra isomorphisms
\[ \TypeACohomology{n} \cong \VG(\A_{\symm{n}}) \quad \quad \TypeBCohomology{n} \cong \VG(\A_{\symmB{n}}), \]
where $\VG(\A)$ is the \emph{(associated graded) Varchenko-Gelfand ring}, defined for any real hyperplane arrangement $\A \subset \R^n$ as the quotient of $\kk[u_i]_{H_{i} \in \A}$
    by an ideal\footnote{In fact, one can take coefficients in $\Z$ rather than $\kk$. However, in what follows, we will want $\kk$ to be a field with characteristic not dividing 2.}
    \[\J_A = \langle u_i^2, \ \   \sum_{j=1}^{c} \epsilon(C,i_j) \cdot u_{i_1} u_{i_2} 
\cdots \widehat{u_{i_j}} \cdots u_{i_{c-1}} u_{i_c} \textrm{ for all } C \subset \A \rangle. \]
Here $C=(C_+,C_-)$ is an {\it oriented matroid} signed circuit  of $\A$, with
  $\epsilon(C,i_j) = \pm 1$, depending on whether $i_j$ lies in $C_+$ or $C_-$. 
\begin{example}\label{ex:typea} \rm
    When $\A = \A_{\symm{n}}$, work of Arnol'd \cite{arnold} and Cohen \cite{cohen} shows that $\TypeACohomology{n}$ has presentation given by
    \[ \TypeACohomology{n} \cong \VG(\A_{\symm{n}}) = \kk[u_{ij}]_{1 \leq i < j \leq n} / \langle u_{ij}^2, \ \   u_{ij} u_{ik} - u_{ij}u_{jk} + u_{ik}u_{jk}\rangle. \]
    Barcelo \cite{barcelo1990action} constructed an elegant \emph{non-broken circuit} monomial basis for $\TypeACohomology{n}$, obtained by taking products with at most one element from each set $U_i$ below: 
 \[U_1 = \{ u_{12} \}, \ U_2 = \{ u_{13}, \  u_{23} \}, \cdots,  \ U_{n-1} =\{ u_{1n}, \  u_{2n},  \ \cdots, \ u_{(n-1),n} \}. \]
\end{example}
In \cite{brauner2022eulerian}, the second author showed that $\VG(\A)$ admits a decomposition by intersection subspaces (i.e. flats) in $\A$. The component of $\VG(\A)_X$ indexed by $X$ is the $\Z$-span of all monomials 
$\{ u_{i_1} \cdots u_{i_\ell} \}$
for which $H_{i_1} \cap \cdots \cap H_{i_\ell}=X$. 

In the case of a reflection arrangement $\A_W$, we can group flats by their $W$-orbits $[X]$, which gives a coarser decomposition of $\VG(\A_W) = \bigoplus \VG(\A_W)_{[X]}$. The flats and flat orbits in $\A_{\symm{n}}$ and $\A_{\symmB{n}}$ have elegant (and useful!) combinatorial descriptions. 

Famously, the flats of $\A_{\symm{n}}$ biject with set partitions of $[n]$. This isomorphism identifies a flat $X$ with the set partition $\pi_X = \{ B_1, \cdots, B_k \}$ where $i$ and $j$ are in the same block $B_\ell$ if and only if $x_i = x_j$ in $X$. The $\symm{n}$-orbits of these flats biject with integer partitions of $n$: the orbit of $\pi_X$ corresponds to the partition $\lambda_X = \{ |B_1|, \cdots, |B_k| \}$. 

Similarly, the flats in $\A_{\symmB{n}}$ can be identified with a set partition on a \emph{subset} $S$ of $[n]^{\pm}:= \{ \overline{1}, \overline{2}, \cdots, \overline{n}, 1, 2, \cdots n \}$, where $S$ does not contain both $i$ and $\overline{i}$. Given a flat $X$, identify $\overline{i}$ with $-x_i$ and let 
$\tau_X = \{ C_1, \cdots C_k \}$ where for $i, j \in [n]$, indices $i$ and $j$ (resp. $i$ and $\overline{j}$) appear in the same block $C_\ell$ if and only if $x_i =  x_j \neq 0$ (resp. if and only if $x_i = -x_j \neq 0$) in $X$. Note that two set partitions related by $i \mapsto \bar{i}$ correspond to the same flat. The $\symmB{n}$ orbit of $\tau_X$ is indexed by a partition $\mu_X= \{ |C_i|, \cdots, |C_k| \}$ of $0 \leq m \leq n$. %For example, if $X = \{ x_1 = -x_2 = x_3 \  ,  \  x_4 = -x_5=  0 \  ,  \  -x_6 \}$ is a flat in $\A_{\symmB{6}}$ then $ \tau_X= \{ \{ 1 , \overline{2}, 3 \}  ,  \{\overline{6} \} \}$ and $\mu_X = (3,1)$. 

We write $\TypeACohomology{\lambda_X}^{(n)}:= \VG(\A_{\symm{n}})_{[\pi_X]}$ and $\TypeBCohomology{\mu_X}^{(n)}:= \VG(\A_{\symmB{n}})_{[\tau_X]}$, giving the decompositions
\[ \TypeACohomology{n} = \bigoplus_{\lambda \vdash n} \TypeACohomology{\lambda}^{(n)} \quad \quad \quad \quad  \TypeBCohomology{n} = \bigoplus_{\mu \vdash 0 \leq m \leq n} \TypeBCohomology{\mu}^{(n)}.\]

\subsection{The Eulerian idempotents and higher Lie characters}

The idempotents $\{ \TypeAEulerianIdempotent{n}{k}\}$ and  $\{ \TypeBEulerianIdempotent{n}{k}\}$ from Section \ref{intro-section} can be defined via the formula in \cite{brauner2022eulerian}:
\[ 
\sum_{k=0}^r  t^k  E_{k}^{W}  = \frac{1}{|W|}\sum_{w \in W} \left( \prod_{i=1}^{\des(w)} (t-e_i) \prod_{i=1}^{r - \des(w)} (t+e_i)  \right) \cdot w,
\]
 which recovers work of Garsia--Reutenauer \cite{garsiareutenauer} for $W = \symm{n}$ and Bergeron--Bergeron \cite{bergeronbergeron} for $W = \symmB{n}$. Here, $r$ is the \emph{rank} of $\A_W$ ($r = n-1$ for $W = \symm{n}$ and $r = n$ for $W = \symmB{n}$) and the $e_i$ are the \emph{exponents} of $W$ ($e_i = i$ for $W = \symm{n}$ and $e_i = 2i-1$ for $W = \symmB{n}$). The \emph{descent number}, $\des(w)$ is the number of simple reflections $s$ of $W$ with $\ell(ws) < \ell(w)$.

The $E_{k}^{W}$ have a refinement due to 
Bergeron--Bergeron--Howlett--Taylor \cite{BBHT}, who introduced families of complete, primitive orthogonal idempotents in $\sol(W)$ for any finite Coxeter group $W$. These idempotents, which we will call the \emph{BBHT idempotents}, are indexed by $W$-flat orbits. We omit the technical definitions, but note that by the discussion in \S \ref{sec:VG},
 for $W = \symm{n}, \symmB{n}$ they can be indexed as $\{ \TypeAEulerianIdempotent{n}{\lambda}: \lambda \vdash n \}$ and $\{ \TypeBEulerianIdempotent{n}{\mu}: \mu \vdash m, m\leq n\}$. 


To recover the $\{ \TypeAEulerianIdempotent{n}{k}\}$ and  $\{ \TypeBEulerianIdempotent{n}{k}\}$, 
group $\{ \TypeAEulerianIdempotent{n}{\lambda}\}$ and  $\{ \TypeBEulerianIdempotent{n}{\mu}\}$ by partition \emph{length} $\ell$:
\begin{equation}\label{eq:eulerianrefinement} 
\TypeAEulerianIdempotent{n}{k} =\sum_{\lambda: \ \ell(\lambda) = k} \TypeAEulerianIdempotent{n}{\lambda} \quad \quad \quad \quad \TypeBEulerianIdempotent{n}{k} =\sum_{\mu: \ \ell(\mu) = k} \TypeBEulerianIdempotent{n}{\mu}. \end{equation}
We can also refine  the isomorphisms in \eqref{type-A-Eulerian-idempotents-give-cohomology-reps} and \eqref{type-B-Eulerian-idempotents-give-cohomology-reps} using the BBHT idempotents: 
\begin{theorem}[Brauner, \cite{brauner2022eulerian}] \label{thm:eulerianrep}
    There are $\symm{n}$ and $\symmB{n}$ representation isomorphisms
    \[\TypeACohomology{\lambda}^{(n)} \cong \left(\kk \symm{n} \right) \TypeAEulerianIdempotent{n}{\lambda} \quad \quad \quad \quad \TypeBCohomology{\mu}^{(n)} \cong \left(\kk \symmB{n} \right) \TypeBEulerianIdempotent{n}{\mu}.  \]
\end{theorem}


In fact, there is more to say in the case of $W = \symm{n}$, relating to the {\it higher Lie representations} $\{ \Lie{\lambda}\}$ of Thrall \cite{thrall}. Let $\CC_{\lambda}$ be the conjugacy class of $\symm{n}$ indexed by the partition $\lambda = (1^{m_1}, 2^{m_2}, \cdots n^{m_n})$. The centralizer $Z_{\lambda}$ of an element of $\CC_{\lambda}$  has isomorphism type
\[ Z_{\lambda} \cong \prod_{j=1}^n \symm{m_j}[\Z_{j}],\]
where $\Z_j$ is the cyclic group of order $j$, and $\symm{m_j}[\Z_{j}]$ is the wreath product. Specifically, the action of $\symm{m_j}$ in this wreath product swaps the $m_j$ blocks of $\lambda$ of size $j$.  

We will be interested in a linear character $\omega_{\lambda}$ on $Z_\lambda$ obtained from extending faithful characters on each $\Z_j$ to $Z_{\lambda}$, where $\omega_{\lambda}$ restricts trivially on the wreath factors $\symm{m_j}$ of $Z_{\lambda}$. 

Write $\uparrow_H^G$ to be the representation induction from a subgroup $H$ of $G$ to $G$. 
\begin{definition}\label{def:thrallhigherlie}\rm
   Give a partition $\lambda \vdash n$, define $\lie_{\lambda}:= \omega_\lambda \uparrow_{Z_{\lambda}}^{\symm{n}}.$
\end{definition}
Thrall proved that $\kk \symm{n} \cong \bigoplus_{\lambda \vdash n} \Lie{\lambda}$.  A beautiful result of Hanlon \cite{hanlon} then shows that $\lie_{\lambda} \cong (\kk \symm{n}) \TypeAEulerianIdempotent{n}{\lambda}$. Using \eqref{eq:eulerianrefinement}, we can thus conclude 
\begin{equation*}
\label{type-A-higher-Lie-decomposition}
\begin{aligned}
 \left(\kk \symm{n} \right) 
 \TypeAEulerianIdempotent{n}{n-1-k} \cong
\bigoplus_{\substack{\lambda \vdash n:\\ \ell(\lambda)=n-k}} \Lie{\lambda} \cong H^{2k} X_n. \quad \quad \quad 
\end{aligned}
\end{equation*}
\begin{example}\rm
    When $\lambda = (n)$, the representation $\lie_n:=\lie_{(n)} $ is isomorphic to the multilinear component of the free Lie algebra, defined and generalized in \S \ref{sec:freejordanalgebra}.
\end{example}

\section{Presentations, Filtrations, and Decompositions of $H^*Y_n$}\label{section:moreonY}
 Our first task is to study the ring $\TypeBCohomology{n}:= H^* Y_n$ in greater detail. It will be important for the remainder of this section to assume that {\bf the field $\kk$ has characteristic larger than $n$}, so that $2 \in \kk^\times$ and $\kk[\symmB{n}]$ is semisimple.  This allows us to make an invertible change-of-variables that diagonalizes the action of the normal subgroup $\Z_2^n$ within $\symmB{n}$.

The presentation of $\TypeBCohomology{n} \cong \VG(\A_{\symmB{n}})$ was first given by Xicotencatl \cite{xico}; it is isomorphic to $\kk[u_{ij}^{+},  u_{ij}^-,  u_i]/ J_{\symmB{n}}$ for $1 \leq i < j \leq n$, with generators corresponding to 
\[ u_{ij}^+ \longleftrightarrow \{ x_i = x_j \} \quad \quad u_{ij}^- \longleftrightarrow \{  x_i = -x_j \} \quad \quad u_i \longleftrightarrow \{ x_i = 0 \} \] respectively. 
The generating relations for $\J_{\symmB{n}}$ are given in Table \ref{table:relations}.  
 
 We will introduce a new basis for $\TypeBCohomology{n}$, a filtration using that basis, and a corresponding associated graded ring. Along the way, we will see several useful decompositions of $\TypeBCohomology{n}$. 

\begin{definition}\label{def:basismap}\rm
For $1 \leq i < j \leq n$, define an isomorphism of graded $\kk$-algebras $\basismap$ by
\[ u_i \longmapsto u_i \quad \quad v_{ij} \longmapsto u_{ij}^{+} + u_{ij}^{-} \quad \quad w_{ij} \longmapsto u_{ij}^{+} - u_{ij}^{-}\]
with inverse given by $\basismap^{-1}(u_i)=u_i,\basismap^{-1}(u_{ij}^+) = \frac{1}{2}(v_{ij} + w_{ij}), \basismap^{-1}(u_{ij}^-) = \frac{1}{2}(v_{ij} - w_{ij})$.
\end{definition}

We wish to rewrite the presentation 
$\TypeBCohomology{n}:=
\kk[ u_{ij}^+, u_{ij}^-, u_i]/\J_{\symmB{n}}$
in terms of these new variables $v_{ij}, w_{ij}$, using a Gr\"obner basis argument.
Introduce a lexicographic monomial ordering $\prec$ on $\kk [v_{ij},w_{ij}, u_{i} ]$, in which the
variables  $u_i, v_{ij}, w_{ij}$ are ordered as
follows:
\begin{equation}\label{vbasisordering} u_1 < u_2 < \cdots u_n < v_{12} < w_{12} < v_{13} < w_{13} < \cdots < v_{(n-1)1} < w_{(n-1)n}.\end{equation}



\begin{theorem}\label{theorem:newpresentation}
The isomorphism $\basismap: \kk[v_{ij}, w_{ij}, u_i] \longrightarrow \kk[u_{ij}^+, u_{ij}^-, u_i]$
induces a graded
$\kk$-algebra isomorphism, where $\I$ is generated by the relations $\GGG$ listed in Table \ref{table:relations} below:
\[ 
\kk [v_{ij},w_{ij}, u_{i} ] / \I 
\overset{\sim}{\longrightarrow}
\kk[ u_{ij}^+, u_{ij}^-, u_i]/\J_{\symmB{n}}  =:\TypeBCohomology{n},
\]

Moreover, $\GGG$ gives a Gr\"obner basis 
for the ideal $\I$ with respect to $\prec$, in which the standard monomial $\kk$-basis for the quotient
$\kk [v_{ij},w_{ij}, u_{i} ] / \I$
is the set of monomials $\vbasis$
obtained 
from taking products with at most one element from each of these sets $V_i$:
 \[V_1 = \{ u_1 \}, \ V_2 = \{ u_2, \  v_{12}, \ w_{12} \}, \ \cdots , \ V_n = \{ u_n, \ v_{1n}, \ w_{1n}, \cdots, v_{(n-1)n}, \ w_{(n-1)n} \}. \]
\end{theorem}


We make two observations about the $\symmB{n}$ action on $\TypeBCohomology{n}$.  First, elements of $\Z_2^n \subset \symmB{n}$ scale all of $u_i, v_{ij}, w_{ij}$ via $\pm 1$; thus Theorem \ref{theorem:newpresentation} will allow us to construct a monomial basis for $H^*\PeakConf{n} \cong (\TypeBCohomology{n})^{\Z_2^n}$ in \S \ref{section:mainresults}. Second, the generators segregate into two $\symmB{n}$-orbits: 
 $\{ u_i \}_{1\leq i\leq n}$ and  $\{ v_{ij}, w_{ij} \}_{1 \leq i < j \leq n}$. 
This leads to a helpful {\it filtration}, as follows.

For $q \in \kk[v_{ij},w_{ij},u_i]$, let $\deg(q)$ be the polynomial degree of $q$,  
$\deg_{\vbasis}(q)$ to be the degree of $q$ in the $v_{ij}$ and $w_{ij}$ variables, and $\deg_u(q)$ be the degree in the $u_i$ variables.  Our key insight is that $\TypeBCohomology{n}$ admits a filtration by $\deg_u$. In particular, define the ideal
\begin{align*}
P^{(i)}:=& \{ q \in \TypeBCohomology{n} \subset \kk[u_i, v_{ij}, w_{ij}]: \deg_u(q) \geq i \}.
  \end{align*}
  
For example, when $n=2$ the ideal $P^{(1)}$ is the $\kk$-span of $\{ u_1, \ u_2, \ u_1 v_{12}, \ u_1 w_{12}, \ u_1 u_2 \}.$


\begin{prop}\label{prop:ufiltration}
    There are $\symmB{n}$-stable ascending filtrations on $\TypeBCohomology{n}$ given by
    \begin{align*} &P^{(n)} \subset P^{(n-1)} \subset \cdots \subset P^{(1)} \subset P^{(0)}. %\\
    \end{align*}
    The associated graded ring $\ag{\TypeBCohomology{n}} = \bigoplus_{i = 0}^{n} P^{(i)}/P^{(i+1)}  $
    has presentation $\kk [v_{ij},w_{ij}, u_{i} ] / \gr(\I) $
for $1 \leq i < j \leq n$, where the relations generating $\gr(\I)$ are given in Table \ref{table:relations}.
\end{prop}


The motivation for introducing and studying the associated graded ring $\ag{\TypeBCohomology{n}}$ is that in our context (i.e. $\kk \symmB{n}$ being a semisimple algebra),  we have $\ag{\TypeBCohomology{n}} \cong \TypeBCohomology{n}$ as $\symmB{n}$-modules. Hence, it suffices to study the basis and representations on $\ag{\TypeBCohomology{n}}$.

We will see that $\ag{\TypeBCohomology{n}}$ has several useful decompositions that make studying the representations on $\TypeBCohomology{n}$ (and eventually $H^*\PeakConf{n}$) far more tractable. 
\newpage
\begin{center}
\begin{table}[!h]
\centering
\setlength{\tabcolsep}{10pt} % Default value: 6pt
\renewcommand{\arraystretch}{1.5} % Default
\begin{tabular}{|l|l|l|}  \hline
\rm
%Generating & Generating & Generating\\
Relations for $\J_{\symmB{n}}$ & Relations for $\I$ & Relations for $\gr(\I)$\\
\hline
$u_i^2$ &   $u_{i}^2  $ &   $u_{i}^2 $  \\
$u_i u_{ij}^+ - u_i u_{ij}^{-} - u_{ij}^+ u_{ij}^-$ &$v_{ij}w_{ij}$ & $v_{ij}w_{ij}$\\
 $u_i u_j - u_i u_{ij}^- - u_j u_{ij}^-$ &$u_{i}w_{ij} -u_{j} v_{ij}$ &  $u_{i}w_{ij} -u_{j} v_{ij}$ \\ 
$(u_{ij}^+)^2$  &$v_{ij}^2 - 2u_{i} w_{ij}$ &  $v_{ij}^2 $ \\
$(u_{ij}^-)^2$  &$w_{ij}^{2} + 2 u_{i} w_{ij}$ &  $w_{ij}^{2}$ \\
 $u_i u_j - u_i u_{ij}^- - u_j u_{ij}^-$&$u_{i} v_{ij} - 2 u_{i} u_{j} - u_{j} w_{ij}$ & $u_{i} v_{ij} - u_{j} w_{ij}$ \\
$u_{ij}^{+}u_{jk}^{+} - u_{ij}^{+}u_{ik}^{+} - u_{ik}^{+}u_{jk}^{+} $ & $v_{ij}w_{jk} - w_{ij}w_{ik} - v_{ik}v_{jk}$ &  $v_{ij}w_{jk} - w_{ij}w_{ik} - v_{ik}v_{jk}$\\
$u_{ij}^{-}u_{jk}^{+} - u_{ij}^{-}u_{ik}^{-} - u_{ik}^{-}u_{jk}^{+}$  &$w_{ij}w_{jk} - v_{ij}w_{ik} - w_{ik}w_{jk}$&  $w_{ij}w_{jk} - v_{ij}w_{ik} - w_{ik}w_{jk}$\\
$-u_{ij}^{-}u_{jk}^{-} + u_{ij}^{-}u_{ik}^{+} - u_{ik}^{+}u_{jk}^{-} $ &$v_{ij}v_{jk} - v_{ij}v_{ik} - v_{ik}w_{jk}$&  $v_{ij}v_{jk} - v_{ij}v_{ik} - v_{ik}w_{jk}$\\
$-u_{ij}^{+}u_{jk}^{-} + u_{ij}^{+}u_{ik}^{-} - u_{ik}^{-}u_{jk}^{-} $ & $w_{ij}v_{jk} - w_{ij}v_{ik} - w_{ik}v_{jk}$ &   $w_{ij}v_{jk} - w_{ij}v_{ik} - w_{ik}v_{jk}$ \\
%  &&\\
 \hline 
\end{tabular}
\caption{Generating relations for the ideals $\J_{\symm{n}}, \I$ and $\gr(\I)$.}
\label{table:relations}
\end{table}
\end{center}
\vspace{-3em}

First, one can show that the flat orbit decomposition from \S \ref{sec:VG} persists in $\ag{\TypeBCohomology{n}}$; we will abuse notation and write $\TypeBCohomology{\mu}^{(n)}$ instead of $\ag{\TypeBCohomology{\mu}}^{(n)}$ since they are isomorphic.  

The second useful decomposition is the following bi-grading:
\[ \TypeBCohomology{k,\ell}^{(n)} := \mathrm{span}_{\kk}\{ q \in \ag{\TypeBCohomology{n}}: \deg(q) = k \quad \deg_{\vbasis}(q) = \ell\}. \]
In fact, this bi-grading can be refined to a third decomposition by signed partitions, which are pairs of partitions $\slambda$ such that $|\lambda^+|+ |\lambda^-|=n$. 
\begin{definition}\label{def:graphs} \rm
Given a monomial in $q \in \Q[u_{i}, v_{ij}, w_{ij}]$, associate to $q$ a signed partition $(\lambda^+_{(q)}, \lambda^-_{(q)})$ as follows:
\begin{enumerate}
    \item Construct a graph $\G(q)$ with vertex set $[n] = \{ 1,2,\cdots, n\}$ by drawing an edge between $i$ and $j$ if $v_{ij}$ or $w_{ij}$ occurs in $q$, and drawing a loop at $i$ if $u_{i}$ occurs in $q$;
    \item Let $\G_1 = (E_1, V_1), \cdots , \G_k = (E_k, V_k)$ be the connected components of $\G(q)$. Then 
    \[ \lambda^+_{(q)} := \{ |V_\ell|: \G_\ell \textrm{ has no loops} \} \quad  \lambda^-_{(q)} := \{ |V_\ell|: \G_\ell \textrm{ has loops} \}. \]
\end{enumerate}
 \end{definition}

\begin{prop}\label{prop:signedpartitiondecomposition}
There is a decomposition of $\ag{\TypeBCohomology{n}} $ by signed partitions $\ag{\TypeBCohomology{n}} = \bigoplus_{\slambda} \TypeBCohomology{\slambda}^{(n)},$ 
where \[ \TypeBCohomology{\slambda}^{(n)}:= \mathrm{span}_{\kk} 
\{ 
\textrm{monomials } q \in \ag{\TypeBCohomology{n}}: (\lambda^+_{(q)}, \lambda^-_{(q)}) = \slambda 
\}.
\]

This decomposition is compatible with the other decompositions of $\ag{\TypeBCohomology{n}}$, in the sense that:
    \[  \TypeBCohomology{\mu}^{(n)} = \bigoplus_{\substack{\slambda: \ \lambda^+ = \mu}} \TypeBCohomology{\slambda}^{(n)} \quad \quad \quad \TypeBCohomology{k,\ell}^{(n)} = \bigoplus_{\substack{\slambda: \ \ell(\lambda^{+}) = n-k\\ \ell(\lambda^+) + \ell(\lambda^-) = n-\ell  }} \TypeBCohomology{\slambda}^{(n)}
\]
\end{prop}

\noindent For example, suppose $n=8$ and $q= w_{12} \cdot u_{5} \cdot v_{56} \cdot u_{7} \cdot v_{24}$. Then $q$ is in the bi-graded piece $\TypeBCohomology{5,3}^{(8)}$ and we have $\lambda^+_{(q)} = \{ 3,1,1 \} $ and $\lambda^-_{(q)} = \{ 2,1 \} $. Thus $q \in  \TypeBCohomology{((3,1,1),(2,1))}^{(8)} \subset \TypeBCohomology{(3,1,1)}^{(8)}$.

\begin{theorem}\label{thm:mapYtoX}
There is a well-defined, $\symm{n}$-equivariant surjection of $\kk$-vector spaces 
\begin{align*}
    \gamma: \ag{\TypeBCohomology{n}} &\longrightarrow \TypeACohomology{n} = \kk[u_{ij}]_{1 \leq i < j \leq n} / \langle u_{ij}^2, \ \   u_{ij} u_{ik} - u_{ij}u_{jk} + u_{ik}u_{jk} \rangle \\ 
    \TypeBCohomology{\slambda}^{(n)} &\longmapsto \TypeACohomology{(\lambda^+ \cup \lambda^-)}^{(n)}, 
\end{align*}
defined by sending $\hspace{2em}  \gamma(u_i) = 1, \hspace{2em} \gamma(w_{ij}) = u_{ij} \hspace{2em} \gamma(v_{ij}) = u_{ij}.$

\end{theorem}
\begin{proof}[Proof idea]
The key observation is that the relations $u_i w_{ij} - u_j v_{ij}$ and $ u_i v_{ij} - u_j w_{ij}$ in $\gr(\I)$ mean that one can give a presentation of $\ag{\TypeBCohomology{n}}$ as a quotient of a subring of $\kk[v_{ij}, w_{ij}, u_i]$, by an ideal $\tilde{\I} \subset \gr(\I)$ that omits the relation $u_i^2$. From this, one can define a surjection of \emph{vector spaces}; note however that $\gamma$ cannot be extended to a map of algebras.  
\end{proof}
\section{Main Results}\label{section:mainresults}
At last, we are ready to analyze the peak representations.
Our investigations began from an observation of Aguiar, Bergeron and Nyman \cite{aguiar2004peak} relating the descent algebras $\Sol{\symm{n}}$ and $ \Sol{\symmB{n}}$ to the {\it peak algebra} $\PeakAlgebra{n}$.  

Recall that one can express the hyperoctahedral group of all signed permutations as 
$
\symmB{n} = \symm{n} \ltimes \Z_2^n
$
where $\Z_2^n$ is the normal subgroup performing arbitrary sign changes in the coordinates.
The quotient map $\symmB{n} \twoheadrightarrow \symmB{n}/\Z_2^n \cong \symm{n}$
of groups, which forgets the signs in a signed permutation, gives rise to a surjective $\kk$-algebra map 
$
\varphi: \kk \symmB{n} \twoheadrightarrow \kk \symm{n}.
$
In \cite{aguiar2004peak}, it was shown that the peak subalgebra $\PeakAlgebra{n}$ is exactly the image under $\varphi$ of $\Sol{\symmB{n}}$, that is, $\varphi$ restricts to an algebra surjection
$
\Sol{\symmB{n}} \overset{\varphi}{\twoheadrightarrow} \PeakAlgebra{n}.
$

As a consequence, one can define a family of {\it peak idempotents} inside $\PeakAlgebra{n} \subset \kk \symm{n}$ via 
\[ \PeakIdempotent{n}{k}:=
\varphi(\TypeBEulerianIdempotent{n}{k}) \textrm{ for }k=0,1,\cdots, n \quad \quad \PeakIdempotent{n}{\mu}:=
\varphi(\TypeBEulerianIdempotent{n}{\mu}) \textrm{ for } \mu \vdash m \leq n.\]
Both families inherit from $\{\TypeBEulerianIdempotent{n}{k}\}$ and $\{\TypeBEulerianIdempotent{n}{\mu}\}$ the property of  being a complete system of orthogonal idempotents in $\kk\symm{n}$, and the $\{ \PeakIdempotent{n}{\mu} \} $ are also primitive if nonzero. Note that some of the $\PeakIdempotent{n}{k}$ and $\PeakIdempotent{n}{\mu}$ will be zero, which we characterize in Theorems \ref{decomposition-of-peak-idempotent-reps-theorem} and \ref{thm:peakcohomology_interpretation}. 

\noindent By construction, one recovers $\PeakIdempotent{n}{k}$ from the ${\PeakIdempotent{n}{\mu}}$ by summing over all $\mu$ of length $k$.

Our goal is to relate the peak idempotents to the ring $\PeakCohomology{n} := H^*\PeakConf{n},$ where 
$$
\PeakConf{n}:=\TypeBConf{n}/\Z_2^n = \Conf_n(\left( \R^3 \setminus \{\origin\} \right)/\Z_2)
=\Conf_n(\RP^2 \times \opensegment)
$$
is the configuration space 
of $n$ ordered points within the quotient $\R^3 \setminus \{\origin\}$ under the $\Z_2$-action via $\xx \mapsto -\xx$, so that 
$
\left( \R^3 \setminus \{\origin\} \right)/\Z_2
\,\, \cong \,\, \RP^2 \times \opensegment.
$  

Note that $(\TypeBCohomology{n})^{\Z_2^n} \cong \PeakCohomology{n}$. The filtration, bigrading, and finer decompositions (by flat orbits and signed partitions) on $\TypeBCohomology{n}$ from Section \ref{section:moreonY} persist when one takes $\Z_2^n$-fixed spaces, giving a bigraded $\symm{n}$-representation on an associated
graded ring $\ag{\PeakCohomology{n}}$: 
\[ \PeakCohomology{k,\ell}^{(n)} := (\TypeBCohomology{k,\ell}^{(n)})^{\Z_2^n}, \quad \quad \quad \PeakCohomology{\mu}^{(n)} := (\TypeBCohomology{\mu}^{(n)})^{\Z_2^n},\quad \quad \quad \PeakCohomology{\slambda}^{(n)}:= (\TypeBCohomology{\slambda}^{(n)})^{\Z_2^n}. \] 

We first construct monomial a basis for $\PeakCohomology{n}$, using the fact that by Theorem \ref{theorem:newpresentation}, the basis $\vbasis$ of $\TypeBCohomology{n}$ diagonalizes the action  of the normal subgroup $\Z_2^n \leq \symmB{n}$ on $\TypeBCohomology{n}$. 

    \begin{definition}\rm
For $1 \leq i < j < k \leq n$, let $\ \I_1: = \{ u_{i} w_{ij} \}, \ \I_2:= \{ w_{ij}w_{ik} \}, \ \I_3:= \{ v_{ij}w_{jk} \} $.

\noindent Let $\tvbasis$ be the monomials obtained from products in $\I_j$ for $j =1,2,3$ that are also in $\vbasis$.
\end{definition}
\begin{theorem}\label{thm:zbasis}
The set $\tvbasis$ is a basis for $\PeakCohomology{n}$ and $\ag{\PeakCohomology{n}}$ that is compatible with the decomposition by signed partitions: $\ag{\PeakCohomology{n}} =  \bigoplus \PeakCohomology{\slambda}^{(n)}$.
\end{theorem}
\begin{proof}[Proof idea]
    We construct a bijection from $\tvbasis$ to the monomial basis of $\TypeACohomology{n}$ from Example \ref{ex:typea}. This involves defining a ``pairing lemma'' to group quadratic terms appearing in $q \in \tvbasis$ and then mapping:  $u_{i}w_{ij}$ to $u_{ij}$, $w_{ij}w_{ik}$ to $u_{ij}u_{ik}$, and $v_{ij}w_{jk}$ to $u_{ij}u_{jk}$.   
    \end{proof}

\begin{example}\rm
  The basis for $ \PeakCohomology{4,2}^{(4)}$ is 
$ \{ (u_1 w_{12})(u_3 w_{34}), \  (u_1 w_{13})(u_2 w_{24}), \ (u_1 w_{14})(u_2 w_{23}) \}. $
\end{example}

Given a partition $\lambda$ of $n$, recall that $\ell(\lambda)$ is its number of parts and $|\lambda|$ is its size. Let $\oddparts(\lambda)$ (resp. $\evenparts(\lambda)$) be the partition obtained by taking only the odd (resp. even) parts of $\lambda$. We call $\lambda$ an \emph{odd partition} if $\oddparts(\lambda) = \lambda$ and an \emph{even partition} if $\evenparts(\lambda) = \lambda$. Write $\odd(\lambda) = \ell(\oddparts(\lambda))$ and $\even(\lambda) = \ell(\evenparts(\lambda))$. 

\begin{theorem}\label{thm:peakcohomology_typeacohomology}
 The space $ \PeakCohomology{\slambda}^{(n)}$ vanishes
unless $\lambda^+$ is an odd partition and $\lambda^-$ is an even partition, while $\PeakCohomology{\mu}^{(n)}$ vanishes unless $\mu$ is an odd partition and $n-|\mu|$ is even.

\noindent Moreover, the map $\gamma$ restricts to an $\symm{n}$-equivariant vector-space isomorphism $ \gamma: \PeakCohomology{n} \longrightarrow \TypeACohomology{n}$:
    \begin{align*} 
    \gamma( \PeakCohomology{\slambda}^{(n)}) = \TypeACohomology{(\lambda^+ \cup \lambda^-)}^{(n)} \quad \quad \gamma^{-1}(\TypeACohomology{\lambda}^{(n)}) = \PeakCohomology{(\oddparts(\lambda),\evenparts(\lambda))}^{(n)}.
    \end{align*}
    
   \noindent Thus, for non-vanishing $\PeakCohomology{\slambda}^{(n)}, \ \PeakCohomology{\mu}^{(n)}, \ $ and $ \ \PeakCohomology{2k,\ell}^{(n)}$, there are  $\symm{n}$-representation isomorphisms 
    \[ \PeakCohomology{\slambda}^{(n)} \cong  \lie_{(\lambda^+ \cup \lambda^-)}, \quad \quad \PeakCohomology{\mu}^{(n)} \cong \bigoplus_{\substack{\lambda: \ \oddparts(\lambda) = \mu 
    %\\ n-|\mu| \equiv 0 \mod 2
    }} \lie_{\lambda}, \quad \quad \PeakCohomology{2k,\ell}^{(n)} \cong \bigoplus_{\substack{\lambda: \ell(\lambda) = n-\ell\\ \odd(\lambda) = n-2k}} \lie_\lambda.  
    \]

\end{theorem}
    \begin{example} \rm When $n=4$, the non-vanishing pieces $\PeakCohomology{\mu}^{(4)}$ are as follows:
        \[ \PeakCohomology{\emptyset}^{(4)} \cong \lie_{(2,2)} \oplus \lie_{(4)} \quad  \PeakCohomology{(1,1)}^{(4)} \cong \lie_{(2,1,1)} \quad \PeakCohomology{(3,1)}^{(4)} \cong \lie_{(3,1)} \quad \PeakCohomology{(1,1,1,1)}^{(4)} \cong \lie_{(1,1,1,1)} . \]
        \newpage
 \noindent  The non-vanishing bi-graded pieces $\PeakCohomology{2k,\ell}^{(4)}$ are
    \[ \PeakCohomology{0,0}^{(4)} \cong \lie_{(1,1,1,1)} \quad \PeakCohomology{2,1}^{(4)} \cong \lie_{(2,1,1)} \quad \PeakCohomology{2,2}^{(4)} \cong \lie_{(3,1)} \quad \PeakCohomology{4,2}^{(4)} \cong \lie_{(2,2)} \quad \PeakCohomology{4,3}^{(4)} \cong \lie_{(4)}. \]
    \end{example}

In fact, we now have all the tools necessary to provide a cohomological interpretation of the $\symm{n}$-representations generated by the Peak idempotents, by analyzing the $\Z_2^n$ fixed spaces of Theorem \ref{thm:eulerianrep} and applying Theorem \ref{thm:peakcohomology_typeacohomology}.

\begin{theorem}\label{thm:peakcohomology_interpretation}
The idempotent $\PeakIdempotent{n}{\mu}$ does not vanish if and only if $\mu$ is an odd partition (including $\mu = \emptyset)$ and $n- |\mu|$ is even. In this case, there are $\symm{n}$-representation isomorphisms
    \[ (\kk \symm{n}) \PeakIdempotent{n}{\mu} \cong \PeakCohomology{\mu}^{(n)} \cong \bigoplus_{\lambda: \ \oddparts(\lambda) = \mu} \lie_{\lambda} .\]

\end{theorem}
Note that combining Proposition \ref{prop:signedpartitiondecomposition} with Theorems \ref{thm:peakcohomology_typeacohomology} and \ref{thm:peakcohomology_interpretation} implies Theorem \ref{decomposition-of-peak-idempotent-reps-theorem}. 
\section{Hilbert series and the free Jordan algebra}\label{section:hilbertandjordan}
Having established the connection between the peak algebra and the ring $\PeakCohomology{n}$, we now develop enumerative and recursive properties of the latter. 

Let $\Lambda$ denote the {\it ring of symmetric functions} (of bounded degree, in infinitely many variables).  
It has a $\Z$-algebra isomorphism known as the {\it Frobenius characteristic map}  $\fch: \oplus_{n \geq 0} \mathrm{Rep}(\symm{n}) \rightarrow \Lambda$, where $\mathrm{Rep}(\symm{n})$ are the {\it virtual characters} of $\symm{n}$. 
We will study the Frobenius characteristic of $\PeakCohomology{2k,\ell}^{(n)}$, using the fact that $\PeakCohomology{2k+1,\ell}^{(n)} = 0$ by Theorem \ref{decomposition-of-peak-idempotent-reps-theorem}.
\begin{definition} \rm \label{def:genfunc}
 Write $\Lambda_{\Z[t,q]}$ to be the ring $\Lambda$ with coefficients in $\Z[t,q]$ and define
$$
\begin{aligned}
M_{n}(t,q) := \sum_{k,\ell} \dim \left( \PeakCohomology{2k,\ell}^{(n)} \right) t^{k} q^{\ell} \in \Z[t,q], \quad \quad \quad \M^{(n)}(t,q):=\sum_{k,\ell}
\fchar \left( \PeakCohomology{2k,\ell}^{(n)}  \right) t^k q^\ell \in \Lambda_{\Z[t,q]}.
\end{aligned}
$$
\end{definition}

\noindent For $w \in \symm{n}$ let
$\even(w), \odd(w)$ denote the number of even-sized and odd-sized cycles of $w$, and $\cyc(w)$ the number of cycles of $w$.

\begin{theorem}\label{thm:generatingfunction}
Write $L_{\lambda}:= \fch(\lie_{\lambda})$. Then one can rewrite $M_{n}(t,q)$ and $\M^{(n)}(t,q)$ as follows:
$$
\begin{aligned}
M_{n}(t,q) =
\sum_{w \in \symm{n}} t^{\frac{n-\odd(w)}{2}} q^{n-\cyc(w)}, \quad \quad \quad
\M^{(n)}(t,q)=
\sum_{\lambda \vdash n} L_\lambda \cdot t^{ \frac{|\lambda|-\odd(\lambda)}{2}} q^{|\lambda|-\ell(\lambda)}.
\end{aligned}
$$
\end{theorem}
Using Theorem \ref{thm:generatingfunction}, we manipulate the symmetric functions in $\M^{(n)}(t,q)$ to give a branching rule recurrence for the bi-graded pieces $\PeakCohomology{2k,\ell}^{(n)}$. Let $\ind$ denote representation induction from $\symm{n}$ to $\symm{n+1}$ and $\res$ denote representation restriction from $\symm{n}$ to $\symm{n-1}$. 

\begin{theorem}
\label{restriction-rep-recursion}
The restriction of $\PeakCohomology{2k,j}^{(n)}$ 
%One has this description of
%the restriction of each bidegree 
from an $\symm{n}$ to an $\symm{n-1}$-module is given by
\[  
\PeakCohomology{2k,\ell}^{(n)} 
\res
= \PeakCohomology{2k,\ell}^{(n-1)} +  
\PeakCohomology{2(k-1),\ell-1}^{(n-2)} \ind + 
\left( \PeakCohomology{2(k-1),\ell-2}^{(n-2)}  \ind \right) * \chi^{(n-2,1)},\]
where $*$ is the Kronecker product and $\chi^{(n-2,1)}$ is the irreducible reflection representation of $\symm{n-1}$.
\end{theorem}

\noindent Theorem \ref{restriction-rep-recursion} implies a recursive formula for $M_n(t,q)$ with interesting specializations:
\begin{align}
  \label{M1q}  M_n(1,q) &= (1+q)(1+2q)\cdots (1+(n-1)q),\\
  \label{Mt1}  M_n(t,1) &= (1+(n-1)q) \cdot M_{n-1}(1,q),
\end{align}
where \eqref{M1q} is the generating function for the {\it Stirling numbers of the first kind}, and \eqref{Mt1} describes the \emph{Sheffer polynomials} \cite{oeis} counting permutations $w$ according to $\odd(w)$.
\subsection{The space of simple Jordan elements}\label{sec:freejordanalgebra}
Finally, we mention an interesting connection between $\PeakCohomology{n}$ and the multilinear part of the {\it space of simple Jordan elements} within the free associative algebra
$\kk\langle \xx\rangle=\kk\langle x_1,\ldots,x_n\rangle$.

Consider a deformation of the Lie bracket on $\kk\langle \xx\rangle$ by $\alpha \in \C$: \ $[x,y]_{\alpha}:= xy - \alpha yx.$
Let $J_{\alpha}$ be the smallest $\kk$-subspace of $\kk \langle \xx \rangle$ containing the generators $\xx$ and closed under $[\cdot,\cdot]_\alpha$.


For example, $J_1 \subset \kk\langle \xx \rangle$ is the free Lie algebra.
Define $V_n(\alpha) \subset J_{\alpha}$ to be the $\kk$-subspace spanned by
these multilinear bracketings 
%$[ \  \cdot \ , \  \cdot \ ]_{\alpha}$ 
of homogeneous degree $n$ for $w \in \symm{n}$:
$$
[[\cdots [ x_{w(1)}, x_{w(2)}]_\alpha, x_{w(3)}]_\alpha, \cdots]_\alpha, x_{w(n)}]_\alpha  
$$
Then $V_n(1) \cong \lie_{n}$ is the multilinear component of the free Lie algebra, while $V_n(-1)$ is the multilinear part of the {\it space of simple Jordan elements}. The following was proved by Robbins in \cite[\S 6, Thm. 7]{robbins1971jordan} and later in \cite[Thm 2.1]{calderbank1994representations} by Calderbank--Hanlon--Sundaram: 
    \begin{equation}\label{eq:freejordan} V_n(-1) \cong \bigoplus_{\substack{\lambda \vdash n \\ \odd(\lambda) = \ell(\lambda)}} \lie_{\lambda}.\end{equation}
   
We combine Theorem \ref{thm:peakcohomology_typeacohomology} and \eqref{eq:freejordan}, to give a cohomological interpretation for $V_n(-1)$.
\begin{cor}
   The space $V_n(-1)$ is isomorphic as an $\symm{n}$-representation to $\bigoplus_{k}\PeakCohomology{2k,2k}^{(n)}$. %\cong \bigoplus_{k} \PeakCohomology{\} \]
\end{cor}
\acknowledgements{The authors are grateful to Sheila Sundaram for bringing our attention to \cite{calderbank1994representations}.}

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