the quantum coordinate ring $\oqspnc$ (or some Hopf algebra containing this)?
\ep
\bp Can $\oqspnc$ be expressed as a quotient of a type-$\msfBC$ quantum matrix
bialgebra such as $\atwonq$, which itself is
a quotient of the noncommutative ring
\begin{equation*}
\zqq \langle x_{i,j} \,|\, i,j \in [-n,-1] \cup [1,n] \rangle ?
\end{equation*}
\ep
\bp
When we specialize $\atwonq$ at $\qp12 = 1$, do we get
a quotient of the commutative polynomial ring
%Here $x$ is the $2n \times 2n$ matrix
%x =
%  x_{\ol{n}, \ol{n}} & x_{\ol{n}, \ol{n-1}} & \cdots & x_{\ol{n}, \ol1} &
%  x_{\ol{n}, 1} & \cdots & x_{\ol{n}, n-1} & x_{\ol{n}, n} \\
%  x_{\ol{n-1}, \ol{n}} & x_{\ol{n-1}, \ol{n-1}} & \cdots & x_{\ol{n-1}, \ol1} &
%  x_{\ol{n-1}, 1} & \cdots & x_{\ol{n-1}, n-1} & x_{\ol{n-1}, n} \\
%  \vdots          & \vdots            &        & \vdots        &
%  \vdots      &        & \vdots        & \vdots \\
%  x_{\ol{1}, \ol{n}} & x_{\ol{1}, \ol{n-1}} & \cdots & x_{\ol{1}, \ol1} &
%  x_{\ol{1}, 1} & \cdots & x_{\ol{1}, n-1} & x_{\ol{1}, n} \\
%  x_{1, \ol{n}} & x_{1, \ol{n-1}} & \cdots & x_{1, \ol1} &
%  x_{1, 1} & \cdots & x_{1, n-1} & x_{1, n} \\
%  \vdots          & \vdots            &        & \vdots        &
%  \vdots      &        & \vdots        & \vdots \\
%  x_{{n-1}, \ol{n}} & x_{{n-1}, \ol{n-1}} & \cdots & x_{{n-1}, \ol1} &
%  x_{{n-1}, 1} & \cdots & x_{{n-1}, n-1} & x_{{n-1}, n} \\
%  x_{n, \ol{n}} & x_{n, \ol{n-1}} & \cdots & x_{n, \ol1} &
%  x_{n, 1} & \cdots & x_{n, n-1} & x_{n, n}
If so, is there a basis of monomials in which variables appear in lexicographic
order?  Is there an easy algorithm to express any other monomial in this basis?
\ep

See Hyashi papers \cite{HayashiCGroups}, \cite{HayashiDet}, and papers
by Jimbo and Fadeev--Reshitikin--Takhtajan cited there.

\ssec{Quantum coordinate ring of type $BC$}

\bp
Does it make sense to define the type-$\msfBC$ immanant space of $\atwonq$
to be a similar span, over $\zqq$?
%\spn \{ x^{e,w} = x_{\ol n,w_{\ol n}} \cdots x_{\ol 1,w_{\ol 1}} x_{1,w_1} \cdots x_{n,w_n} \,|\, w \in \bn  \}?
%That is, $w \in \bn$ is equivalent to $|w_1| \cdots |w_n| \in \sn$.
%Also, we can
%Here we are thinking of $x$ as the $2n \times 2n$ matrix
%and for any function $f$ on $\bn$ we would define
%$\bnimm{f}(x) = \sum_{w \in \bn} f(w) x^{e,w}$.
for any function $f: \hbnq \rightarrow \zqq$ we would define
$\bnimm{f}(x) = \sum_{w \in \bn} f(T_w) \qiew x^{e,w}$.
%where $\qew = \qm{\ell(w)}2$.
\ep

\bp Describe the left- and right-actions of $\hbnq$ on the quantum
immanant space.
\ep

\bp Is there a $q$-extension of the linear functional - immanant evaluation
%(and a $q$-extension of this)
which allows one to compute
%$\theta(D)$ for $g \in \mathbb C[\bn]$
$\theta_q(D)$ for $g \in \hbnq$ as
%$\sigma_B(\imm{\theta}(x))$ (or
where $B$ is some kind of weighted path matrix for a network corresponding
to $g$?
\ep

%\section{Type-$C$ symmetric functions}
%Define the ring of type-$\msfBC$ symmetric functions to be the graded ring
%  \Lambda^C = \oplus_{n \geq 0} \Lambda_n^C,
%where $\Lambda_n^C$ is defined in terms of type-$\msfA$ symmetric functions by
%  \Lambda_n^C = \oplus_{k=0}^n \Lambda_k(x) \otimes \Lambda_{n-k}(y).
%Assume the coefficient ring to be $\mathbb Z$.
%Thus a bounded-degree power series $g$ in variables
%$\{ x_i \,|\, i > 0 \} \cup \{ y_i \,|\, i > 0 \}$ is type-$\msfBC$ symmetric if
%swapping any pair of variables of the form $(x_i,x_j)$ or $(y_i,y_j)$ leaves
%$g$ unchanged.

%Bases of $\Lambda_n^C$ may be defined in terms of bases of
%$\{\Lambda_k \,|\, 0 \leq k \leq n \}$ with or without plethysm.

%  Let $\{ a_\lambda \,|\, \lambda \vdash k \}$
%  and $\{ b_\lambda \,|\, \lambda \vdash k \}$ be bases of $\Lambda_k$.
%  Then $\{ a_\lambda(x)b_\mu(y) \,|\, (\lambda,\mu) \vdash n \}$
%  is a basis of $\Lambda_n^C$.
%  This follows from the definition of type-$\msfBC$ symmetric functions. (right?)

%$\star$ Define plethysm, in particular $g(x+y)$ and $g(x-y)$.

%Let $\{ a_\lambda \,|\, \lambda \vdash k \}$
%  and $\{ b_\lambda \,|\, \lambda \vdash k \}$ be bases of $\Lambda_k$.
%Show that $\{ a_\lambda(x+y)b_\mu(x-y) \,|\, (\lambda,\mu) \vdash n \}$
%  is a basis of $\Lambda_n^C$.

%Corresponding to the $10$ bipartitions $(\lambda,\mu)$ of $3$, we have
%the elementary basis $\{(ee)_{\lambda,\mu} \,|\, (\lambda,\mu) \vdash 3 \}$.
%  \begin{aligned}
%    m_3(x)m_\emptyset(y) &= x_1^3 + x_2^3 + x_3^3 + \cdots, \\
%    m_{21}(x)m_\emptyset(y) &= x_1^2x_2 + x_1x_2^2 + x_1^2x_3 + x_1x_3^2 + \cdo%ts, \\
%    m_{111}(x)m_\emptyset(y) &= x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + \cdots, \\
%    m_2(x)m_1(y) &= x_1^2y_1 + x_1^2y_2 + x_1^2y_3 + \cdots \\
%                & \quad + x_2^2y_1 + x_2^2y_2 + x_2^2y_3 + \cdots \\
%                & \quad + \cdots,\\
%    m_{11}(x)m_1(y) &= x_1x_2y_1 + x_1x_2y_2 + x_1x_2y_3 + \cdots \\
%                   & \quad + x_1x_3y_1 + x_1x_3y_2 + x_1x_3y_3 + \cdots \\
%                   & \quad + \cdots,\\
%    m_1(x)m_2(y) &= x_1y_1^2 + x_1y_2^2 + x_1y_3^2 + \cdots \\
%                &= x_2y_1^2 + x_2y_2^2 + x_2y_3^2 + \cdots \\
%                & \quad + \cdots,\\
%    m_1(x)m_{11}(y) &= x_1y_1y_2 + x_1y_1y_3 + x_1y_2y_3 + \cdots \\
%                   &= x_2y_1y_2 + x_2y_1y_3 + x_2y_2y_3 + \cdots \\
%                   & \quad + \cdots,\\
%    m_\emptyset(x)m_3(y) &= y_1^3 + y_2^3 + y_3^3 + \cdots, \\
%    m_\emptyset(x)m_{21}(y) &= y_1^2y_2 + y_1y_2^2 + y_1^2y_3 + y_1y_3^2 + \cdo%ts, \\
%    m_\emptyset(x)m_{111}(y) &= y_1y_2y_3 + y_1y_2y_4 + y_1y_3y_4 + \cdots.
%  \end{aligned}
%Other monomial-forgotten bases are
%$\{ m_\lambda(x)f_\mu(y) \,|\, (\lambda,\mu) \vdash n \}$,
%$\{ (fm)_{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$,
%$\{ (f\ntnsp f)_{\lambda,\mu} \,|\, (\lambda,\mu) \vdash n \}$.

\section{Hessenberg varieties}\label{s:hessenberg}

\ssec{Type $A$}

Let $P$ be a natural unit interval order, and let
$\mathbf m = \mathbf m(P) = (m_1,\dotsc,m_n)$
be the sequence of cardinalities of complements of dual order ideals of $i$,
for $i = 1,\dotsc,n$,
\begin{equation*}
m_i = \# \{ j \,|\, j \not >_P i \}.
\end{equation*}
This is the number of ones in rows $1,\dotsc,n$
of the antiadjacency matrix of $P$.
If $G = (V,E)$ is the incomparability graph of $P$, then we can equivalently
define $\mathbf m = \mathbf m(D)$ by
\begin{equation*}
m_i = \# \{ j \in V \,|\, j < i \text{ or } (i,j) \in E \}.
\end{equation*}
If $P$ corresponds to the $312$-avoiding permutation $w$,
we can equivalently define $\mathbf m = \mathbf m(w)$ by
\begin{equation*}
m_i = \max \{ w_1, \dotsc, w_i \}.
\end{equation*}
The sequence $\mathbf m$ may be thought of as a function from $[n]$ to $[n]$,
and is called a {\em Hessenberg function}.
It is characterized by the properties
\begin{equation*}
i \leq m_i \leq m_{i+1}, \quad \text{ for } i =1,\dotsc,n-1, \qquad m_n = n.
\end{equation*}

Let $\fl n$ be the flag variety,
\begin{equation*}
\fl n = \{ F = (V_1,\dotsc,V_n = \mathbb C^n) \,|\, \dim V_i = i, V_i  \subset V_{i+1}, \, i = 1,\dotsc, n-1 \}.
\end{equation*}
Let $S$ be any complex matrix having $n$ distinct eigenvalues,
and let it act on $\mathbb C^n$, and therefore $\fl n$, by left multiplication.
Define the {\em regular semisimple Hessenberg variety of type $A$
corresponding to $\mathbf m$} to be
\begin{equation*}
\hess(m) = \{ F \in \fl n \,|\, SV_i \subset V_{m_i}, i = 1,\dotsc,n-1 \}.
\end{equation*}
We may assume $S$ to be diagonal, since conjugating it by
$g \in GL_n(\mathbb C)$
does not change the (geometry of the) variety.

Tymoczko [66], [67] has used Goresky--Kottwitz--MacPherson
theory to describe the cohomology $H^*(\hess m)$ in terms of
a certain {\em moment graph} $M(P)$ whose vertices are elements of $\sn$
and whose edges $(v,w)$ are those pairs of permutations related by
$(i,j)v = w$ for some $i,j$ which are incomparable in $P$.
($\star$ Check to see that Tymoczko's  left multiplication and mine are the
same.)  The action of $\sn$ on itself by right (?) multiplication
is an action of $\sn$ on (the vertices of ?) $M(P)$ and can be used to
describe an action of $\sn$ on $H^*(\hess m)$.  The character of this action
corresponds by the Frobenius characteristic map to the symmetric function
$\omega X_P$,
\begin{equation}\label{eq:chromFchar}
\omega X_P = \frobch H^*(\hess m).
\end{equation}
($\star$ Is this the correct statement when $q=1$?)
More precisely, we have for general $q$ that
\begin{equation}\label{eq:chromFcharq}
\omega X_{P,q} = \sum_{j=0}^{\ell(w)} \frobch H^{2j}(\hess m)q^j.
\end{equation}

\ssec{Type $C$}

$\star$ Define the type-$\msfBC$ Hessenberg variety.

$\star$ Is there a $\bn$ action on a moment graph
that gives a $\bn$ action on the cohomology ring of the variety?

\section{Wreath products $C_k \wr \sn$}\label{s:wreath}

One might expect many facts about the hyperoctahedral group
$W \cong C_2 \wr \sn$
to generalize to other wreath products
$C_k \wr \sn$.

\bp
Generalize the first eight subsections of Section~\ref{s:coxetergp}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:planarnet}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:hecke}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:symmfn}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:coordring}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:repns}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:chrom}
to $C_k \wr \sn$.
\ep

\bp
Generalize Section~\ref{s:hessenberg}
to $C_k \wr \sn$.
\ep

\section{Hecke algebra}

$\star$ Start here.

\ssec{Modules for induced sign-trivial characters of $\hbnq$}

There are $(q,Q)$-analogs of $\trsp(\bn)$-bases which span
%the trace space of

\bp What are natural modules for the characters $\epsilon_{q,Q}^{\lambda,\mu}$,
$(\epsilon\eta)_{q,Q}^{\lambda,\mu}$, $(\eta\epsilon)_{q,Q}^{\lambda,\mu}$,
For which (if any) of these characters $\theta_{q,Q}^{\lambda,\mu}$
can $\theta_{q,Q}^{\lambda,\mu}(T_w)$ be computed
by an action of $\hbnQ$ on coset sums of Young subalgebras?
Conjecturally, if $\lambda \multiu \mu = \nu$, then each of these
representations should have dimension equal to the number of row-strict
Young tableaux of shape $\nu$.
\ep
($\star$ In particular, for $\lambda \vdash n$, $\mu \vdash n$, do we obtain
$\epsilon_{q,Q}^{\lambda,\emptyset}$ by letting $\hbnQ$
act by left multiplication on the space
\begin{equation}
\begin{aligned}
\spn
\bigg \{
T_u \ntksp\sum_{w \in \mfb \lambda}\ntksp
\delta_{q,Q}(T_w)(\delta\epsilon)_{q,Q}(T_w)T_w
&= T_u \ntksp\sum_{w \in \mfb \lambda}
\frac{1_{q,Q}(T_w)}{\epsilon_{q,Q}(T_w)} T_w \\
&= T_u \ntksp\sum_{w \in \mfb \lambda}
(-q)^{\ell_s(w)} T_w
\,\biggr\rvert\, u \in \mfb \lambda^-
\bigg \},
\end{aligned}
\end{equation}
or something similar?
%For $\mu \vdash n$
Do we obtain
$\epsilon_{q,Q}^{\emptyset,\mu}$ by letting $\hbnQ$
act by left multiplication on the space
\begin{equation}
\begin{aligned}
\spn
\bigg \{
T_u \ntksp\sum_{w \in \mfb \lambda}\ntksp
\delta_{q,Q}(T_w)\epsilon_{q,Q}(T_w)T_w
&= T_u \ntksp\sum_{w \in \mfb \lambda}
\frac{1_{q,Q}(T_w)}{(\delta\epsilon)_{q,Q}(T_w)} T_w \\
&= T_u \ntksp\sum_{w \in \mfb \mu} (-q)^{\ell_s(w)}(-Q)^{\ell_t(w)} T_w
\,\biggr\rvert\, u \in \mfb \mu^- \bigg \},
\end{aligned}
\end{equation}
or something similar?
%For $\lambda \vdash n$, d
Do we obtain
$\eta_{q,Q}^{\lambda,\emptyset}$ by letting $\hbnQ$
act by left multiplication on the space
\begin{equation}
\begin{aligned}
\spn
\bigg \{
T_u \ntksp\sum_{w \in \mfb \lambda}\ntksp
&= T_u \ntksp\sum_{w \in \mfb \lambda}
\frac{1_{q,Q}(T_w)}{1_{q,Q}(T_w)} T_w \\
&=T_u \ntksp\sum_{w \in \mfb \lambda} \ntksp T_w
\,\biggr\rvert\, u \in \mfb \lambda^- \bigg \},
\end{aligned}
\end{equation}
or something similar?
%For $\mu \vdash n$ d
Do we obtain
$\eta^{\emptyset,\mu}$ by letting $\hbnQ$ act by left multiplication on the space
\begin{equation}
\begin{aligned}
\spn
\bigg \{
T_u \ntksp\sum_{w \in \mfb \lambda}\ntksp
\epsilon_{q,Q}(T_w)(\delta\epsilon)_{q,Q}(T_w)T_w
&= T_u \ntksp\sum_{w \in \mfb \lambda}
\frac{1_{q,Q}(T_w)}{\delta_{q,Q}(T_w)} T_w \\
&= T_u \ntksp \sum_{w \in \mfb \mu} \ntksp (-Q)^{\ell_t(w)} T_w
\,\biggr\rvert\, u \in \mfb \mu^- \bigg \},
\end{aligned}
\end{equation}
or something similar?)

For $(\lambda,\mu)\vdash n$,
do we obtain $\epsilon_{q,Q}^{\lambda,\mu}$
by letting $\hbnQ$ act by left multiplication on the space
\begin{equation}
\spn
\bigg \{
T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_{vw})}{\epsilon_{q,Q}(T_v)(\delta\epsilon)_{q,Q}(T_w)} T_{vw}
= T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-q)^{\ell_s(v)} (-q)^{\ell_s(w)}(-Q)^{\ell_t(w)} T_{vw}
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
\begin{equation*}
T_u \Big( \ntksp \sum_{v \in \mfb \lambda}
\frac{1_{q,Q}(T_v)}{\epsilon_{q,Q}(T_v)} T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_w)}{(\delta\epsilon)_{q,Q}(T_w)}T_w \Big)
= T_u \Big( \ntksp \sum_{v \in \mfb \lambda} (-q)^{\ell_s(v)} T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell(w)} q^{\ell_s(w)} Q^{\ell_t(w)} T_w \Big)
%    (-q)^{\ell_s(w)} (-Q)^{\ell_t(w)} T_w \Big).
\end{equation*}
or something similar?
Do we obtain $(\epsilon\eta)_{q,Q}^{\lambda,\mu}$
by letting $\hbnQ$ act by left multiplication on the space
\begin{equation}
\spn
\bigg \{
T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_{vw})}{\epsilon_{q,Q}(T_v)\delta_{q,Q}(T_w)} T_{vw}
= T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-q)^{\ell_s(v)} (-Q)^{\ell_t(w)} T_{vw}
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
%or something similar?  Each basis element above can also be written as
\begin{equation*}
T_u \Big( \ntksp \sum_{v \in \mfb \lambda}
\frac{1_{q,Q}(T_v)}{\epsilon_{q,Q}(T_v)} T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_w)}{\delta_{q,Q}(T_w)}T_w \Big)
= T_u \Big( \ntksp \sum_{v \in \mfb \lambda} (-q)^{\ell_s(v)} T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-Q)^{\ell_t(w)} T_w \Big)
\end{equation*}
or something similar?
Do we obtain $(\eta\epsilon)_{q,Q}^{\lambda,\mu}$
by letting $\hbnQ$ act by left multiplication on the space
\begin{equation}
\spn
\bigg \{
T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_{vw})}{1_{q,Q}(T_v)(\delta\epsilon)_{q,Q}(T_w)} T_{vw}
= T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-q)^{\ell_s(w)}(-Q)^{\ell_t(w)} T_{vw}
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
%or something similar? Each basis element above can also be written as
\begin{equation*}
T_u \Big( \ntksp \sum_{v \in \mfb \lambda}
T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_w)}{(\delta\epsilon)_{q,Q}(T_w)}T_w \Big)
= T_u \Big( \ntksp \sum_{v \in \mfb \lambda}
T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell(w)} q^{\ell_s(w)} Q^{\ell_t(w)} T_w \Big),
%    (-q)^{\ell_s(w)} (-Q)^{\ell_t(w)} T_w \Big).
\end{equation*}
or something similar?
Do we obtain $\eta_{q,Q}^{\lambda,\mu}$
by letting $\hbnQ$ act by left multiplication
on the space
\begin{equation}
\spn
\bigg \{
T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_{vw})}{1_{q,Q}(T_v)\delta_{q,Q}(T_w)}
= T_u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-Q)^{\ell_t(w)} T_{vw}
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
%or something similar? Each basis element above can also be written as
\begin{equation*}
T_u \Big( \ntksp \sum_{v \in \mfb \lambda}
T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
\frac{1_{q,Q}(T_w)}{\delta_{q,Q}(T_w)}T_w \Big)
= T_u \Big( \ntksp \sum_{v \in \mfb \lambda}
T_v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-Q)^{\ell_t(w)} T_w \Big)
\end{equation*}
or something similar?

\section{Type $B_n$ gen functions}

\ssec{Modules for induced one-dimensional characters of $\bn$}

Give $\mathbb R[G]$ a $G$-module structure by letting $G$ act by
left multiplication.
It can be helpful to realize $G$-characters as the characters of
sub-$G$-modules of $\mathbb R[G]$.

\begin{prop}\label{p:inductsum}
Let $H$ be a subgroup of a finite group $G$ and let
$\rho$ be a one-dimensional real representation of $H$
with character $\theta$ ($= \rho$).
Let $F = (f_1,\dotsc,f_r)$
be a transversal of representatives of cosets of $H$ in $G$.
Let $G$ act by left multiplication on the submodule
\begin{equation}\label{eq:vdefn}
V := \spn_{\mathbb R} \bigg\{ f_i \sum_{h \in H} \theta(h)h \,\biggr\rvert \,
1 \leq i \leq r \bigg\}
\end{equation}
of $\mathbb R[G]$.  Then $V$ is a $G$-module with
character $\theta \upparrow_H^G$.
\end{prop}
\begin{proof}
To see that $V$ is a $G$-module,
%  To compute the character of the action of the $G$-module $V$,
consider the action of $g \in G$ on the $j$th element of
the defining basis of $V$.
Let $f_iH$ be the unique coset satisfying
$gf_jH = f_iH$, i.e., $f_i^{-1}gf_j \in H$. Then we have
\begin{equation}
\begin{aligned}
g f_j \sum_{h\in H} \theta(h)h
%  &= f_i f_i^{-1} g f_j \sum_{h \in H} \theta(h)h
&= f_i \sum_{h\in H} \theta(h) f_i^{-1}gf_j h
= f_i \sum_{h' \in H} \theta( f_j^{-1}g^{-1}f_ih') h'\\
&= \theta(f_j^{-1}g^{-1}f_i) f_i \sum_{h'\in H} \theta(h') h' \in V.
\end{aligned}
\end{equation}
%which is a multiple of
%The character of this representation is the trace of the matrix
It follows that in the $j$th column of the matrix representing $g$,
all components are $0$ except for the $i$th,
which is $\theta(f_j^{-1}g^{-1}f_i)$.  Thus the trace of this matrix is
\begin{equation}\label{eq:sumthetaginv}
%  \begin{aligned}
\sumsb{1\leq i\leq r\\f_i^{-1}gf_i\in H} \theta(f_i^{-1}g^{-1}f_i)
= \sumsb{1\leq i\leq r\\f_i^{-1}gf_i\in H} \theta(g^{-1}).
%    \end{aligned}
\end{equation}

On the other hand, for $g \in G$,
the matrix of $\rho \upparrow_H^G(D)$ is the $r \times r$
matrix $A = (a_{i,j})$,
%  \begin{equation*}
%%  \begin{bmatrix}
%     A_{1,1} & \cdots & A_{1,\ell} \\
%     \vdots &        & \vdots   \\
%     A_{\ell,1} & \cdots & A_{\ell,\ell}
%  \end{bmatrix}
%  \end{equation*}
\begin{equation*}
a_{i,j} = \begin{cases}
\theta(f_i^{-1} g f_j) &\text{if $f_i^{-1}gf_j \in H$},\\
0 &\text{otherwise},
\end{cases}
\end{equation*}
The trace of this matrix is
\begin{equation}\label{eq:sumthetag}
%  \begin{aligned}
\theta \upparrow_H^G(D)
= \sumsb{1\leq i\leq r\\f_i^{-1}gf_i\in H} \theta(f_i^{-1}gf_i)
= \sumsb{1\leq i\leq r\\f_i^{-1}gf_i\in H} \theta(D),
%  \end{aligned}
\end{equation}
since $\theta = \rho$ is a homomorphism.

%  The coordinate (column) vector of the product
%$g f_j \sum_{k\in K} \theta(k)k$
%  has a $1$ in position $i$ if $g f_j K = f_i K$,
%  i.e., if we have $f_i^{-1}gf_j \in K$.
%  $(i,i)$ entry of this matrix is
%   produces a matrix with respect
%   to the given
%   acts by left multiplication on the $i$th
%   basis element of $V$, we obtain
%($\star$ Do we need to assume that the character is $1$-dimensional and real
%so that $\theta(g^{-1}) = \theta(D)$? Or did I make a mistake?)
Since $H$ is finite, $\theta(D)$ must be a root of unity.
Since $\theta(D) \in \mathbb R$, we must have $\theta(D) \in \{ \pm 1 \}$.
Thus $\theta(g^{-1}) = \theta(D)^{-1} = \theta(D)$,
and the expressions (\ref{eq:sumthetaginv}), (\ref{eq:sumthetag}) are equal.
\end{proof}

\begin{cor}\label{c:inductsum}
Let $H_1$, $H_2$, $H_1H_2 \cong H_1 \times H_2$ be subgroups
of a finite group $G$, and
%let $\rho_1$, $\rho_2$ be real representations of $H_1$, $H_2$, with
let $\theta_1$, $\theta_2$, be the characters of
one-dimensional real representations of $H_1$, $H_2$, respectively.
%and let $\theta_1$, $\theta_2$ be an $H_1$-character and an $H_2$-character.
Let $\theta_1 \otimes \theta_2$ be the (one-dimensional) $H_1H_2$-character
\begin{equation*}
(\theta_1 \otimes \theta_2)(h_1h_2) = \theta_1(h_1)\theta_2(h_2)
\end{equation*}
for $h_1 \in H_1$, $h_2 \in H_2$.  (See \cite[Thm.\,1.11.2]{Sag}.)
Let $F$ be a transversal of cosets of $H_1H_2$ in $G$, and let
$G$ act by left multiplication by on the submodule
\begin{equation}\label{eq:1diminduct}
V = \spn_{\mathbb R}
\bigg\{
g \bigg( \sum_{h_1 \in H_1} \theta_1(h_1)h_1 \bigg)
\bigg(\sum_{h_2 \in H_2} \theta_2(h_2)h_2 \bigg)
\,\biggr\rvert \, g \in F
\bigg\}
\end{equation}
of $\mathbb R[G]$.
Then $V$ is a $G$-module with character
$(\theta_1 \otimes \theta_2) \upparrow_{H_1H_2}^G$.
\end{cor}
\begin{proof}
Since $H_1H_2 \cong H_1 \times H_2$, we may express $V$ as
\begin{equation*}
\spn_{\mathbb R}
\bigg\{ g \bigg( \sum_{h \in H_1H_2} (\theta_1 \otimes \theta_2)(h) h \bigg)
\,\biggr\rvert \, g \in F
\bigg\}.
\end{equation*}
This is a $G$-module with the required character by Prop.~\ref{p:inductsum}.
\end{proof}

%($\star$ Try to generalize Jongwon's lemmas to arbitrary subgroups of
%a finite group $G$.)

For $(\lambda,\mu) \vdash n$, $\lambda \vdash a$,
%let $\mfb {\lambda,\mu}^-$
%be a transversal of
define the submodules
\begin{equation}\label{eq:indsubmodules}
\begin{aligned}
E^{\lambda,\mu} &= \spn_{\mathbb Z}
\bigg \{
u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTtksp
\epsilon(v)(\delta\epsilon)(w) vw
= u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTksp
(-1)^{\ell_s(v)} (-1)^{\ell(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},\\
(EH)^{\lambda,\mu} &= \spn_{\mathbb Z}
\bigg \{
u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTtksp
\epsilon(v)\delta(w) vw
= u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTksp
(-1)^{\ell_s(v)} (-1)^{\ell_t(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},\\
(HE)^{\lambda,\mu} &= \spn_{\mathbb Z}
\bigg \{
u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTksp
(\delta\epsilon)(w) vw
= u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTksp
(-1)^{\ell(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},\\
H^{\lambda,\mu} &= \spn_{\mathbb Z}
\bigg \{
u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTtksp
\delta(w)
= u \nTksp \sumsb{v\in \mfb \lambda\\w \in \mfb {0^a\mu}} \nTksp
(-1)^{\ell_t(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \}
\end{aligned}
\end{equation}
of $\mathbb Z[\bn]$.

%($\star$ Think of four different notations for versions of the sum
%$T_{\mfb {\lambda,\mu}}$
%that have coefficients corresponding to the
%four $1$-dimensional characters.  Maybe
%$T_{\mfb {\lambda,\mu}}^{\epsilon,\delta\epsilon}$,
%$T_{\mfb {\lambda,\mu}}^{\epsilon,\delta}$,
%$T_{\mfb {\lambda,\mu}}^{1,\delta\epsilon}$,
%$T_{\mfb {\lambda,\mu}}^{1,\delta}$?)

For a one-dimensional character $\theta$ of $H$,
let $T_H^\theta$ denote the sum appearing in Proposition~\ref{p:inductsum},
\begin{equation}\label{eq:Tdef}
T_H^\theta = \sum_{h\in H}\theta(h)h.
\end{equation}
Then we may express the submodules (\ref{eq:indsubmodules}) as
\begin{equation}\label{eq:indsubmodules2}
\begin{aligned}
E^{\lambda,\mu} &= \spn_\mathbb{Z}
\{ u T_{\mfb {\lambda,\mu}}^{\epsilon \otimes \delta\epsilon} \,|\, u \in \mfb {\lambda,\mu}^- \},\\
(EH)^{\lambda,\mu} &= \spn_\mathbb{Z}
\{ u T_{\mfb {\lambda,\mu}}^{\epsilon \otimes \delta} \,|\, u \in \mfb {\lambda,\mu}^- \},\\
(HE)^{\lambda,\mu} &= \spn_\mathbb{Z}
\{ u T_{\mfb {\lambda,\mu}}^{1 \otimes \delta\epsilon} \,|\, u \in \mfb {\lambda,\mu}^- \},\\
H^{\lambda,\mu} &= \spn_\mathbb{Z}
\{ u T_{\mfb {\lambda,\mu}}^{1 \otimes \delta} \,|\, u \in \mfb {\lambda,\mu}^- \}.
\end{aligned}
\end{equation}

\begin{thm}\label{t:1diminduct}
Left multiplication by $\bn$ makes the submodules
$E^{\lambda,\mu}$, $(EH)^{\lambda,\mu}$,
$(HE)^{\lambda,\mu}$, $H^{\lambda,\mu}$ of $\mathbb Z[\bn]$
into $\bn$-modules with characters
\begin{equation*}
\begin{gathered}
\epsilon^{\lambda,\mu} \defeq
(\epsilon \otimes \delta\epsilon) \upparrow_{\mfb {\lambda,\mu}}^{\bn},
\qquad
(\epsilon\eta)^{\lambda,\mu} \defeq
(\epsilon \otimes \delta) \upparrow_{\mfb {\lambda,\mu}}^{\bn},\\
%  \qquad
(\eta\epsilon)^{\lambda,\mu} \defeq
(1 \otimes \delta\epsilon) \upparrow_{\mfb {\lambda,\mu}}^{\bn},
\qquad
\eta^{\lambda,\mu} \defeq
(1 \otimes \delta) \upparrow_{\mfb {\lambda,\mu}}^{\bn},
\end{gathered}
\end{equation*}
\end{thm}
\begin{proof}
These modules have the form required by Corollary~\ref{c:inductsum}:
$\mfb {\lambda,\mu}^-$ is a transversal for cosets of
$\mfb {\lambda,\mu} = \mfb \lambda \mfb {0^a\mu} \cong \mfb \lambda \times \mfb {0^a\mu}$ in $\bn$,
%these modules have the form (\ref{eq:1diminduct}), with
$(\theta_1,\theta_2) \in \{(\epsilon, \delta\epsilon), (\epsilon, \delta),
(1, \delta\epsilon), (1, \delta) \}$.
\end{proof}

\ssec{Generating functions for induced one-dimensional characters of $\bn$}

To define generating functions in $\zx$
(defined as in Section~\ref{s:coordring})
for induced one-dimensional characters of $\bn$,
we look closely at the $\bn$-module $V$ in Proposition~\ref{p:inductsum}
and the matrix representing each element $g \in G$.

\begin{lem}
Let groups $G$, $H$, transversal $F = (f_1,\dotsc,f_r)$,
$H$-character $\theta$, and $G$-module $V$ be as in
and let $A = A(D) = (a_{i,j})$
%  Let $\rho(D)$
be the matrix of $g \in G$
with respect to the defining basis (\ref{eq:vdefn}) of $V$.
Let $\chi$ be the character of $V$.
%the $G$-module $V$ in
%, the $(i,j)$-entry of the matrix $\rho(D)$,
Then $a_{i,j}$ equals the coefficient of $g$ in
%  \begin{equation}\label{eq:sumconj}
$f_i T_H^\theta f_j^{-1}$.
%  \end{equation}
In particular we have the identity
\begin{equation}\label{eq:gpalggenfn}
\sum_{i=1}^r f_i T_H^\theta f_i^{-1} = \sum_{g \in G} \chi(D) g.
\end{equation}
in $\mathbb R[G]$.
\end{lem}
\begin{proof}
By the proof of Proposition~\ref{p:inductsum},
we have $a_{i,j} = \theta(f_j^{-1}g^{-1}f_i)$
if some $h \in H$ satisfies
$g = f_ihf_j^{-1}$,
and is $0$ otherwise.  On the other hand, we have
\begin{equation}\label{eq:findg}
f_i T_H^\theta f_j^{-1} = \sum_{h\in H} \theta(h) f_ihf_j^{-1}.
\end{equation}
If there is no $h \in H$ satisfying $g = f_ihf_j^{-1}$, then the coefficient
of $g$ in (\ref{eq:findg}) is $0$.  Otherwise, the coefficient of $g$ is
\begin{equation*}
\theta(h) = \theta(f_i^{-1}gf_j) = \theta(f_j^{-1}g^{-1}f_i)^{-1},
\end{equation*}
which equals $\theta(f_j^{-1}g^{-1}f_i)$ by our assumptions on $G$ and $\theta$.
It follows that $a_{i,i}$
%the matrix representing $g$
is equal to the coefficient of $g$ in $f_i T_H^\theta f_i^{-1}$,
and $\chi(D) = \sum_i a_{i,i}$
%, the trace of $\rho(D)$,
is equal to the coefficient of $g$ in $\sum_i f_i T_H^\theta f_i^{-1}$.
\end{proof}

Now we have a ``generating function'' in $\mathbb Z[\bn]$
for each induced one-dimensional character $\chi$ of $\bn$.
\begin{cor}\label{c:indchargenfn}
For each one-dimensional character $\theta$ of $\mfb {\lambda,\mu}$
and the corresponding induced character
$\chi = \theta \upparrow_{\mfb {\lambda,\mu}}^{\bn}$ of $\bn$,
%  $\theta$ of $\mfb {\lambda,\mu}$
%  $\theta \in \{ \epsilon^{\lambda,\mu}, (\epsilon\eta)^{\lambda,\mu},
%  (\eta\epsilon)^{\lambda,\mu}, \eta^{\lambda,\mu} \}$ of $\bn$,
we have
\begin{equation*}
\sum_{u \in \mfb {\lambda,\mu}^-} \nTksp u T_{\mfb {\lambda,\mu}}^\theta u^{-1} =
\sum_{w \in \bn} \chi(w) w.
\end{equation*}
\end{cor}
More explicitly, for
$\theta \in \{ \epsilon \otimes \delta\epsilon, \epsilon \otimes \delta,
1 \otimes \delta\epsilon, 1 \otimes \delta \}$,
we have
\begin{equation*}
\begin{gathered}{2}
\sum_{u \in \mfb {\lambda,\mu}^-} \nTksp u T_{\mfb {\lambda,\mu}}^{\epsilon \otimes \delta\epsilon} u^{-1}
= \sum_{w \in \bn} \epsilon^{\lambda,\mu}(w) w,
\qquad
\sum_{u \in \mfb {\lambda,\mu}^-} \nTksp u T_{\mfb {\lambda,\mu}}^{\epsilon \otimes \delta} u^{-1}
= \sum_{w \in \bn} (\epsilon\eta)^{\lambda,\mu}(w) w, \\
\sum_{u \in \mfb {\lambda,\mu}^-} \nTksp u T_{\mfb {\lambda,\mu}}^{1\otimes \delta\epsilon} u^{-1}
= \sum_{w \in \bn} (\eta\epsilon)^{\lambda,\mu}(w) w,
\qquad
\sum_{u \in \mfb {\lambda,\mu}^-} \nTksp u T_{\mfb {\lambda,\mu}}^{1 \otimes \delta} u^{-1}
= \sum_{w \in \bn} \eta^{\lambda,\mu}(w) w.
\end{gathered}
\end{equation*}

\section{Multiplicative characters of type $BC$}\label{s:character}

For every weight $\gamma \in \mathfrak h^*$, we define a map
$M \mapsto M^\gamma$ from $T$ to $\mathbb C$ by
$\exp(X)^\gamma \defeq e^{\gamma(X)}$, ($X \in \mathfrak h$)
and call this a {\em character}.
Specifically, we have
\begin{equation*}
\exp(H_i)^{\epsilon_j} = \exp(H_{\alpha_i})^{\omega_j} = e^{\delta_{i,j}} =
\begin{cases}
e  &\text{if $i=j$},\\
1  &\text{otherwise}.
\end{cases}
%  \qquad
%  \exp(H_{\alpha_i})^{\omega_j} = e^{\delta_{i,j}}.
%  \begin{cases}
%    e  &\text{if $i=j$},\\
%    1  &\text{otherwise}.
%  \end{cases}
\end{equation*}
Observe that since all $X, Y \in \mathfrak h$ commute, we have
$\exp(X+Y) = \exp(X)\exp(Y)$.  Thus the character is a homomorphism:
\begin{equation*}
(\exp(X)\exp(Y))^\gamma = \exp(X+Y)^\gamma = e^{\gamma(X+Y)} = e^{\gamma(X)+\gamma(Y)}
= e^{\gamma(X)} e^{\gamma(Y)}.
\end{equation*}
It follows that the character is a one-dimensional representation of $T$
and simultaneously the trace of that representation,
justifying the terminology {\em character}.
More generally, we have the formulas
\begin{equation}\label{eq:toralchar}
\begin{aligned}
%\sum_{d_0\omega_0 + \cdots + d_{n-1}\omega_{n-1}$
\gamma = \sum_{j=1}^{n} a_j\epsilon_j, \
H = \prod_{j=1}^{n} \exp(c_j H_j)
\qquad &\Longrightarrow \qquad
%    H^\gamma = \sum_{j=1}^{n} a_jc_j, \\
H^\gamma = e^{a_1c_1 + \cdots + a_nc_n}, \\
\gamma = \sum_{j=0}^{n-1} b_j\omega_j, \
H = \prod_{j=0}^{n-1} \exp( d_j H_{\alpha_j})
\qquad &\Longrightarrow \qquad
%    H^\gamma = \sum_{j=0}^{n-1} b_jd_j.
H^\gamma = e^{b_0d_0 + \cdots + b_{n-1}d_{n-1}},
\end{aligned}
\end{equation}
and for $j < k$ we have
\begin{equation*}
\begin{aligned}
( \mathrm{diag}(e^{c_n}, \dotsc, e^{c_1}, e^{c_{-1}}, \dotsc, e^{c_{-n}}) )^{\omega_j - \omega_k}
&= ( \mathrm{diag}(e^{c_n}, \dotsc, e^{c_1}, e^{c_{-1}}, \dotsc, e^{c_{-n}}) )^{\epsilon_{j+1} + \cdots + \epsilon_k} \\
&= e^{c_{j+1}} \cdots e^{c_k}.
\end{aligned}
%( \mathrm{diag}(e^{c_n}, \dotsc, e^{c_1}, e^{c_{-1}}, \dotsc, e^{c_{-n}}) )^{\omega_j}
%  &= ( \mathrm{diag}(e^{c_n}, \dotsc, e^{c_1}, e^{c_{-1}}, \dotsc, e^{c_{-n}}) )^{\epsilon_{j+1} + \cdots + \epsilon_n} \\
%  &= e^{c_{j+1}} \cdots e^{c_n}.
\end{equation*}

%and $H = \exp(b_0H_{\alpha_0} + \cdots + b_{n-1}H_{\alpha_{n-1}})$,
%then $H^\gamma = d_0b_0 + \cdots + d_{n-1}b_{n-1}$.

Let $\gamma = \gamma_1 + \gamma_2 \in \mathfrak h$.
Since $\gamma(X) = \gamma_1(X) + \gamma_2(X)$, the characters form a group
isomorphic to the weight lattice.
$\bn$ acts on the group of characters (weight lattice)
by sending the character $M \mapsto M^\gamma$ to the character
\begin{equation*}
M \mapsto M^{w(\gamma)} \defeq (w^{-1}Mw)^{\gamma}.
\end{equation*}

\bp Is it easy to describe $w(\gamma)$ in terms of $w$ and $\gamma$
without using the group $T$?  Is this a new action on the weight lattice, or
just a restatement of the action defined in
\ep

\section{Bruhat decomposition and total nonnegativity}\label{s:bruhattnn}

Let $G = \mathrm{SP}_{2n}(\mathbb C)$ and let $\bn$ be the Weyl group.
%We have three
%{\em Bruhat decompositions}
Three partitions of $G$ into disjoint subsets indexed by Weyl group elements
are called {\em Bruhat decompositions}:
\begin{equation*}
G = \biguplus_{w \in \bn} B \dot w B
= \biguplus_{w \in \bn} B_- \dot w B_-
= \biguplus_{w \in \bn} B_- \dot w B.
\end{equation*}
Any matrix belonging to the block $G_0 \defeq B_- \dot e B = B_-B$
of the third parition is said to have {\em Gaussian decomposition}.
Thus $G_0$ is the subset (not subgroup) $N_- T N$ of $G$.
%consisting of elements
%$factoring as

\bp Is there a $B \dot w B_-$ Bruhat decomposition as well?
\ep

Define the {\em double Bruhat cells} $\{ G^{u,v} \,|\, u,w \in \bn \}$ by
\begin{equation*}
G^{u,v} = B \dot u B \cap B_- \dot v B_-.
\end{equation*}

Observe that the diagonal entries of the matrix (\ref{eq:torus1})
are real and positive if and only if we have $c_1,\dotsc,c_n \in \mathbb R$.
Similarly, the diagonal entries of the matrix (\ref{eq:torus2})
are real and positive if and only if we have $d_0,\dotsc,d_{n-1} \in \mathbb R$.
Thus we have
\begin{equation}\label{eq:realspanh}
\mathrm{exp}(\spn_{\mathbb R}\{H_1,\dotsc,H_n\}) =
\mathrm{exp}(\spn_{\mathbb R}\{H_{\alpha_0},\dotsc,H_{\alpha_{n-1}}\}).
\end{equation}
Define $T_{>0}$ to be the subgroup of $T$ of
all matrices
of the form (\ref{eq:torus1}) with $c_1,\dotsc,c_n \in \mathbb R_{\geq 0}$,
or equivalently of the form (\ref{eq:torus2})
with $d_0, \dotsc, d_{n-1} \in \mathbb R_{\geq 0}$.
Equivalently, $T_{>0}$ is the set of all $M \in T$ such
that $M^\gamma \in \mathbb R_{>0}$ for all $\gamma$ in the weight lattice
$\spn_{\mathbb Z} (\omega_0,\dotsc, \omega_{n-1})$.

Define $G_{\geq 0}$, the {\em set of totally nonnegative elements} of $G$,
to be semigroup generated by $T_{>0}$ and
$\{ \exp(E_{\alpha_j}) \,|\, 0 \leq j \leq n-1 \}
\cup
\{ \exp(E_{-\alpha_j}) \,|\, 0 \leq j \leq n-1 \}$, that is, the semigroup
generated by all matrices of the forms
\begin{equation}\label{eq:tnngen}
\begin{gathered}
\begin{bmatrix}
I &   &   &       & \\
& t &   &       & \\
&   & I &       & \\
&   &   & t^{-1} & \\
&   &   &       & I
\end{bmatrix},
\quad
%where $t \in \mathbb R_{>0}$,
\begin{bmatrix}
I &            &   &             & \\
& \abdiag{t} &   &             & \\
&            & I &             & \\
&            &   & \abdiag{-t} & \\
&            &   &             & I
\end{bmatrix},
\quad
\begin{bmatrix}
I &            &   &             & \\
& \bediag{t} &   &             & \\
&            & I &             & \\
&            &   & \bediag{-t} & \\
&            &   &             & I
\end{bmatrix},\\
\begin{bmatrix}
I &            &   \\
& \abdiag{t} &   \\
&            & I
\end{bmatrix},
\quad
\begin{bmatrix}
I &            &   \\
& \bediag{t} &   \\
&            & I
\end{bmatrix},
\end{gathered}
\end{equation}
where $t \in \mathbb R_{>0}$.

(Alternatively, should the 2nd and 5th matrices be something like
\begin{equation}\label{eq:tnngen}
\begin{gathered}
\begin{bmatrix}
I &             &   &             & \\
& \abdiag{-t} &   &             & \\
&             & I &             & \\
&             &   & \abdiag{t}  & \\
&             &   &             & I
\end{bmatrix},
\quad
\begin{bmatrix}
I &             &   \\
& \bediag{-t} &   \\
&             & I
\end{bmatrix},
\end{gathered}
\end{equation}
with the $t$ and $-t$ swapped?)

Defining the {\em totally positive part} $G_{>0}^{u,v} = G_{\geq 0} \cap G^{u,v}$
of the double Bruhat cell $G^{u,v}$, we have
\begin{equation*}
G_{>0} = \biguplus_{u,v \in \bn} G_{>0}^{u,v}.
\end{equation*}

There are natural planar networks for the  generators above, each consisting
of $2n$ horizonal edges and $0$, $1$, or $2$ diagonal edges:
%To generators $t, s_{1}', \dotsc, s_{n-1}'$,
%or each generator,
%we associate the wiring diagrams
\begin{equation}\label{eq:hyperbigwiring diagrams}
%  \begin{tikzpicture}[scale=.5,baseline=-20]
\begin{tikzpicture}[scale=.6,baseline=0]
\node at (0,6.5) {$(n)$};            %\node at (1,5.5) {$n$};
%  \node at (0, 5.8) {$\cdot$};
%  \node at (0, 5.5) {$\cdot$};
%  \node at (0, 5.2) {$\cdot$};
\node at (0,4.5) {$(j+1)$};            %\node at (1,1.5) {$2$};
\node at (0,3.5) {$(j)$};            %\node at (1,1.5) {$2$};
\node at (0,2.5) {$(j-1)$};            %\node at (1,1.5) {$2$};
%  \node at (0, 1.8) {$\cdot$};
%  \node at (0, 1.5) {$\cdot$};
%  \node at (0, 1.2) {$\cdot$};
\node at (0,0.5) {$(1)$};            %\node at (1,0.5) {$1$};
\node at (0,-0.5) {$(\ol 1)$};       %\node at (1,-0.5) {$\ol 1$};
%  \node at (0, -1.2) {$\cdot$};
%  \node at (0, -1.5) {$\cdot$};
%  \node at (0, -1.8) {$\cdot$};
\node at (0,-2.5) {$(\ol{j-1})$};            %\node at (1,1.5) {$2$};
\node at (0,-3.5) {$(\ol j)$};            %\node at (1,1.5) {$2$};
\node at (0,-4.5) {$(\ol{j+1})$};            %\node at (1,1.5) {$2$};
%  \node at (0, -5.2) {$\cdot$};
%  \node at (0, -5.5) {$\cdot$};
%  \node at (0, -5.8) {$\cdot$};
\node at (0,-6.5) {$(\ol n)$};            %\node at (1,5.5) {$n$};
%  \node at (0, -6.5) {$s_{n-1}$};
\end{tikzpicture}
\qquad \qquad
\begin{tikzpicture}[scale=.6,baseline=0]
\draw[-] (-.5,6.5) -- (.5,6.5);
\node at (0, 5.8) {$\cdot$};
\node at (0, 5.5) {$\cdot$};
\node at (0, 5.2) {$\cdot$};
\node at (0, 3.8) {$\overset{t^{-1}}{}$};
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\draw[-] (-.5,2.5) -- (.5,2.5);
\node at (0, 1.8) {$\cdot$};
\node at (0, 1.5) {$\cdot$};
\node at (0, 1.2) {$\cdot$};
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\node at (0, -1.2) {$\cdot$};
\node at (0, -1.5) {$\cdot$};
\node at (0, -1.8) {$\cdot$};
\node at (0, -3.2) {$\overset{t}{}$};
\draw[-] (-.5,-2.5) -- (.5,-2.5);
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (0, -5.2) {$\cdot$};
\node at (0, -5.5) {$\cdot$};
\node at (0, -5.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-6.5) -- (.5,-6.5);
%\node at (0, -6.5) {$t$};
\end{tikzpicture}
\ ,\quad
\begin{tikzpicture}[scale=.6,baseline=0]
\draw[-] (-.5,6.5) -- (.5,6.5);
\node at (0, 5.8) {$\cdot$};
\node at (0, 5.5) {$\cdot$};
\node at (0, 5.2) {$\cdot$};
\node at (-.5, 4) {$\overset{-t}{}$};
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\draw[-] (-.5,3.5) -- (.5,4.5);
\draw[-] (-.5,2.5) -- (.5,2.5);
\node at (0, 1.8) {$\cdot$};
\node at (0, 1.5) {$\cdot$};
\node at (0, 1.2) {$\cdot$};
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\node at (0, -1.2) {$\cdot$};
\node at (0, -1.5) {$\cdot$};
\node at (0, -1.8) {$\cdot$};
\draw[-] (-.5,-2.5) -- (.5,-2.5);
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (-.3, -4) {$\overset{t}{}$};
\node at (0, -5.2) {$\cdot$};
\node at (0, -5.5) {$\cdot$};
\node at (0, -5.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-6.5) -- (.5,-6.5);
%\node at (0, -6.5) {$t$};
\end{tikzpicture}
\ ,\quad
\begin{tikzpicture}[scale=.6,baseline=0]
\draw[-] (-.5,6.5) -- (.5,6.5);
\node at (0, 5.8) {$\cdot$};
\node at (0, 5.5) {$\cdot$};
\node at (0, 5.2) {$\cdot$};
%\node at (0, 2.8) {$\vdots$};
\node at (.35, 4) {$\overset{-t}{}$};
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,4.5) -- (.5,3.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\draw[-] (-.5,2.5) -- (.5,2.5);
\node at (0, 1.8) {$\cdot$};
\node at (0, 1.5) {$\cdot$};
\node at (0, 1.2) {$\cdot$};
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\node at (0, -1.2) {$\cdot$};
\node at (0, -1.5) {$\cdot$};
\node at (0, -1.8) {$\cdot$};
\draw[-] (-.5,-2.5) -- (.5,-2.5);
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-3.5) -- (.5,-4.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (.35, -4) {$\overset{t}{}$};
\node at (0, -5.2) {$\cdot$};
\node at (0, -5.5) {$\cdot$};
\node at (0, -5.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-6.5) -- (.5,-6.5);
%\node at (0, -6.5) {$t$};
\end{tikzpicture}
\ ,\quad
\begin{tikzpicture}[scale=.6,baseline=0]
\draw[-] (-.5,6.5) -- (.5,6.5);
\node at (0, 5.8) {$\cdot$};
\node at (0, 5.5) {$\cdot$};
\node at (0, 5.2) {$\cdot$};
%\node at (0, 2.8) {$\vdots$};
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\draw[-] (-.5,2.5) -- (.5,2.5);
\node at (0, 1.8) {$\cdot$};
\node at (0, 1.5) {$\cdot$};
\node at (0, 1.2) {$\cdot$};
\node at (-.3, 0) {$\overset{t}{}$};
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (-.5,-0.5) -- (.5,0.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\node at (0, -1.2) {$\cdot$};
\node at (0, -1.5) {$\cdot$};
\node at (0, -1.8) {$\cdot$};
\draw[-] (-.5,-2.5) -- (.5,-2.5);
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (0, -5.2) {$\cdot$};
\node at (0, -5.5) {$\cdot$};
\node at (0, -5.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-6.5) -- (.5,-6.5);
%\node at (0, -6.5) {$t$};
\end{tikzpicture}
\ ,\quad
\begin{tikzpicture}[scale=.6,baseline=0]
\draw[-] (-.5,6.5) -- (.5,6.5);
\node at (0, 5.8) {$\cdot$};
\node at (0, 5.5) {$\cdot$};
\node at (0, 5.2) {$\cdot$};
%\node at (0, 2.8) {$\vdots$};
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\draw[-] (-.5,2.5) -- (.5,2.5);
\node at (0, 1.8) {$\cdot$};
\node at (0, 1.5) {$\cdot$};
\node at (0, 1.2) {$\cdot$};
\node at (.35, 0) {$\overset{t}{}$};
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (-.5,0.5) -- (.5,-0.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\node at (0, -1.2) {$\cdot$};
\node at (0, -1.5) {$\cdot$};
\node at (0, -1.8) {$\cdot$};
\draw[-] (-.5,-2.5) -- (.5,-2.5);
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (0, -5.2) {$\cdot$};
\node at (0, -5.5) {$\cdot$};
\node at (0, -5.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-6.5) -- (.5,-6.5);
%\node at (0, -6.5) {$t$};
\end{tikzpicture}
\ .
\end{equation}

\bp Does the concatenation of these planar networks give us a planar network
for $A$ and an analog of Lindstr\"om's Lemma
(and Cryer-Brenti Theorem~\cite{BrentiCTP}, \cite{CryerProp})
for TNN elements of $\mathrm{SP}_{2n}(\mathbb R)$?
\ep
\bp What is the analog of a minor of $A \in \mathrm{SP}_{2n}(\mathbb C)$?
Is $A$ TNN if and only if all of its type-$\msfBC$ minors are nonnegative?
\ep
See Fomin-Zelevinsky paper \cite{FominDouble}.

For each element $g \in G_0$, write its Gaussian decomposition as
$g = g_- g_0 g_+$, and for each fundamental weight $\omega_i$,
define the minor $\Delta^{\omega_i}: G_0 \rightarrow \mathbb C$ by
$\Delta^{\omega_i}(D) = g_0^{\omega_i}$. Apparently there is a unique extension
of this function to a regular function on $G$ \cite[p.\,340]{FominDouble}.
($\star$ Do examples.)

\section{mincosetreps and \bn actions}

\begin{prop}
Fix $(\lambda,\mu) \vdash n$ and define the composition $\alpha$ of $n$
to be the concatenation of $\lambda$ and $\mu$.
Let $H \cong \sn$ be the subgroup of $\bn$ generated by $s'_1, \dotsc, s'_{n-1}$,
and let $H_\alpha \cong \mfs \alpha$ be the
Young subgroup of $H$ indexed by $\alpha$.
Then the type-$\msfBC$ Bruhat-minimal elements of cosets $u\mfb {\lambda,\mu} \subset W$
%of the type-$\msfBC$ double Young subgroup $\mfb {\lambda,\mu}$
are precisely the type-$\msfA$ Bruhat-minimal elements of
cosets $uH_\alpha \subset H$.
%same as the Bruhat-minimal elements of cosets
may be characterized as follows.
\begin{equation}\label{eq:minrepdefns}
\begin{aligned}
\mfb {\lambda,\mu}^-
&= \{ w \in H \,|\, ws'_i > w \text{ for all } s'_i \in J(\lambda,\mu) \} \\
&= \{ w \in H \,|\, \text{$i$ appears before $i+1$
in $w_1 \cdots w_n$ for all $s'_i \in J(\lambda,\mu)$} \} \\
&= \{ w \in H \,|\, s'_iw^{-1} > w^{-1} \text{ for all } s'_i \in J(\lambda,\mu) \} \\
&= \{ w \in H \,|\, w^{-1}_i < w^{-1}_{i+1} \text{ for all } s'_i \in J(\lambda,\mu) \},
\end{aligned}
\end{equation}
where $w_1 \cdots w_n$ is the short one-line notation of $w$ in $\bn$.
\end{prop}

\ssec{Action of $\bn$ on the root system}
\begin{thm}
$\bn$ is isomorphic to the Weyl group defined in Section~\ref{s:rootsystem},
and its Coxeter graph (\ref{eq:coxeter})
is the Dynkin diagram (\ref{eq:dynkin}) with
undirected edges.
\end{thm}
In particular, $\bn$ acts on the root system $\Phi$
(and similarly on the entire weight lattice) as follows.
\begin{enumerate}
\item For $i = 1, \dotsc, n-1$, $s'_i$ is the reflection $\sigma_{\alpha_i}$
in the hyperplane perpendicular to $\alpha_i = \epsilon_{i+1} - \epsilon_{i}$.
\item $t = s'_0$ is the reflection $s_{\alpha_0}$
in the hyperplane perpendicular to $\alpha_0 = 2 \epsilon_1$.
\end{enumerate}
Equivalently, for $\gamma$ in the weight lattice, we define
\begin{equation}\label{eq:wactonweightlattice}
%  s'_i(\gamma) = \gamma - \gamma(H_i) \alpha_i,
s'_i(\gamma) = \gamma - \frac{2(\gamma,\alpha_i)}{(\alpha_i,\alpha_i)} \alpha_i
= \gamma - \gamma(H_{\alpha_i}) \alpha_i.
\end{equation}

\ssec{Action of $\bn$ on the maximal torus}
The group $\mathrm{Norm}_{\mathrm{SP}_{2n}(\mathbb C)}(T)$
acts on $T$ by conjugation:
for $g \in \mathrm{Norm}_{\mathrm{SP}_{2n}(\mathbb C)}(T)$ and $h \in T$
we have
$g \circ h \defeq g h g^{-1}$.
%for $g \in \mathrm{Norm}_{\mathrm{SP}_{2n}(\mathbb C)}(T)$.
The set of cosets acts on $T$ similarly:
$gT \circ h \defeq gT \cdot h \cdot g^{-1}T = ghg^{-1}T$.
To view this as an action of $\bn$ on $T$, we replace $w \in \bn$ with
%let $\dot w$ be
any representative in $\mathrm{Norm}_{\mathrm{SP}_{2n}(\mathbb C)}(T)$
of its corresponding coset in
$\mathrm{Norm}_{\mathrm{SP}_{2n}(\mathbb C)}(T)/T$.
%$w \in \bn$.
Thus we have
$w \circ h \defeq \dot w h \ddot w^{-1} = \ddot w h \dot w^{-1}$.
(This is not the same as $\dot w h \dot w^{-1} = \ddot w h \ddot w^{-1}$.)
To view this action on matrices of the form (\ref{eq:torus1}), we
define the word $c_n \cdots c_1 c_{\ol 1} \cdots c_{\ol n}$ by
$c_{\ol i} \defeq - c_i$.  Then we have
the formula
\begin{equation*}
w \circ \mathrm{diag}(e^{c_n}, \dotsc, e^{c_1}, e^{c_{\ol 1}}, \dotsc, e^{c_{\ol n}}) =
\mathrm{diag}(e^{c_{w_n}}, \dotsc, e^{c_{w_1}}, e^{c_{w_{\ol 1}}}, \dotsc, e^{c_{w_{\ol n}}}),
\end{equation*}
which is the same as $h \mapsto P(w)hP(w^{-1})$, performed inside
$\mathrm{GL}_{2n}(\mathbb C)$.

%\bp Is there a canonical way to choose a $2n \times 2n$ matrix
%$\dot w$ corresponding to $w$?
%Two matrices which look a lot like $P(t)$ and do belong to
%$\mathrm{SP}_{2n}(\mathbb C)$ have the middle
%$(\{ \ol3, \ol2, \ol1, 1, 2, 3 \}, \{ \ol3, \ol2, \ol1, 1, 2, 3 \})$
%submatrix replaced by one of the submatrices
%  \begin{equation*}
%  \begin{bmatrix}
%    1 & 0 & 0 & 0 & 0 & 0\\
%    0 & 1 & 0 & 0 & 0 & 0\\
%    0 & 0 & 0 & -1 & 0 & 0\\
%    0 & 0 & 1 & 0 & 0 & 0\\
%    0 & 0 & 0 & 0 & 1 & 0\\
%    0 & 0 & 0 & 0 & 0 & 1
%  \end{bmatrix},
%  \quad
%  \begin{bmatrix}
%    1 & 0 & 0 & 0 & 0 & 0\\
%    0 & 1 & 0 & 0 & 0 & 0\\
%    0 & 0 & 0 & 1 & 0 & 0\\
%    0 & 0 & -1 & 0 & 0 & 0\\
%    0 & 0 & 0 & 0 & 1 & 0\\
%    0 & 0 & 0 & 0 & 0 & 1
%  \end{bmatrix}.

\section{notations}

\ssec{Big and small matrix notations in $\bn$}
The association of a (big) $2n \times 2n$ matrix $P(w)$ to $w \in \bn$
is inherited from $\snn$.
Thus the matrix $P(t) = P(s_0)$ is exactly as in (\ref{eq:s2nmatrix}),
and the matrices
%$P(s_i') = P(s_i s_{\ol i})$, $i = 1, \dotsc, n-1$
%$P(s_2') = P(s_2s_{\ol2})$
\begin{equation}\label{eq:wmatrix}
P(s_i') = P(s_i s_{\ol i}) =
\begin{bmatrix}
I &       &   &       &  \\
& \sreg &   &       &  \\
&       & I &       &  \\
&       &   & \sreg &  \\
&       &   &       & I
\end{bmatrix}
\end{equation}
match the $2n \times 2n$
identity matrix everywhere except in rows $\{\ol{i+1}, \ol i, i, i+1\}$.
The group of big matrices $\{P(w) \,|\, w \in \bn \}$ is not a subgroup
of $\mathrm{SP}_{2n}(\mathbb C)$: while the generating matrices
$P(s_1'), \dotsc, P(s_{n-1}')$
satisfy the condition (\ref{eq:sp2ndefn}),
the generating matrix $P(t)$ does not.

The symmetry of entries in the matrix $P(w)$
imply that it can easily be recovered
from the (small) $n \times n$ matrix $M(w) = (m_{i,j})_{i,j=1}^n$ defined by
\begin{equation*}
\begin{aligned}
m_{i,j} &= p_{i,j} - p_{\ol i, j}\\
&= \begin{cases} p_{i,j} &\text{if $p_{i,j} = 1$},\\
-p_{\ol i, j} &\text{if $p_{\ol i,j} = 1$}.
\end{cases}
\end{aligned}
\end{equation*}
Thus $M(t) = \mathrm{diag}(-1,1, \dotsc, 1)$, and
\begin{equation*}
M(s'_i) =
\begin{bmatrix}
I &       &   \\
& \sreg &   \\
&       & I
\end{bmatrix},
\end{equation*}
differing from the $n \times n$ identity matrix in rows $i, i+1$.

The matrices $\{ M(w) \,|\, w \in \bn \}$ are often thought of as the
{\em wreath product} of the cyclic group $\{\pm 1\}$
with the symmetric group $\sn$:
they are the $n \times n$ permutation matrices
in which the entries equal to $1$ are replaced
by $1$ or $-1$ in all possible ways.
Thus by taking the absolute values of the entries of $M(w)$,
%this small matrix,
we obtain a permutation matrix
for the permutation $|w_1| \cdots |w_n|$ in $\sn$.
Call this matrix $Q(w)$, for $w \in \bn$.

%\bp Describe what
The homomorphism $\varphi$
%does to big and small matrix notation.
$M(\varphi(w)) = Q(w)$, i.e., $Q(w)$ is simply
%Is $Q(w)$
the permutation matrix in $\sn$ for $\varphi(w)$.

It follows that the big and small matrix notations for the reversal
$s'_{[a,b]}$ ($a > 0$) are
\begin{equation*}
P(s'_{[a,b]}) =     \begin{bmatrix}
I &         &   &         &  \\
& \smallj &   &         &  \\
&         & I &         &  \\
&         &   & \smallj &  \\
&         &   &         & I
\end{bmatrix},
\qquad
M(s'_{[a,b]}) =
\begin{bmatrix}
I &         &  \\
& \smallj &  \\
&         & I
\end{bmatrix},
\end{equation*}
differing from the $2n \times 2n$ identity matrix in rows
$[\ol b, \ol a] \cup [a,b]$ and $[a,b]$, respectively.
%$\{\ol b, \dotsc, \ol a, a, \dotcs, b \}$ and $[
%while t
The big and small matrix notations
for the reversal $s'_{[\ol a,a]}$ are
\begin{equation*}
P(s'_{[\ol a, a]}) = \begin{bmatrix}
I &         &  \\
& \smallj &  \\
&         & I
\end{bmatrix},
\qquad
\qquad
M(s'_{[\ol a, a]}) = \begin{bmatrix} -I & 0 \\ 0 & I \end{bmatrix},
\end{equation*}
differing from the $2n \times 2n$ identity matrix in rows
$[\ol a, a]$ and $[1,a]$, respectively.

\ssec{Long and short cycle notations in $\bn$}
The association of a (long) cycle notation to $w \in \bn$
is inherited from $\snn$.  This is an arrangement of the letters
$\ol n, \dotsc, \ol 1, 1, \dotsc, n$ into cycles $C_1, \dotsc, C_r$
such that each cycle satisfies (exactly) one of the following conditions:
\begin{enumerate}
\item $\ol C_i = C_i$.
Equivalently, for some letter $a$ in $C_i$,
the letter $\ol a$ also appears in $C_i$.
Equivalently, for each letter $a$ in $C_i$,
the letter $\ol a$ also appears in $C_i$.
Equivalently, the permutation defined by $C_i$ belongs to $\bn$.
\item  $\ol C_i = C_j$ for some $j \neq i$ and no two letters of $C_i$ have
the same absolute value.
\end{enumerate}
Permutations satisfying the first condition must be even;
permutations satisfying the second condition may be odd or even.
Thus the long cycle type of $w \in \bn$ is a partition of $2n$ in which
all odd parts appear with even multiplicity.  Each such cycle type may occur.

%\bp \label{p:cycledesc}
%Characterize the possible (long) cycle notations of elements of $\bn$.
%Is it true that cycles come in pairs $(C_i, C_j)$ that satisfy
%$\ol{C_i} = C_j$ with the possibility that $i = j$,
%at most one cycle $C_k$ pairing with itself,
%i.e., $\ol{C_i} = C_i$?
%\bp Describe what the homomorphism $\varphi$ does to long cycle notation.
%Assuming that cycles come in pairs $(C_i, C_j)$ as in
%Problem~\ref{p:cycledesc}, do we have
We have
\begin{equation*}
\begin{cases}
\varphi(C_iC_j) = |C_i| = |C_j| &\text{if $\ol{C_i} = C_j; i \neq j$},\\
\varphi(C_i) = |C_i| &\text{if $\ol{C_i} = C_i$},
\end{cases}
\end{equation*}
where $|C_i|$ means the cycle in which every letter is replaced
by its absolute value.  ($\star$ Find a better notation that doesn't look like
Note that when $\ol C_i = C_j$ with $i \neq j$,
we do not have $\varphi(C_iC_j) = \varphi(C_i) \varphi(C_j)$,
since $C_i, C_j \not \in \bn$ in this case.  Call a cycle {\em positive}
if it has an even number of negative letters, and {\em negative} otherwise.

We define the short cycle notation of $w \in \bn$ in terms of the $n \times n$
matrices $M(w)$ and $Q(w)$.
Specifically, it is the cycle notation of the permutation $\varphi(w) \in \sn$,
whose permutation matrix is $Q(w)$,
with a bar over each letter $i$ for which $m_{i,j} = -1$.
%\bp Is this
This is equivalent to saying that the cycles are
\begin{equation*}
(w(j), w(|w(j)|), w(|w(|w(j)|)|), \dotsc, \pm j)
\end{equation*}
for some $j \in [n]$.
($\star$ Include Angela's or other example.)

The short cycle type of $w \in \bn$, $\sct(w)$, is defined to be
a pair $(\lambda, \mu)$ of partitions,
with $\lambda$ recording the cycle sizes of positive cycles,
and $\mu$ recording with cycle sizes of negative cycles.
All possible pairs with $|\lambda| + |\mu| = n$ may occur.

For $w \in \bn$, we obtain the cycle notation of $\varphi(w) \in \sn$
from the short cycle notation of $w$ simply by
replacing each letter by its absolute value.
We obtain the short cycle type of $\varphi(w)$ from the short cycle type
of $w$ simply by rearranging the parts of $\lambda$ and $\mu$ into
nonincreasing order.
%moving signs.

%\bp Describe what the homomorphism $\varphi$ does to short cycle notation.
%Does it simply remove all of the negative signs?

%$\star$ Say something about cycles having an even or odd
%number of negative elements, and short cycle type.

%\bp Explain what connection exists between the
%long and short cycle notations of $w \in \bn$.
%In theory one determines the other.
%Observe that if $u, w \in \bn$ are any elements whose short one-line notations
%satisfy $|u_i| = |w_i|$, then the multisets of parts in their cycle types
%will be equal.

To compute long cycle notation from short, we map each short
cycle $C = (a_1, \dotsc, a_k)$ to the long cycle(s)
\begin{equation*}
\begin{cases}
(a_1, \dotsc, a_k) (\ol{a_1}, \dotsc, \ol{a_k}) &\text{if $C$ is positive},\\
(a_1, \dotsc, a_k, \ol{a_1}, \dotsc, \ol{a_k}) &\text{if $C$ is negative}.
\end{cases}
\end{equation*}
To invert this map, we consider each letter $a$ with $w^{-1}(a) > 0$ and
$a$ and $\ol a$ not yet assigned to a short cycle, and create the short cycle
\begin{equation*}
(a, w(|a|), w(|w(|a|)|), \dotsc, w^{-1}(a) ).
\end{equation*}
%  ($\star$ Fix this: not every short cycle contains a letter $> 0$.)

%  ($\star$ Organize this example better.)
When $n=2$ we have the following reduced expressions in generators
($s \defeq s'_1$),
one-line notations, cycle notations, and cycle type.
\begin{equation*}
\begin{matrix}
\begin{matrix}\text{reduced}\\ \text{expression}\end{matrix}
& \begin{matrix}\text{long}\\ \text{one-line}\end{matrix}
& \begin{matrix}\text{short}\\ \text{one-line}\end{matrix}
& \begin{matrix}\text{long}\\ \text{cycle}\end{matrix}
& \begin{matrix}\text{short}\\ \text{cycle}\end{matrix}
& \begin{matrix}\text{cycle}\\ \text{type}\end{matrix}\\
& & & & & \\
e & \ol2\ol112 & 12 & (\ol2)(\ol1)(1)(2) & (1)(2) & (11,\emptyset)\\
s & \ol1\ol221 & 21 & (\ol1,\ol2)(1,2)   & (1,2)  & (2, \emptyset)\\
t & \ol21\ol12 & \ol12 & (1,\ol1)(2)(\ol2) & (\ol1)(2) & (1,1)\\
st & 1\ol22\ol1 & 2\ol1 & (1,2,\ol1,\ol2) & (\ol1,2) & (\emptyset,2)\\
ts & \ol12\ol21 & \ol21 & (1,\ol2,\ol1,2) & (1,\ol2) & (\emptyset, 2)\\
sts & 2\ol11\ol2 & 12 & (1)(\ol1)(2,\ol2) & (1)(\ol2) & (1,1)\\
tst & 12\ol2\ol1 & \ol2\ol1 & (1,\ol2)(2,\ol1) & (\ol1,\ol2) & (2,\emptyset)\\
stst=tsts & 21\ol1\ol2 & \ol1\ol2 & (1,\ol1)(2,\ol2) & (\ol1)(\ol2) & (\emptyset, 11)
\end{matrix}
\end{equation*}
Observe that if $v$, $w$ satisfy $\varphi(v) = \varphi(w)$
then the short cycle types of $v$ and $w$ are rearrangements of one another.
For example, $12$, $\ol12$, $1\ol2$, $\ol1\ol2$ have short cycle types
$(11,\emptyset)$, $(1,1)$, $(1,1)$, $(\emptyset, 11)$, respectively,
while $21$, $\ol21$, $2\ol1$, $\ol2\ol1$ have short cycle types
$(2, \emptyset)$, $(\emptyset, 2)$, $(\emptyset, 2)$, $(2, \emptyset)$,
%When $n = 2$, the correspondence between long and short cycle notations
%of $w \in \bn$ is
%(1, \ol1)(2, \ol2) \mapsto (\ol 1)(\ol 2), \quad
%(1, \ol2)(\ol1, 2) \mapsto (\ol1, \ol2), \quad
%(1)(\ol1)(2, \ol2) \mapsto (1)(\ol 2), \quad
%(1, 2, \ol1, \ol2) \mapsto (\ol1, 2), \\
%(1, \ol2, \ol1, 2) \mapsto (1, \ol 2), \quad
%(2)(\ol2)(1, \ol1) \mapsto (\ol1)(2), \quad
%(\ol1, \ol2)(1, 2) \mapsto (1, 2), \quad
%(\ol2)(\ol1)(1)(2) \mapsto (1)(2).
%These $8$ elements have cycle types
%  \begin{gathered}
%    (\emptyset, 11), \qquad \qquad
%    (2, \emptyset), \qquad \qquad
%    (1,1), \qquad \qquad
%    (\emptyset, 2)\\
%    (\emptyset, 2), \qquad \qquad
%    (1,1), \qquad \qquad
%    (2, \emptyset), \qquad \qquad
%    (11, \emptyset).
%  \end{gathered}

\section{s2n}

In ($\snn$) cycle notation we have $t = (1, \ol 1)$ and
$s'_i = (i,i+1)(\ol i,\ol{i+1})$ for $i \geq 1$.

Observe that we can rewrite the four classes of relations above as
\begin{equation*}
\begin{alignedat}2
\smash{(s'_is'_i)^1} &= e &\quad &\text{for $i = 0, \dotsc, n-1$,}\\
\smash{(s'_0s'_1)^4} &= e &\quad & \\
%    ts_i &= s_it &\quad &\text{for $i=2,\dotsc,n-1$,}\\
\smash{(s'_is'_j)^2} &= e &\quad &\text{for $i,j \geq 0$ and $|i-j| \geq 2$,}\\

\smash{(s'_is'_j)^3} &= e &\quad &\text{for $i,j \geq 1$ and $|i-j| = 1$.}
\end{alignedat}
\end{equation*}
This information is often summarized in the {\em Coxeter graph}
\begin{equation}\label{eq:coxeter}
%  \begin{tikzpicture}[scale=.7,baseline=-25]
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (0,0) -- (2,0) node[above, midway, yshift=1mm] {4};
\draw[-] (2,0) -- (4,0);
\draw[-] (4,0) -- (5,0);
%  \draw[-] (5.5,0) -- (6.5,0);
\draw[-] (7,0) -- (8,0);
\draw[-] (8,0) -- (10,0);
\draw[-,fill] (0,0) circle (3pt);
\draw[-, fill] (2,0) circle (3pt);
\draw[-, fill] (4,0) circle (3pt);
\draw[-, fill] (5.5,0) circle (1pt);
\draw[-, fill] (6,0) circle (1pt);
\draw[-, fill] (6.5,0) circle (1pt);
\draw[-, fill] (8,0) circle (3pt);
\draw[-, fill] (10,0) circle (3pt);
\node at (0,-.8) {$s'_0$};
\node at (2,-.8) {$s'_1$};
\node at (4,-.8) {$s'_2$};
\node at (8,-.8) {$s'_{n-2}$};
\node at (10,-.8) {$s'_{n-1}$};
\end{tikzpicture} ,
\end{equation}
in which the label $m(\smash{s'_i,s'_j})$ on an edge $(\smash{s'_i, s'_j})$
means that $(\smash{s'_is'_j})^{m(\smash{s'_i,s'_j})} = e$.
Edges are drawn without labels
when $m(\smash{s'_i,s'_j}) = 3$ and edges are omitted altogether when
$m(s'_i,s'_j) \in \{1,2\}$.
This information can also be summarized in the
{\em Coxeter matrix} $(m(\smash{s'_i,s'_j}))_{i,j=0}^{n-1}$:
\begin{equation*}
\begin{bmatrix}
1      & 4      & 2      & \hdots & 2 \\
4      & 1      & 3      & \ddots & \vdots \\
2      & 3      & \ddots & \ddots & 2 \\
\vdots & \ddots & \ddots & \ddots & 3 \\
2      & \hdots & 2      & 3      & 1
\end{bmatrix}.
\end{equation*}

To generators $t, s_{1}', \dotsc, s_{n-1}'$,
%or each generator,
we associate the simple wiring diagrams
\begin{equation}\label{eq:hyperbigwiring diagrams}
%  \begin{tikzpicture}[scale=.5,baseline=-20]
\begin{tikzpicture}[scale=.5,baseline=0]
\node at (0,4.5) {$(n)$};            %\node at (1,5.5) {$n$};
\node at (0,3.5) {$(n-1)$};          %\node at (1.75,4.5) {$n-1$};
%  \node at (0, 2.8) {$\cdot$};
%  \node at (0, 2.5) {$\cdot$};
%  \node at (0, 2.2) {$\cdot$};
\node at (0,1.5) {$(2)$};            %\node at (1,1.5) {$2$};
\node at (0,0.5) {$(1)$};            %\node at (1,0.5) {$1$};
\node at (0,-0.5) {$(\ol 1)$};       %\node at (1,-0.5) {$\ol 1$};
\node at (0,-1.5) {$(\ol 2)$};       %\node at (1,-1.5) {$\ol 2$};
\node at (0,-3.5) {$(\ol{n-1})$}; %\node at (1.75,-4.5) {$\ol{n-1}$};
\node at (0,-4.5) {$(\ol n)$};       %\node at (1,-5.5) {$\ol n$};
%  \node at (0, -6.5) {$s_{n-1}$};
\end{tikzpicture}
\qquad \qquad
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\node at (0, 2.8) {$\cdot$};
\node at (0, 2.5) {$\cdot$};
\node at (0, 2.2) {$\cdot$};
%\node at (0, 2.8) {$\vdots$};
\draw[-] (-.5,1.5) -- (.5,1.5);
\draw[-] (-.5,0.5) -- (.5,-0.5);
\draw[-] (-.5,-0.5) -- (.5,0.5);
\draw[-] (-.5,-1.5) -- (.5,-1.5);
\node at (0, -2.2) {$\cdot$};
\node at (0, -2.5) {$\cdot$};
\node at (0, -2.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (0, -6.5) {$t$};
\end{tikzpicture}
\ ,\;
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (-.5,4.5) -- (.5,4.5);
\draw[-] (-.5,3.5) -- (.5,3.5);
\node at (0, 2.8) {$\cdot$};
\node at (0, 2.5) {$\cdot$};
\node at (0, 2.2) {$\cdot$};
%\node at (0, 2.8) {$\vdots$};
\draw[-] (-.5,1.5) -- (.5,0.5);
\draw[-] (-.5,0.5) -- (.5,1.5);
\draw[-] (-.5,-0.5) -- (.5,-1.5);
\draw[-] (-.5,-1.5) -- (.5,-0.5);
\node at (0, -2.2) {$\cdot$};
\node at (0, -2.5) {$\cdot$};
\node at (0, -2.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-3.5) -- (.5,-3.5);
\draw[-] (-.5,-4.5) -- (.5,-4.5);
\node at (0, -6.5) {$s_1'$};
\end{tikzpicture}
\ ,\dotsc, \
\begin{tikzpicture}[scale=.5,baseline=0]
\draw[-] (-.5,4.5) -- (.5,3.5);
\draw[-] (-.5,3.5) -- (.5,4.5);
\node at (0, 2.8) {$\cdot$};
\node at (0, 2.5) {$\cdot$};
\node at (0, 2.2) {$\cdot$};
%\node at (0, 2.8) {$\vdots$};
\draw[-] (-.5,1.5) -- (.5,1.5);
\draw[-] (-.5,0.5) -- (.5,0.5);
\draw[-] (-.5,-0.5) -- (.5,-0.5);
\draw[-] (-.5,-1.5) -- (.5,-1.5);
\node at (0, -2.2) {$\cdot$};
\node at (0, -2.5) {$\cdot$};
\node at (0, -2.8) {$\cdot$};
%\node at (0, -2.8) {$\vdots$};
\draw[-] (-.5,-3.5) -- (.5,-4.5);
\draw[-] (-.5,-4.5) -- (.5,-3.5);
\node at (0, -6.5) {$s_{n-1}'$};
\end{tikzpicture}
\end{equation}
and to products of generators, we associate the corresponding
concatenations of simple wiring diagrams, which we call {\em wiring diagrams}.

\section{multibasis char gen fn}

space $\Lambda_n(x)$,
each symmetric function can be expressed as
of homogeneous degree-$n$ symmetric functions
\begin{equation*}
\sum \epsilon^\lambda m_\lambda = \sum \phi^\lambda e_\lambda
= \sum \chi^{\lambda^\tr} s_\lambda = \cdots ?
\end{equation*}
Do we have both
\begin{equation*}
\begin{gathered}
\sum_{\lambda,\mu} \epsilon^{\lambda,\mu} m_\lambda \tilde m_\mu
= \sum_{\lambda,\mu} (\epsilon \eta)^{\lambda,\mu} m_\lambda \tilde f_\mu
= \sum_{\lambda,\mu} (\eta \epsilon)^{\lambda,\mu} f_\lambda \tilde m_\mu
= \sum_{\lambda,\mu} \eta^{\lambda,\mu} f_\lambda \tilde f_\mu\\
= \sum_{\lambda,\mu} \chi^{\lambda^\tr,\mu^\tr} s_\lambda \tilde s_\mu
= \sum_{\lambda,\mu} \frac{(-1)^{\ell(\lambda)+\ell(\mu)}}{z_\lambda z_\mu}\psi^{\lambda,\mu} p_\lambda(x) p_\mu(y) \\
= \sum_{\lambda,\mu} \phi^{\lambda,\mu} e_\lambda \tilde e_\mu
= \sum_{\lambda,\mu} (\phi \gamma)^{\lambda,\mu} e_\lambda \tilde h_\mu
= \sum_{\lambda,\mu} (\gamma \phi)^{\lambda,\mu} h_\lambda \tilde e_\mu
= \sum_{\lambda,\mu} \gamma^{\lambda,\mu} h_\lambda \tilde h_\mu
\end{gathered}
\end{equation*}
(or something similar) and
\begin{equation*}
\begin{gathered}
\sum_{\lambda,\mu} \epsilon^{\lambda,\mu} (mm)_{\lambda,\mu}
= \sum_{\lambda,\mu} (\epsilon \eta)^{\lambda,\mu} (mf)_{\lambda,\mu}
= \sum_{\lambda,\mu} (\eta \epsilon)^{\lambda,\mu} (fm)_{\lambda,\mu}
= \sum_{\lambda,\mu} \eta^{\lambda,\mu} (f\ntnsp f)_{\lambda,\mu}\\
= \sum_{\lambda,\mu} \chi^{\lambda^\tr,\mu^\tr} (ss)_{\lambda,\mu}
= \sum_{\lambda,\mu} \frac{(-1)^{\ell(\lambda)+\ell(\mu)}}{z_\lambda z_\mu}\psi^{\lambda,\mu} p_{\lambda,\mu}^+(x,y) p_{\lambda,\mu}^-(x,y) \\
= \sum_{\lambda,\mu} \phi^{\lambda,\mu} (ee)_{\lambda,\mu}
= \sum_{\lambda,\mu} (\phi \gamma)^{\lambda,\mu} (eh)_{\lambda,\mu}
= \sum_{\lambda,\mu} (\gamma \phi)^{\lambda,\mu} (he)_{\lambda,\mu}
= \sum_{\lambda,\mu} \gamma^{\lambda,\mu} (hh)_{\lambda,\mu}
\end{gathered}
\end{equation*}
(or something similar)?  The power sum coefficients should probably be
different in the two identities.

\section{Permanents}

%  and contains
%  one path corresponding to a grounded element $j$ of $Q$, then there are two
%  \begin{equation*}\label{eq:detPBpathfam}
%    \pi = (\pi_{\ol k}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_k)
%  \end{equation*}
%  of type $e$ in $Z_{I,I}$ in which no paths intersect;
%  $\det(B^+)$ counts path families (\ref{eq:detDBPBpathfam})
%  of type $e$ or $t$ in which
%  only paths $\pi_1$, $\pi_{\ol 1}$ may intersect one another.
%  for some $u$
%  with $\ell(u)$ odd ($\ell_s(u)$ even and $\ell_t(u)$ odd).
%  Again $\pi$ and $\sigma$ contribute (\ref{eq:ellwellu}) to (\ref{eq:detDB}).
%  Thus after cancellation, all positive contributions to (\ref{eq:detDB})
%  come from path families with no intersections of paths.
%  These path families necessarily have type $e$.
%  \end{proof}

\begin{thm}\label{t:detPB}
We have that $\det(B^+)$ counts path families
$\pi = (\pi_{\ol k}, \dotsc, \pi_{\ol 1}, \pi_1, \dotsc, \pi_k)$
of type $e$ in $Z_{I,I}$,
in which paths $\pi_1,\dotsc,\pi_k$ are nonintersecting,
and in which path $\pi_1$ may be circled if it passes
through the central star of $Z$.
\end{thm}
\begin{proof}
We have
%Each term of $\det(B^-)$ is
\begin{equation}\label{eq:detPB}
\det(D(P)) = \sum_{w \in \mfb k}
(-1)^{\ell_s(w)}b_{\ol k, w_{\ol k}} \cdots b_{\ol 1, w_{\ol 1}} b_{1,w_1} \cdots b_{k,w_k},
\end{equation}
%  for $w \in \bn$ ($k$th hyperoctahedral group),
and all path families $\pi$ of type $w$ contribute to this term.
Suppose that
for $0 < i < j$
some path family $\pi$ of type $w$ has a pair of intersecting paths.
Choose a vertex $v$ of intersection which is
at or above the horizontal line of symmetry of $Z_{I,I}$,
and which is maximal with respect to the left-to-right order.
From the pairs of paths which intersect at $v$,
choose one pair whose sink indices are
lexicographically least in absolute value.
($\star$ Wait.  If there are intersections between paths
with different sink indices, we must take care of these first.)
%  Suppose first that the sink indices are
Now create a path family $\sigma$ by swapping the $v$-to-sink
portions of the two paths.
(If there are no two intersecting paths having different sink indices,)
If the two sink indices are $i$, $j$ with $|i| \neq |j|$,
then $\type(\sigma) = wu$ for some $u$
with $\ell_s(u)$ odd (and $\ell_t(u)$ even).
($\star$ Write a lemma about type-$\msfBC$ transpositions justifying this.)
Thus $\pi$ and $\sigma$ contribute
\begin{equation}\label{eq:ellwellu}
(-1)^{\ell_s(w)} + (-1)^{\ell_s(w) + \ell_s(u)} = (-1)^{\ell_s(w)} -(-1)^{\ell_s(w)} = 0
\end{equation}
to (\ref{eq:detPB}).
On the other hand, if the two sink indices are $i$, $\ol i$,
then they must be $1$, $\ol 1$. ($\star$ Explain this better.)
Then $\type(\sigma) = wu$ for some $u$
with $\ell(u)$ odd ($\ell_s(u)$ even and $\ell_t(u)$ odd).
Now $\pi$ and $\sigma$ both contribute (\ref{eq:ellwellu})
to (\ref{eq:detPB}).
By ($\star$ some other lemma) ?
Thus after cancellation, all positive contributions to (\ref{eq:detDB})
come from path families with no intersections of paths.
These path families necessarily have type $e$.
\end{proof}

\section{Answered questions about Merris-Watkins extension to type C}

\bp
%= x^{e,e} \circ g_4.
%Describe the $\hbnq$ elements $g_1, g_2, g_3, g_4$ which satisfy
%  \begin{gathered}
%  \imm{\epsilon^{\lambda}}(P) = x^{e,e} \circ g_1,
%  \qquad
%  \imm{\epsilon^{\lambda}}(D) = x^{e,e} \circ g_2,\\
%  \imm{\eta^{\lambda}}(P) = x^{e,e} \circ g_3,
%  \qquad
%  \imm{\eta^{\lambda}}(D) = x^{e,e} \circ g_4.
%  \end{gathered}
Let $I = (I_1,\dotsc,I_m)$ be a set partition of $[n]$ of
type $\lambda = (\lambda_1,\dotsc,\lambda_m)$,
and let $u = u(I) \in \sn$ be the permutation in which the $m$ subwords
\begin{equation*}
u_1 \cdots u_{\lambda_1}, \qquad
u_{\lambda_1 + 1} \cdots u_{\lambda_1 + \lambda_2},
\quad \dotsc, \quad
u_{n-\lambda_m+1} \cdots u_n
\end{equation*}
consist of the increasing rearrangements of $I_1, \dotsc, I_m$, respectively.
Find simple summation formulas for
%= x^{e,e} \circ g_1,
%  \qquad
%= x^{e,e} \circ g_2,\\
%= x^{e,e} \circ g_3,
%  \qquad
in terms of $I$, $\lambda$, $u$.  In particular, do we have
%Do we have
\begin{equation*}
\begin{gathered}
\perm(P_{I_1,I_1}) \cdots \perm(P_{I_m,I_m})
= \sum_{y \in \mfb \lambda} x^{u,yu}
= \nTksp \sum_{y \in u^{-1}\mfb \lambda u} \nTksp x^{u,uy},\\
\det(P_{I_1,I_1}) \cdots \det(P_{I_m,I_m})
= \sum_{y \in \mfb \lambda} (-1)^{\ell_s(y)} x^{u,yu}
= \nTksp \sum_{y \in u^{-1}\mfb \lambda u} \nTksp (-1)^{\ell_s(y)} x^{u,uy},\\
\perm(D_{I_1,I_1}) \cdots \perm(D_{I_m,I_m})
= \sum_{y \in \mfb \lambda} (-1)^{\ell_t(y)} x^{u,yu}
= \nTksp \sum_{y \in u^{-1}\mfb \lambda u} \nTksp (-1)^{\ell_t(y)} x^{u,uy},\\
\det(D_{I_1,I_1}) \cdots \det(D_{I_m,I_m})
= \sum_{y \in \mfb \lambda} (-1)^{\ell(y)} x^{u,yu}
= \nTksp \sum_{y \in u^{-1}\mfb \lambda u} \nTksp (-1)^{\ell(y)} x^{u,uy},
\end{gathered}
\end{equation*}
or something similar?
\ep

\bp
Fix $(\lambda,\mu) \vdash n$ with
$\lambda = (\lambda_1, \dotsc, \lambda_k)$,
$\mu = (\mu_1, \dotsc, \mu_m)$, and let
$(I,J) = (I_1,\dotsc,I_k, J_1, \dotsc, J_m)$
be a set partition of $[n]$ of
type $(\lambda,\mu)$.
%= (\lambda_1,\dotsc,\lambda_m)$,
Let $u = u(I,J) \in \sn$ be the permutation in which the $k+m$ subwords
\begin{equation*}
\begin{gathered}
u_1 \cdots u_{\lambda_1}, \qquad
u_{\lambda_1 + 1} \cdots u_{\lambda_1 + \lambda_2},
\quad \dotsc, \quad
u_{|\lambda|-\lambda_k+1} \cdots u_{|\lambda|},\\
u_{|\lambda| + 1} \cdots u_{|\lambda|+ \mu_1}, \qquad
u_{|\lambda| + \mu_1 + 1} \cdots u_{|\lambda| + \mu_1 + \mu_2},
\quad \dotsc, \quad
u_{n-\mu_m+1} \cdots u_n
\end{gathered}
\end{equation*}
consist of the increasing rearrangements of
$I_1, \dotsc, I_k, J_1,\dotsc,J_m$, respectively.
Find simple summation formulas for the four right-hand sides in
In particular, do we have
\begin{equation*}
\begin{gathered}
\begin{aligned}
\perm(P_{I_1,I_1})\cdots\perm(P_{I_k,I_k})\perm(D_{J_1,J_1})\cdots\perm(D_{J_m,J_m})
&= \ntksp \sumsb{v \in \mfb \lambda\\ w \in \mfb {|\lambda|+\mu}} \ntksp
(-1)^{\ell_t(w)} x^{u,vwu}\\
&= \nTTksp \sumsb{v \in u^{-1}\mfb \lambda u\\w \in u^{-1}\mfb {{|\lambda|+\mu}}u}\nTTksp
(-1)^{\ell_t(w)} x^{u,uvw},
\end{aligned}\\
\begin{aligned}
\perm(P_{I_1,I_1})\cdots\perm(P_{I_k,I_k})\det(D_{J_1,J_1})\cdots\det(D_{J_m,J_m})
&= \ntksp \sumsb{v \in \mfb \lambda\\ w \in \mfb {|\lambda|+\mu}} \ntksp
(-1)^{\ell(w)} x^{u,vwu}\\
&= \nTTksp \sumsb{v \in u^{-1}\mfb \lambda u\\w \in u^{-1}\mfb {{|\lambda|+\mu}}u}\nTTksp
(-1)^{\ell(w)} x^{u,uvw},
\end{aligned}\\
\begin{aligned}
\det(P_{I_1,I_1})\cdots\det(P_{I_k,I_k})\perm(D_{J_1,J_1})\cdots\perm(D_{J_m,J_m})
&= \ntksp \sumsb{v \in \mfb \lambda\\ w \in \mfb {|\lambda|+\mu}} \ntksp
(-1)^{\ell_s(v)} (-1)^{\ell_t(w)} x^{u,vwu}\\
&= \nTTksp \sumsb{v \in u^{-1}\mfb \lambda u\\w \in u^{-1}\mfb {{|\lambda|+\mu}}u}\nTTksp
(-1)^{\ell_s(v)} (-1)^{\ell_t(w)} x^{u,uvw},
\end{aligned}\\
\begin{aligned}
\det(P_{I_1,I_1})\cdots\det(P_{I_k,I_k})\det(D_{J_1,J_1})\cdots\det(D_{J_m,J_m})
&= \ntksp \sumsb{v \in \mfb \lambda\\ w \in \mfb {|\lambda|+\mu}} \ntksp
(-1)^{\ell_s(v)} (-1)^{\ell(w)} x^{u,vwu}\\
&= \nTTksp \sumsb{v \in u^{-1}\mfb \lambda u\\w \in u^{-1}\mfb {{|\lambda|+\mu}}u}\nTTksp
(-1)^{\ell_s(v)} (-1)^{\ell(w)} x^{u,uvw},
\end{aligned}
\end{gathered}
\end{equation*}
or something similar?
\ep

\section{a proof}

By the last line of (\ref{eq:minrepdefns}),
%we have that
$u^{-1}$ belongs to $\mfb {\lambda,\mu}^-$.  Furthermore,
ordered set partitions of $[n]$ of type $\alpha$
are in bijection with elements of $\mfb {\lambda,\mu}^-$.

%  and letters
%  $a + \mu_1 + \cdots + \mu_{i-1} + 1, \dotsc, a + \mu_1 + \cdots + \mu_k$
%  in positions $J_{

such that
positions $1, \dotsc, n$ of the one-line notation of $u^{-1}$
($\star$ Continue here.)

Now fix one such ordered set partition
%$(J_1,\dotsc,J_{r+k})$ of $[n]$
%of type $\alpha = \lambda \circ \mu$
and consider the polynomial
\begin{equation}\label{eq:lmwexpand2}
\sum_{(J_1,\dotsc,J_{r+k})} \nTksp F_1(z_{J_1,J_1}) \cdots F_1(z_{J_r,J_r})
F_2(z_{J_{r+1},J_{r+1}}) \cdots F_2(z_{J_{r+k},J_{r+k}})
\in \mathbb Z[z_{1,1},\dotsc,z_{n,n}].
\end{equation}
%  Fix an $a$-element subset $I$ of $[n]$ and consider the polynomial
%  \begin{equation}\label{eq:immprod}
%    \sum_I \imm{\chi_1}(z_{I,I})\imm{\chi_2}(z_{[n]\ssm I, [n]\ssm I}).
%  \end{equation}
%  By
%  we may expand this polynomial in the basis
%  $\{ z^{e,v} = z_{1,v_1} \cdots z_{n,v_n} \,|\, v \in \sn \}$ by
%  summing over ordered set partitions $(J_1, \dotsc, J_{r+k})$ of $[n]$,
%  where
%  We may describe the terms in the sum
%  choosing an ordered set partition
%  $(J_1,\dotsc,J_r)$ is an ordered set partition of $I$ of type $\lambda$, and
%  and an ordered set partition
%  $(J_{r+1},\dotsc,J_{r+k})$ is an ordered set partition of $[n] \ssm I$
%  of type $\mu$.  Specifically, (\ref{eq:immprod}) is equal to
%  and we write
%  by first using the
%  It is clear that
When we expand this in the basis
$\{ z^{e,w} = z_{1,w_1} \cdots z_{n,w_n} \,|\, w \in \sn \}$,
each resulting monomial
%appearing in (\ref{eq:lmwexpand2})
has the form $z^{v^{-1},yv^{-1}}$ where $v^{-1}$ is the permutation in $\sn$
formed by concatenating the increasing arrangements of $J_1,\dotsc,J_{r+k}$,
and $y$ belongs to
\begin{equation*}
\mfs{\alpha} \defeq
\mfs{[1,\alpha_1]} \times \mfs{[\alpha_1 + 1, \alpha_1 + \alpha_2]}
\times \cdots \times \mfs{[n - a - \alpha_{r+k}, n]}.
\end{equation*}
%+ \alpha_2 + 1
%\mfs{\lambda,\mu} \defeq
%\mfs{[1,\lambda_1]} \times \mfs{[\lambda_1+1,\lambda_1+\lambda_2]}
%\times \cdots \times
%\mfs{[a+1,a+\mu_1]} \times \mfs{[a+\mu_1+1,a+\mu_1+\mu_2]}
%\times \cdots \times
%  \end{equation*}
%  \times \mfs_{\lambda,\mu}$.
By (\ref{eq:coordring3}),
we may express the monomial as
%$z^{v^{-1},yv} = vyv^{-1} \circ z^{e,e}$,
$vyv^{-1} \circ z^{e,e}$.
By the last line of (\ref{eq:minrepdefns}), we have that
$v \in \mfb {\lambda,\mu}^-$, and furthermore that
ordered set partitions of $[n]$ of type $\alpha$
are in bijection with elements of $\mfb {\lambda,\mu}^-$.
$y = y^{(1)} y^{(2)}$ with
\begin{equation*}
\begin{aligned}
y^{(1)} &\in \mfs{[1,\alpha_1]}
\times
\mfs{[\alpha_1+1, \alpha_1 + \alpha_2]}
\times \cdots \times
\mfs{[a - \alpha_r+1,a]},\\
y^{(2)} &\in \mfs{[a+1,a+\alpha_{r+1}]}
\times
\mfs{[a+\alpha_{r+1}+1, a+\alpha_{r+1} + \alpha_{r+2}]}
\times \cdots \times
\mfs{[n - \alpha_{r+k}+1,n]},
\end{aligned}
\end{equation*}
%  $y = y^{(1,1)} \cdots y^{(1,r)} y^{(2,1)} \cdots y^{(2,k)}$, with
%  \begin{equation*}
%    \begin{alignedat}{2}
%    y^{(1,j)} &\in \mfs{[\lambda_1 + \cdots + \lambda_{j-1} + 1,\; \lambda_1 + \cdots \lambda_j]} &\qquad &\text{for $j = 1,\dotsc,r$},\\
%    y^{(2,j)} &\in \mfs{[a + \mu_1 + \cdots + \mu_{j-1} + 1,\; a + \mu_1 + \cdo%ts \mu_{j}]} &\qquad &\text{for $j = 1,\dotsc,k$},
%    \end{alignedat}
%    \end{equation*}
%  $y = y^{(1)} \cdots y^{(r+k)}$ with
%  $y^{(j)} \in
%  \mfs{[\alpha_1 + \cdots + \alpha_{j-1}+1,\; \alpha_1 + \cdots + \alpha_j]}$,
we may express
%the monomial is equal to $vyv^{-1} \circ z^{e,e}$,
the coefficient of this monomial
%of $\pm 1$ is given
%  in terms of the factorization
%$\varepsilon_1 \cdots \varepsilon_{r+k}$,
%  $\theta_1(y^{(1)} \cdots y^{(r)}) \theta_2(y^{(r+1)} \cdots y^{(r+k)})$.
$\theta_1(y^{(1)}) \theta_2(y^{(2)})$.
%  $\varepsilon_{1,1} \cdots \varepsilon_{1,r} \varepsilon_{2,1} \cdots \varepsilon_{2,k} \in \{ \pm 1 \}$, where
%  \begin{equation*}
%    \varepsilon_{i,j} = \begin{cases}
%      1 &\text{if $\theta_i = \mathrm{triv}$},\\
%%      \sgn(y^{(i,j)}) &\text{if $\theta_i = \sgn$}.
%      \sgn(y^{(j)}) &\text{if $i=1$ and $\theta_1 = \sgn$},\\
%      \sgn(y^{(r+j)}) &\text{if $i=2$ and $\theta_2 = \sgn$}.
%      \end{cases}
%  \end{equation*}
Using this factorization, we may rewrite
%  a precise formula
%  for the sum
(\ref{eq:lmwexpand}) as
\begin{equation*}
\sum_{v \in \mfb {\lambda,\mu}^-}
\sumsb{y \in \mfs{\alpha}\\ y = y^{(1)}y^{(2)}} \nTksp
\theta_1(y^{(1)})\theta_2(y^{(2)}) vyv^{-1} \circ z^{e,e}.
\end{equation*}

($\star$ Finish this by plugging $P$, $D$ into $z$.)
\end{proof}

\section{}

For $(\lambda,\mu)\vdash n$,
do we obtain $\epsilon^{\lambda,\mu}$
by letting $\bn$ act by left multiplication on the space
\begin{equation}
\spn
\end{equation}
where each basis element above can also be written as
\begin{equation*}
u \Big( \ntksp \sum_{v \in \mfb \lambda}
\epsilon(v) v \Big)
\Big( \nTtksp \sum_{w \in \mfb {0^a\mu}} \nTtksp
(\delta\epsilon)(w)w \Big)
= u \Big( \ntksp \sum_{v \in \mfb \lambda} (-1)^{\ell_s(v)} v \Big)
\Big( \nTtksp \sum_{w \in \mfb {0^a\mu}} \nTtksp
(-1)^{\ell(w)} w \Big)
%    (-q)^{\ell_s(w)} (-Q)^{\ell_t(w)} T_w \Big).
\end{equation*}
or something similar?
Do we obtain $(\epsilon\eta)^{\lambda,\mu}$
by letting $\bn$ act by left multiplication on the space
\begin{equation}
\spn
\bigg \{
u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
\epsilon(v)\delta(w) vw
= u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell_s(v)} (-1)^{\ell_t(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
%or something similar?  Each basis element above can also be written as
\begin{equation*}
u \Big( \ntksp \sum_{v \in \mfb \lambda}
\epsilon(v) v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
\delta(w)w \Big)
= u \Big( \ntksp \sum_{v \in \mfb \lambda} (-1)^{\ell_s(v)} v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell_t(w)} w \Big)
\end{equation*}
or something similar?
Do we obtain $(\eta\epsilon)^{\lambda,\mu}$
by letting $\bn$ act by left multiplication on the space
\begin{equation}
\spn
\bigg \{
u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(\delta\epsilon)(w) vw
= u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
%or something similar? Each basis element above can also be written as
\begin{equation*}
u \Big( \ntksp \sum_{v \in \mfb \lambda}
v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(\delta\epsilon)(w)w \Big)
= u \Big( \ntksp \sum_{v \in \mfb \lambda}
v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell(w)} w \Big),
%    (-q)^{\ell_s(w)} (-Q)^{\ell_t(w)} T_w \Big).
\end{equation*}
or something similar?
Do we obtain $\eta^{\lambda,\mu}$ by letting $\bn$ act by left multiplication
on the space
\begin{equation}
\spn
\bigg \{
u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
\delta(w)
= u \nTtksp \sumsb{v\in \mfb \lambda\\w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell_t(w)} vw
\,\biggr\rvert\, u \in \mfb {\lambda,\mu}^- \bigg \},
\end{equation}
where each basis element above can also be written as
%or something similar? Each basis element above can also be written as
\begin{equation*}
u \Big( \ntksp \sum_{v \in \mfb \lambda}
v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
\delta(w)w \Big)
= u \Big( \ntksp \sum_{v \in \mfb \lambda}
v \Big)
\Big( \nTtksp \sum_{w \in \mfb {|\lambda|+\mu}} \nTtksp
(-1)^{\ell_t(w)} w \Big),
\end{equation*}
or something similar?

\bp What are natural modules for the characters $\epsilon^{\lambda,\mu}$,
$(\epsilon\eta)^{\lambda,\mu}$, $(\eta\epsilon)^{\lambda,\mu}$, $\eta^{\lambda,\mu}$?
For which (if any) of these characters $\theta^{\lambda,\mu}$
can $\theta^{\lambda,\mu}(w)$ be computed
by an action of $\bn$ on coset sums of Young subgroups?
Conjecturally, if $\lambda \multiu \mu = \nu$, then each of these
representations should have dimension equal to the number of row-strict
Young tableaux of shape $\nu$.
\ep
($\star$ In particular, for $\lambda \vdash n$, $\mu \vdash n$, do we obtain
$\epsilon^{\lambda,\emptyset}$ by letting $\bn$ act by left multiplication on the space
\begin{equation*}
\spn
\bigg \{ u \ntksp \sum_{w \in \mfb \lambda} \ntksp \epsilon(w) w
= u \ntksp \sum_{w \in \mfb \lambda} (-1)^{\ell_s(w)} w
\,\biggr\rvert \, u \in \mfb \lambda^- \bigg \},
\end{equation*}
or something similar?
%For $\mu \vdash n$ d
Do we obtain
$\epsilon^{\emptyset,\mu}$ by letting $\bn$ act by left multiplication on the space
\begin{equation*}
\spn
\bigg \{ u \ntksp \sum_{w \in \mfb \mu}\ntksp \delta\epsilon(w)w
= u \ntksp \sum_{w \in \mfb \lambda}\ntksp (-1)^{\ell(w)} w
\,\biggr\rvert\, u \in \mfb \mu^- \bigg \},
\end{equation*}
or something similar?
%For $\lambda \vdash n$, d
Do we obtain
$\eta^{\lambda,\emptyset}$ by letting $\bn$ act by left multiplication on the space
\begin{equation*}
\spn
\bigg \{ u \ntksp \sum_{w \in \mfb \lambda}\ntksp 1(w) w
= u \ntksp \sum_{w \in \mfb \lambda} \ntksp w
\,\biggr\rvert\, u \in \mfb \lambda^- \bigg \},
\end{equation*}
or something similar?
%For $\mu \vdash n$ d
Do we obtain
$\eta^{\emptyset,\mu}$ by letting $\bn$ act by left multiplication on the space
\begin{equation*}
\spn
\bigg \{ u \ntksp \sum_{w \in \mfb \mu} \ntksp \delta(w)w
=  u \ntksp \sum_{w \in \mfb \lambda} \ntksp (-1)^{\ell_t(w)} w
\,\biggr\rvert\, u \in \mfb \mu^- \bigg \},
\end{equation*}
or something similar?)

\begin{alignat*}{6}
&\begin{matrix}\text{reduced}\\ \text{expression}\end{matrix} &\quad
&\begin{matrix}\text{long}\\ \text{one-line}\end{matrix} &\quad
&\begin{matrix}\text{short}\\ \text{one-line}\end{matrix} &\quad
&\begin{matrix}\text{long}\\ \text{cycle}\end{matrix} &\quad
&\begin{matrix}\text{short}\\ \text{cycle}\end{matrix} &\quad
&\begin{matrix}\text{cycle}\\ \text{type}\end{matrix}\\
&e &\quad &\ol2\ol112 &\quad &12 &\quad
&(\ol2)(\ol1)(1)(2) &\quad &(1)(2) &\quad &(11,\emptyset)\\
&s &\quad &\ol1\ol221 &\quad &21 &\quad
&(\ol1,\ol2)(1,2) &\quad &(1,2) &\quad &(2, \emptyset)\\
&t &\quad &\ol21\ol12 &\quad &\ol12 &\quad
&(1,\ol1)(2)(\ol2) &\quad &(\ol1)(2) &\quad &(1,1)\\
&st &\quad &1\ol22\ol1 &\quad &2\ol1 &\quad
&(1,2,\ol1,\ol2) &\quad &(\ol1,2) &\quad &(\emptyset,2)\\
&ts &\quad &\ol12\ol21 &\quad &\ol21 &\quad
&(1,\ol2,\ol1,2) &\quad &(1,\ol2) &\quad &(\emptyset, 2)\\
&sts &\quad &2\ol11\ol2 &\quad &12 &\quad
&(1)(\ol1)(2,\ol2) &\quad &(1)(\ol2) &\quad &(1,1)\\
&tst &\quad &12\ol2\ol1 &\quad &\ol2\ol1 &\quad
&(1,\ol2)(2,\ol1) &\quad &(\ol1,\ol2) &\quad &(2,\emptyset)\\
&stst &\quad &21\ol1\ol2 &\quad &\ol1\ol2 &\quad
&(1,\ol1)(2,\ol2) &\quad &(\ol1)(\ol2) &\quad &(\emptyset, 11)
\end{alignat*}