%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[submission]{FPSAC2019}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

\articlenumber{42}
\addbibresource{42.bib}


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%The following set of commands is cut and paste from
%the macros file of my dissertations. Much can be
%removed most likely.


%  \newtheorem{thm}{Theorem}[section]
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
%  \newtheorem{lem}[thm]{Lemma}
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%
%
%\documentclass[12pt]{amsart}
%%\documentclass{elsarticle}
%%\documentclass set to mimic other paper by Charlie
%
%%\textheight 8in
%%\textwidth 6in
%\topmargin = 0 in
%\textwidth = 6.5 in
%\textheight = 8.8 in
%\oddsidemargin = 0.0 in
%\evensidemargin = 0.0 in







%% define your title in the usual way
\title{Total Nonnegativity and Evaluations of Hecke Algebra Characters}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Adam Clearwater%\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}
\addressmark{1} \and Mark Skandera\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of Pennsylvania \\ \addressmark{2}Department of Mathematics, Lehigh University\\}

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

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{Let $\mfs{[i,j]}$ be the subgroup of the symmetric group $\sn$ generated by
	adjacent transpositions $(i,i+1), \dotsc, (j-1,j)$. 
        %assuming $1 \leq i < j \leq n$.
	We give a combinatorial rule for
	evaluating induced sign characters of the type $A$ Hecke algebra $\hnq$
	at all elements of the form $\sum_{w \in \mfs{[i,j]}} T_w$ and at all products
	of such elements. 
        %This includes evaluation at some elements $C'_w(q)$
	%of the Kazhdan-Lusztig basis.
	By a result of Stembridge, our result connects $\hnq$ trace evaluation to
	%values of 
        immanants of totally nonnegative matrices.
	%  We also give a combinatorial rule for evaluating ribbon skew characters 
	%  at elements $C'_w(q)$ indexed by permutations $w$ \avoidingp.
}
%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Hecke algebra trace, total nonnegativity}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations, ie. [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
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%\addbibresource{fpsac19.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands








\begin{document}
	
	
%	\author{Adam Clearwater}
%	%\author{Justin Lambright}%\corref{cor}}   needed for elsarticle
%	%\ead{jjlambright@anderson.edu}
%	\author{Mark Skandera}
%	%\ead{mas906@lehigh.edu,mark.skandera@gmail.com}
%	\title%[Evaluations of $\hnq$ characters]
%	{Total nonnegativity and evaluations of Hecke algebra characters}
%	%at Kazhdan-Lusztig
%	% basis elements}
%	%\title[Induced sign characters]
%	%{Induced sign characters and wiring diagram 
%	%bases of the Hecke algebra}
%	%{A generalization of Kazhdan and Lusztig's R-polynomials}
%	%\cortext[cor]{Corresponding author}      needed for elsarticle
%	%\address{Department of Mathematics, Anderson University, ??? , Anderson, IN 4601?, United States.}
%	%\address{Department of Mathematics, Lehigh University, 14 East Packer Avenue, Christmas-Saucon Hall, Bethlehem, PA 18015, United States.}
%	%\begin{keyword}
%	%Kazhdan-Lusztig \sep Hecke algebra \sep immanant \sep symmetric group
%	%\end{keyword}       needed for elsarticle??
%	
%	
%%	\bibliographystyle{../my}
%	
%	\date{\today}
%	
%	
	
	
	\maketitle

	

\section{Introduction}\label{s:intro}
Related to the study of totally nonnegative matrices,
those matrices having only nonnegative minors,
%Concepts of total nonnegativity, first explored by
%Gantmacher and Krein~\cite{GantKreinOsc}
%have found their way into many areas of mathematics,
%and also into 
is the study of polynomials $p(x_{1,1},\dotsc,x_{n,n})$
satisfying $p(a_{1,1}, \dotsc, a_{n,n}) \geq 0$ for every totally nonnegative
$n \times n$ matrix $A = (a_{i,j})$.
We call these {\em totally nonnegative (TNN)} polynomials.
In particular, work of Lusztig~\cite{LusztigTP} implies
that if we view $\zx := \zxn$
as a free $\mathbb Z$-module, then
certain elements which are related to the dual canonical basis of
the quantum group $\oqslnc$ are TNN polynomials.

In practice, it is sometimes possible to use cluster algebras and
a computer to demonstrate that a polynomial is TNN by expressing it as
a subtraction-free rational expression in matrix minors~\cite{FominTPTest}.
%(which are the cluster variables belonging to any chosen cluster).
%On the other hand, it is not known that every cluster
%produces such an expression for every TNN polynomial.
%to
On the other hand, no simple characterization of TNN polynomials is known.
To improve our understanding of TNN polynomials,
one might begin by investigating the {\em immanant subspace}
$\spn_{\mathbb Z} \{ \permmon xw \,|\, w \in \sn \}$ of $\zx$,
where $\sn$ is the symmetric group,
especially the generating functions
\begin{equation}\label{eq:imm}
\imm{\theta}(x) := \sum_{w \in \sn} \theta(w) \permmon xw
\end{equation}
for class functions $\theta: \sn \rightarrow \mathbb Z$.
Or, since some published results concern 
%the 
Hecke algebra 
%$\hnq$ and its 
traces, linear functions $\theta_q: \hnq \rightarrow \zqq$
satisfying $\theta_q(gh) = \theta_q(hg)$,
one might investigate 
%known results about 
these.
%The $\hnq$ traces, linear functions 
% for all $g, h \in \hnq$,
%and specializing at $\qp12 =1$ to $\sn$ class functions,
%similar generating functions
%\begin{equation}\label{eq:qimm}
%  \imm{\theta}(x) := \sum_{w \in \sn} \qiew \theta_q(T_w) \permmon xw
%\end{equation}
%in the noncommutative ring $\anq$, which specialize at $\qp12 = 1$ to the
%generating functions $(\ref{eq:imm})$.

In particular, let $\{ \wtc wq \,|\, w \in \sn \}$ be the (modified, signless)
{\em Kazhdan-Lusztig basis} of $\hnq$, defined by
\begin{equation*}
\wtc wq := \qp{\ell(w)}2  C'_w(q) = \sum_{v \leq w} P_{v,w}(q) T_v,
\end{equation*}
where $\{ T_w \,|\, w \in \sn \}$ is the natural basis of $\hnq$,
$\{ P_{v,w}(q) \,|\, v,w \in \sn \}$ are the
Kazhdan-Lusztig polynomials~\cite{KLRepCH}, and $\leq$ denotes the Bruhat order.
Specializing at $\qp12 = 1$ we have $T_v \mapsto v$ and $H_n(1) \cong \zsn$.
%is the symmetric group algebra $\zsn$.
For $1 \leq a < b \leq n$, let $s_{[a,b]} \in \sn$ be the {\em reversal}
whose one-line notation is
$1 \cdots (a-1) b (b-1) \cdots (a+1) a (a+2) \cdots n$.
Stembridge~\cite{StemConj} showed that
for any linear function $\theta: \zsn \rightarrow \mathbb Z$, the immanant
$\imm{\theta}(x)$ is TNN if for all sequences $J_1, \dotsc, J_r$ of
subintervals of $[1,n]$, we have
\begin{equation}\label{eq:cprod1}
\theta(\wtc{s_{J_1}}1 \cdots \wtc{s_{J_r}}1) \geq 0.
\end{equation}
Furthermore, by Lindstr\"om's Lemma
%~\cite{LinVRep}
and its converse~\cite{BrentiCTP}, 
%\cite{CryerProp},
a combinatorial interpretation of the above
expression would immediately yield a combinatorial interpretation of
the number $\imm{\theta}(A)$ for $A$ a TNN matrix.
Now if $\theta_q: \hnq \rightarrow \zqq$ specializes at $\qp12 = 1$ to $\theta$,
then (\ref{eq:cprod1}) is clearly a consequence of the condition
%its $q$-extension
\begin{equation}\label{eq:cprodq}
\theta_q(\wtc{s_{J_1}}q \cdots \wtc{s_{J_r}}q) \in \mathbb N[q],
\end{equation}
and a combinatorial interpretation of the coefficients
of the resulting polynomial would yield
combinatorial interpretations of the earlier expressions.
Haiman~\cite[Appendix]{HaimanHecke} observed that (\ref{eq:cprodq}) in turn
follows from the condition that for all $w \in \sn$ we have
\begin{equation}\label{eq:cnoprodq}
\theta_q(\wtc wq) \in \mathbb N[q],
\end{equation}
since products of Kazhdan-Lusztig basis
elements belong to $\spn_{\mathbb N[q]}\{ \wtc wq \,|\, w \in \sn \}$.
%On the other hand, it is not clear that the coefficients in the resulting
%linear combination would lead to a combinatorial interpretation of
%the expression in (\ref{eq:cprodq}).

Stembridge~\cite{StemImm} and Haiman~\cite{HaimanHecke} proved
%without combinatorial interpretation,
that for $\theta$ equal to any irreducible character $\chi^\lambda$ of $\sn$
($\chi_q^\lambda$ of $\hnq$)
%partitions $\lambda \vdash n$,
the evaluations (\ref{eq:cprod1}) and (\ref{eq:cnoprodq})
belong to $\mathbb N$ and $\mathbb N[q]$, respectively.
%and the evaluations (\ref{eq:cprod1}) belong to $\mathbb N$
%$\imm{\chi^\lambda}(x) \in \zx$ is TNN and the polynomials
%$\{\chi_q^\lambda(\wtc wq) \}$ in $\mathbb Z[q]$ have nonnegative coefficients.
They conjectured the same~\cite{StemConj}, \cite{HaimanHecke}
for functions $\phi^\lambda$ ($\phi_q^\lambda$, respectively),
called {\em monomial traces}, related to irreducible characters by
the inverse Kostka numbers.  None of these results or conjectures
included a combinatorial interpretation.
To better
%For the purposes of understanding 
understand TNN polynomials of the form (\ref{eq:imm}),
it would be desirable to solve the following problem.
\begin{prob}\label{prob:interp}
	Give combinatorial interpretations of
	all of the expressions in (\ref{eq:cprod1}) or (\ref{eq:cprodq})
	when $\theta_q$ varies over all elements of any basis of the
	$\hnq$ trace space.
	%is an irreducible character $\chi_q^\lambda$,
	%or another trace such as an induced sign character $\epsilon_q^\lambda$.
\end{prob}
So far, only some special cases have such interpretations.
In the case that $w$ \avoidsp, the Kazhdan-Lusztig basis element
$\wtc wq$ is closely
related to a product of the form appearing in (\ref{eq:cprodq}).
%In this case,
Combinatorial interpretations of the corresponding expressions
(\ref{eq:cprodq}) and (\ref{eq:cnoprodq})
were given in \cite{CHSSkanEKL} for
$\theta_q \in \{\chi_q^\lambda \,|\, \lambda \vdash n\}$,
%\epsilon_q^\lambda \}$,
and for $\theta_q$ belonging to several other bases of the $\hnq$ trace space,
including the basis $\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$ of
induced sign characters.
Also in this case,
combinatorial interpretations for $\theta_q = \phi_q^\lambda$
were given only when
when $\lambda$ has at most two parts,
or when $\lambda$ has rectangular shape and $q=1$~\cite[Thm.\,2.8]{StemConj}.
In the case that all permutations $s_{J_1},\dotsc,s_{J_r}$ in (\ref{eq:cprodq})
are adjacent transpositions ($s_1, \dotsc, s_{n-1}$),
combinatorial interpretations of the corresponding expressions
in (\ref{eq:cprodq}) were given in \cite{KLSBasesQMBIndSgn}
for $\theta_q = \epsilon_q^\lambda$.

We solve \cref{prob:interp} for the trace space basis
$\{ \epsilon_q^\lambda \,|\, \lambda \vdash n \}$
and state our solution in \cref{s:main}.
% of the $\hnq$ trace space,
In \cref{s:hnqanq} we introduce our computational tools:
%using computational tools belonging to 
the quantum matrix bialgebra,
%The two algebras and trace space are defined in Section~\ref{s:hnqanq}.
%we define these algebras,
%In Section~\ref{s:snet} we introduce 
combinatorial structures
called star networks,
and our general evaluation theorem
which links the two.
%se networks to elements of the above algebras.
%Finally in Section~\ref{s:main} we state our solution of
%Problem~\ref{prob:interp}.

%\section{The Hecke algebra and quantum matrix bialgebra}\label{s:hnqanq}
\section{The quantum matrix bialgebra and star networks}\label{s:hnqanq}

%Define the {\em symmetric group algebra} $\zsn$
%and the {\em (type $A$ Iwahori-) Hecke algebra} $\hnq$
%to be the algebras with
%multiplicative identity elements $e$ and $T_e$,
%respectively,
%generated over $\mathbb Z$ and $\zqq$
%by elements
%$s_1,\dotsc, s_{n-1}$ and 
%$T_{s_1},\dotsc, T_{s_{n-1}}$, 
%subject to the relations
%\begin{equation}\label{eq:hnqdef}
%\begin{alignedat}{3}
%s_i^2 &= e &\qquad
%T_{s_i}^2 &= (q-1) T_{s_i} + qT_e &\qquad
%&\text{for $i = 1, \dotsc, n-1$},\\
%s_is_js_i &= s_js_is_j &\qquad
%T_{s_i}T_{s_j}T_{s_i} &= T_{s_j}T_{s_i}T_{s_j} &\qquad
%&\text{for $|i - j| = 1$},\\
%s_is_j &= s_js_i &\qquad
%T_{s_i}T_{s_j} &= T_{s_j}T_{s_i} &\qquad
%&\text{for $|i - j| \geq 2$}.
%\end{alignedat}
%\end{equation}
%Analogous to the natural basis $\{ w \,|\, w \in \sn \}$ of $\zsn$
%is the natural basis $\{ T_w \,|\, w \in \sn \}$ of $\hnq$,
%where we define
%$T_w = T_{s_{i_1}} \ntksp \cdots T_{s_{i_\ell}}$
%whenever $\sprod i\ell$
%is a reduced (short as possible)
%expression for $w$ in $\sn$.  We call $\ell$ the {\em length}
%of $w$ and write $\ell = \ell(w)$.
%We define the {\em one-line notation} $w_1 \cdots w_n$ of $w \in \sn$ by
%letting any expression for $w$ act on the word $1 \cdots n$,
%where each generator $s_j$ acts on an $n$-letter word by
%swapping the letters in positions $j$ and $j+1$, i.e.,
%$s_j \circ v_1 \cdots v_n = v_1 \cdots v_{j-1} v_{j+1} v_j v_{j+2} \cdots v_n$.
%It is known that $\ell(w)$ is equal
%to $\inv(w)$, the number of inversions 
%in the one-line notation $w_1 \cdots w_n$ of $w$.
%We call a linear function $\theta: H_n(q) \rightarrow \zqq$ 
%a {\em Hecke algebra trace} if it satisfies $\theta(gh) = \theta(hg)$ 
%for all $g, h \in H_n(q)$.  
%Such functions form a free $\zqq$-module of rank equal to the number of
%partitions of $n$, and include the characters of all finite representations 
%of $H_n(q)$.

Define the \emph{quantum matrix bialgebra} (See, e.g.,~\cite{ManinQNoncommG}) $\A = \anq$
to be the associative algebra with unit $1$ generated over $\zqq$
by $n^2$ variables $x=(x_{1,1},\dots,x_{n,n})$,
subject to the relations
\begin{equation}\label{eq:anqdef}
\begin{alignedat}{2}
x_{i,\ell}x_{i,k} &= \qp12x_{i,k} x_{i,\ell}, &\qquad 
x_{j,k} x_{i,\ell} &= x_{i,\ell}x_{j,k},\\
x_{j,k}x_{i,k} &= \qp12x_{i,k} x_{j,k}, &\qquad 
x_{j,\ell} x_{i,k} &= x_{i,k} x_{j,\ell} + (\qdiff) x_{i,\ell}x_{j,k},
\end{alignedat}
\end{equation}
for all indices $ 1 \leq i < j \leq n $ and 
$ 1 \leq k < \ell \leq n $.
The counit map $\varepsilon(x_{i,j}) = \delta_{i,j}$, 
and coproduct map
$\Delta(x_{i,j}) = \sum_{k=1}^n x_{i,k} \otimes x_{k,j}$
%\end{equation*}
%The counit and coproduct maps
%\begin{equation*}
%\varepsilon(x_{i,j}) = \delta_{i,j}, \qquad 
%\Delta(x_{i,j}) = \sum_{k=1}^n x_{i,k} \otimes x_{k,j}
%\end{equation*}
give $\A$ a bialgebra structure.  
While not a Hopf algebra, 
%two Hopf algebras 
$\A$ is closely related to 
the quantum group
%quantum coordinate ring of $SL_n(\mathbb C)$,
%and $GL_n(\mathbb C)$,
%Each of these is isomorphic to a quotient of $\A$,
%\begin{equation*}
$\oqslnc \cong \mathbb C \otimes \A/(\qdet(x)-1)$,
%\qquad
%\oqglnc \cong \mathbb C \otimes \A[t]/(\qdet(x)t-1),
%\end{equation*}
where
\begin{equation}\label{eq:qdetdef}
\qdet(x) \defeq \sum_{v \in \sn} (-\qm 12)^{\ell(v)} \permmon xv
= \sum_{v \in \sn} (-\qm 12)^{\ell(v)} x_{v_1,1} \cdots x_{v_n,n}
\end{equation}
is the ($n \times n$) {\em quantum determinant} of the matrix $x = (x_{i,j})$.  
(The second equality holds in $\A$ but not in the noncommutative ring
$\zqq\langle x_{1,1}, \dotsc, x_{n,n} \rangle$.)
%, which follows from the relations (\ref{eq:anqdef}),
%does not hold in an arbitrary noncommutative ring in $n^2$ variables.)
The antipode map of this Hopf algebra is 
%\begin{equation*}
$\mathcal S (x_{i,j}) = (-\qp12)^{j-i} \qdet(x_{[n]\ssm\{j\}, [n]\ssm \{i\}})$,
%\qquad
%\mathcal S (x_{i,j}) = 
%\frac{(-\qp12)^{j-i}\qdet(x_{[n]\ssm\{j\}, [n]\ssm \{i\}})}{\qdet(x)},
%\end{equation*}
%respectively,
where
%we define
\begin{equation}\label{eq:submatrixdef}
%  \begin{gathered}
[n] \defeq \{ 1, \dotsc, n \}, \qquad
x_{L,M} \defeq (x_{\ell,m})_{\ell \in L, m \in M},
\end{equation}
%and where $x_{I,J}$
%$x_{[n] \ssm \{j\}, [n] \ssm \{i\} }$ is the $(n-1) \times (n-1)$
%submatrix of $x$ obtained by deleting row $j$ and column $i$,
and %the $(n-1) \times (n-1)$ quantum determinant
$\qdet(x_{L,M})$ is defined analogously to (\ref{eq:qdetdef}),
assuming $|L| = |M|$.
%in the obvious way.  
%($\star$ Cite Manin? or Soibelman and Vaksman?)
%More generally, for any row subset $I$ and column subset $J$,
%we will denote the $I,J$ submatrix of $x$ by $x_{I,J}$.
Specializing $\A$ at $\qp12 = 1$, we obtain 
%Notice that $ \mathcal{A}(n;1) $ is
the commutative 
%polynomial 
ring $\mathbb{Z}[x_{1,1}, \dotsc, x_{n,n}]$.
%While $\anq$ is not a Hopf algebra,

$\A$ has a natural $\zqq$-basis 
$\{ x_{1,1}^{a_{1,1}} \cdots x_{n,n}^{a_{n,n}} \,|\, 
a_{1,1}, \dotsc, a_{n,n} \in \mathbb{N} \}$
of monomials in which variables 
appear in lexicographic order, and 
%  We can use
the relations (\ref{eq:anqdef})
provide an algorithm for expressing any other
% any non-lexicographically ordered 
monomial in terms of this basis.
%(See, e.g., \cite[Lem.\,2.1]{ZhangHowe}.)
%We can use the relations (\ref{eq:anqdef}) to express any monomial as a linear
%combination of monomials in lexicographic order, which we shall call
%\emph{standard}. ($\star$ Do we actually use that term later, or not?)
%Thus as a $\mathbb{C}[\qp12, \qm12]$-module, 
%$\anq$ is spanned by monomials whose variables appear in lexicographic order. 
%$\A$ has a natural grading by degree,
%\begin{equation}
%\A = \bigoplus_{r \geq 0} \A_r,
%\end{equation}
%where $\A_r$
%consists of polynomials which are homogeneous of degree $r$.
%We may further decompose each
%homogeneous component 
%$\arnq$ by considering pairs $(L,M)$ of multisets of $r$
%integers, written as weakly increasing words
%\begin{equation*}
%L = \ell_1 \cdots \ell_r, \quad M = m_1 \cdots m_r.
%\end{equation*}
%This leads to the multigrading
%\begin{equation}
%\A = \bigoplus_{r \geq 0} \bigoplus_{L,M} \ALM,
%\end{equation}
%where $\A_{L,M}$ is the $\zqq$-span of monomials whose row indices and
%column indices (with multiplicity) are equal to the multisets $L$
%and $M$, respectively. 
The submodule $\annnq$
%Thus the multigraded component $\annnq$ is
spanned by the monomials
%\begin{equation}
$\{ x^{u,v} \defeq x_{u_1,v_1} \cdots x_{u_n,v_n} \mid u,v \in \sn \}$
%\{x_{1,w_1} \cdots x_{n,w_n} \mid w \in \sn \}.
%\end{equation}
%is called the {\em zero-weight space} of $\A$ and
%By (\ref{eq:anqdef})
%By, e.g., \cite{Brundan}
%This component 
has rank $n!$ and natural basis $\{ x^{e,w} \,|\, w \in \sn \}$.
%More generally, $\ALM$ is 
%spanned by the monomials
%\begin{equation}\label{eq:natspan}
%%\{x_{\ell_1,m_{w_1}} \cdots x_{\ell_r,m_{w_r}} \mid w \in \sr \} 
%\{x_{\ell_{u_1},m_{v_1}} \cdots x_{\ell_{u_r},m_{w_r}} \mid u,v \in \sr \} 
%%= \{(x_{L,M})^{e,w} \mid w \in \sr \},
%= \{(x_{L,M})^{u,v} \mid u,v \in \sr \},
%\end{equation}
%where $x_{L,M}$ denotes the {\em generalized submatrix}
%%$ x$ is defined by
%\begin{equation}\label{eq:gensubmatrix}
%x_{L,M} =
%\begin{bmatrix}
%x_{\ell_1,m_1} & x_{\ell_1,m_2} & \cdots & x_{\ell_1,m_r} \\
%x_{\ell_2,m_1} & x_{\ell_2,m_2} & \cdots & x_{\ell_2,m_r} \\
%\vdots & \vdots & & \vdots \\
%x_{\ell_r,m_1} & x_{\ell_r,m_2} & \cdots & x_{\ell_n,m_r}
%\end{bmatrix}
%\end{equation}
%of $x$ determined by the multisets $L$, $M$.
%We may express a natural basis of $\ALM$ in terms of double cosets of the form
%$W_IwW_J$ where
%\begin{equation}\label{eq:IJLM}
%I = \{ s_i \mid \ell_i = \ell_{i+1} \},
%\quad
%J = \{ s_i \mid m_i = m_{i+1} \},
%\end{equation}
%and $W_I$, $W_J$ are the subgroups of $\sr$ generated by $I$, $J$, respectively.
%In particular, 
%the set 
%$\{ (x_{L,M})^{e,v} \mid v \in \Wijp \}$ is a natural basis for $\ALM$.
%%($\star$ necessary?) Let us call $\{ (x_{L,M})^{e,v} \mid v \in ? \}$ the
%%{\em natural} basis of $\almnq$.
%%($\star$ Justify this later?  Better here?)
%%($\star$ Use overkill to explain when monomials are equal.  Then justify
%%linear independence.)
%(See, e.g., \cite[Lem.\,2.1]{ZhangHowe} for more information.)
%\section{Monomial bases of the zero-weight space}\label{s:mbases}
%
%Let us first consider bases of the graded component $\Ann$,
%called the \emph{zero-weight space}
%Let us also consider other monomials in $\Ann$.
%We will call elements of
%this component
%\emph{(quantum) immanants} and we will call the 
%component
%itself the
%or \emph{immanant space}
%of $\A$. 
%We will find it convenient to define the notation
%\begin{equation}
%%\begin{gathered}
%%{[n]} \defeq \{1, \dotsc, n \},\\
%\euv \defeq (-1)^{\ell(v) - \ell(u)},\qquad
%\quv \defeq (\qp12)^{\ell(v) - \ell(u)},
%%\end{gathered}
%\end{equation}
%where $\ell$ is the length function on $\sn$.
%($\star$ necessary? $\quv$ is defined earlier, and is $\euv$ even used in
%this paper?)

%: We combinatorially evaluate several families of Hecke algebra
%characters at certain Kazhdan-Lusztig basis elements $C'_w(q)$
%and products of these.  Combinations include the evaluation of induced
%sign characters at products $C'_{w_{(1)}}(q) \cdots C'_{w^{r}}(q)$
%indexed by maximal permutations of parabolic subgroups of the symmetric group,
%the evaluation of skew characters indexed by ribbon shapes 
%elements of $\epsilon_q^\lambda(\qp{\ell(w)}2 C'_w(q))$

%The Hecke algebra $H_n(q)$ of type A is a noncommutative ring
%which has dimension $n!$ as a $\mathbb{Z}[q]$-module.  
%, form a $\mathbb Z[q]$-module $\mathcal T$  having dimension 
%equal to the number of integer partitions of $n$.
%We will choose a basis $B$ of $\mathcal T$ and a spanning set $C$ of $H_n(q)$,
%and will give a combinatorial formula for $\theta(g)$ for all $\theta \in B$ and
%$g \in C$.  This is the first subtraction-free result of its kind.

%\section{Earlier}
%Call sequence $(v^{(1)}, \dotsc, v^{(m)})$ a {\em generalized subexpression}
%  of sequence $(w^{(1)}, \dotsc, w^{(m)})$ if we have $v^{(i)} \leq w^{(i)}$
%  for all $i$.

%Define $\wtc wq = q_w C'_w(q)$.

%Define $\mfsmin{[i,j]}$ to be the set of minimum length representatives
%of the cosets $w \mfs{[i,j]}$, i.e., the elements $w \in \sn$ satisfying
%$ws > w$ for all $s \in \mfs{[i,j]}$.
%Fact: For $u \in \mfs{[i,j]}$ we have
%$T_u \wtc{s_{[i,j]}\ntnsp}q = q^{\ell(u)} \wtc{s_{[i,j]}\ntnsp}q.$
%Let $w$ factor as $w^- w^*$ with $w^- \in \mathfrak S^{[i,j]}_-$ and
%$w* \in \mfs{[i,j]}$, and $\ell(w^-) + \ell(w*) = \ell(w)$.
%Then we have
%\begin{equation}\label{eq:tcprod}
%  T_w \wtc{s_{[i,j]}\ntnsp}q = T_{w^-} T_{w^*} \wtc{s_{[i,j]}}q
%  = q^{\ell(w^*)} T_{w^{-}} \wtc{s_{[i,j]}}q.
%  \end{equation}

%$\star$ Define $\anq$ and $\annnq$. (Is the second one necessary?)
%$\star$ Define $\qdet$.
%$\star$ Define notation $x^{e,w}$, $x^{u,v}$.

%In the quantum matrix bialgebra we have
%\begin{equation}\label{eq:xuvew}
%  x^{u,v} = x_{e,u^{-1}v} + \sum_{w > u^{-1}v} r_{u,v,e,w}(\qdiff)x^{e,w}.
%\end{equation}

%\section{Star networks and the $q$-immanant evaluation theorem}\label{s:snet}

To evaluate
induced sign characters at elements
$\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q$
%$(1+T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}})$
of $\hnq$,
%having the form (\ref{eq:wdelt}),
we will associate to each such element
%represent ($\star$ or better word?)
%the elements by
a graph 
%$G$ 
called a {\em star network},
%{\em wiring diagram},
a related matrix $B$,
%to represent
%
%($\star$ something like) To each wiring diagram we associate a matrix
%$B$
and a map $\sigma_B: \Ann \rightarrow \zqq$.
A generating function $\imm{\epsilon_q^\lambda}(x) \in \Ann$ for
$\{\epsilon_q^\lambda(T_w) \,|\, w \in \sn \}$ will then allow us to compute
%satisfy
\begin{equation}\label{eq:evalwithsigma}
%  \epsilon_q^\lambda((1+T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}}))
\epsilon_q^\lambda( \wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)
= \sigma_B(\imm{\epsilon_q^\lambda}(x))
\end{equation}
and to combinatorially interpret the resulting polynomial.
%and will yield the desired interpretation in terms of $\lambda$ and $G$.

%\ssec{Star networks and the classical immanant evaluation identity}

%($\star$ Instead of doing a classical version and a quantum version,
%just do the quantum version?)

%Define a {\em simple star diagram} (of order $n$) to be a planar directed
%graph with

For $1 \leq a < b \leq n$,
%$[a,b]$ a nontrivial subinterval of $[n]$,
% = \{1, \dotsc, n\}$, 
let $G_{[a,b]}$ be the directed planar graph
on $2n+1$ vertices defined as follows.
\begin{enumerate}
	\item In a column on the left, $n$ vertices are labeled
	{\em source $1$} $, \dotsc, $ {\em source $n$}, from bottom to top;
%	\item A column of 
in a column on the right, $n$ more vertices are labeled
	{\em sink $1$} $, \dotsc, $ {\em sink $n$}, from bottom to top.
	\item For $i = 1, \dotsc, a-1$ and $i = b+1, \dotsc, n$ a directed edge
	begins at source $i$ and terminates at sink $i$.
	\item An interior vertex is placed between the sources and sinks.  
For $i = a, \dotsc, b$, a directed edge begins at source $i$ and
	terminates at the interior vertex, and another directed edge begins
	at the interior vertex and terminates at sink $i$.
\end{enumerate}
For $a = 1,\dotsc, n$ we define $G_{[a,a]}$
to be the similar directed planar graph on $n$ sources and
$n$ sinks, with one edge from source $i$ to sink $i$ for $i = 1, \dotsc, n$.
Call each of the above graphs a {\em simple star network}.
Define a {\em star network} to be the concatenation of finitely many
simple star networks.
We write
$G \circ H$ for the network in which
sink $i$ of $G$ is identified with source $i$ of $H$, for $i = 1,\dotsc, n$.
In figures we will not explicitly draw vertices or show edge orientations
(assumed to be from left to right).
For $n = 4$, there are seven simple star networks:
$G_{[1,4]}$, 
$G_{[2,4]}$, 
$G_{[1,3]}$, 
$G_{[3,4]}$, 
$G_{[2,3]}$, 
$G_{[1,2]}$, 
%and 
$G_{[1,1]} = \cdots = G_{[4,4]}$.
Drawing these and two more star networks
$G_{[1,2]} \circ G_{[2,4]}\circ G_{[1,2]}$ and
$G_{[2,4]}\circ G_{[1,3]}$, 
we have
%respectively,
%($\star$ Make tikz figures for these.)
\begin{equation}\label{eq:simplestarnets}
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,-3);
\draw[-] (0,-1) -- (1,-2);
\draw[-] (0,-2) -- (1,-1);
\draw[-] (0,-3) -- (1,0);
\end{tikzpicture}
\;, \quad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,-2);
\draw[-] (0,-1) -- (1,-1);
\draw[-] (0,-2) -- (1,0);
\draw[-] (0,-3) -- (1,-3);
\end{tikzpicture}
\;, \quad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,0);
\draw[-] (0,-1) -- (1,-3);
\draw[-] (0,-2) -- (1,-2);
\draw[-] (0,-3) -- (1,-1);
\end{tikzpicture}
\;, \quad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,-1);
\draw[-] (0,-1) -- (1,0);
\draw[-] (0,-2) -- (1,-2);
\draw[-] (0,-3) -- (1,-3);
\end{tikzpicture}
\;, \quad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,0);
\draw[-] (0,-1) -- (1,-2);
\draw[-] (0,-2) -- (1,-1);
\draw[-] (0,-3) -- (1,-3);
\end{tikzpicture}
\;, \quad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,0);
\draw[-] (0,-1) -- (1,-1);
\draw[-] (0,-2) -- (1,-3);
\draw[-] (0,-3) -- (1,-2);
\end{tikzpicture}
\;, \quad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,0);
\draw[-] (0,-1) -- (1,-1);
\draw[-] (0,-2) -- (1,-2);
\draw[-] (0,-3) -- (1,-3);
\end{tikzpicture}
\;; \qquad \qquad
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,0) -- (2,-2) -- (3,-3);
\draw[-] (0,-1) -- (3,-1);
\draw[-] (0,-2) -- (1,-3) -- (2,-3) -- (3,-2);
\draw[-] (0,-3) -- (1,-2) -- (2,0) -- (3,0);
\end{tikzpicture}\;, \quad
% G_{[2,4]}\circ G_{[1,3]}= 
\begin{tikzpicture}[scale=.4,baseline=-20]
\draw[-] (0,0) -- (1,-2) -- (2,-2);
\draw[-] (0,-1) -- (1,-1) -- (2,-3);
\draw[-] (0,-2) -- (1,0) -- (2,0);
\draw[-] (0,-3) -- (1,-3) -- (2,-1);
\end{tikzpicture}\;.
\end{equation}

%\begin{equation*}
%\raisebox{-6mm}{
%\includegraphics[height=12mm]{xfigures/stars}}.
%%\includegraphics[width=50mm]{xfigures/stars.eps}}.
%\end{equation*}

%to be the concatenation of $G$ and $H$, 
%where for $i = 1,\dotsc, n$, sink $i$ of $G$ is dropped,
%source $i$ of $H$ is dropped, and 
%the unique edge in $G$ from vertex $x$ to sink $i$ 
%and the unique edge in $H$ from source $i$ to vertex $y$ 
%are merged to form a single edge from $x$ to $y$ in $G \circ H$.
%Note that for star networks $G_{[c_1,d_1]}$, $G_{[c_2,d_2]}$ 
%indexed by nonintersecting intervals, the two concatenations
%$G_{[c_1,d_1]} \circ G_{[c_2,d_2]}$ and 
%$G_{[c_2,d_2]} \circ G_{[c_1,d_1]}$ are isomorphic.
% as directed graphs.
%For example, when $n = 4$ we have
%\begin{equation}\label{eq:starnetsexample}
%G_{[1,2]} \circ G_{[2,4]}\circ G_{[1,2]}= 
%\begin{tikzpicture}[scale=.4,baseline=-20]
%\draw[-] (0,0) -- (1,0) -- (2,-2) -- (3,-3);
%\draw[-] (0,-1) -- (3,-1);
%\draw[-] (0,-2) -- (1,-3) -- (2,-3) -- (3,-2);
%\draw[-] (0,-3) -- (1,-2) -- (2,0) -- (3,0);
%\end{tikzpicture}\;, \qquad
% G_{[2,4]}\circ G_{[1,3]}= 
%\begin{tikzpicture}[scale=.4,baseline=-20]
%\draw[-] (0,0) -- (1,-2) -- (2,-2);
%\draw[-] (0,-1) -- (1,-1) -- (2,-3);
%\draw[-] (0,-2) -- (1,0) -- (2,0);
%\draw[-] (0,-3) -- (1,-3) -- (2,-1);
%\end{tikzpicture}\;.
%\end{equation}

Let $\pi = (\pi_1,\dotsc,\pi_n)$ be a sequence of source-to-sink paths
in a
%an $\sn$
star network $G$.  We call $\pi$
a {\em path family} if there exists a permutation
$w = w_1 \cdots w_n \in \sn$ such that 
$\pi_i$ is a path from source $i$ to sink $w_i$.
In this case, we say more specifically that $\pi$ has {\em type $w$}.
We say that the path family {\em covers} $G$ if it contains every edge
exactly once.

One can enhance a star network by
%One often enhances a planar network by
associating to each edge
%we will place {\em weights} on their edges.
%To each edge of a planar network we associate
a {\em weight} belonging to some ring $R$, and by defining
the {\em weight of a path} to be the product of its edge weights.
If $R$ is noncommutative,
%then weights are multiplied
then one multiplies weights
%we will multiply weights
in the order that the corresponding edges appear in the path.
For a {\em family} $\pi = (\pi_1,\dotsc,\pi_n)$ of $n$
paths in a planar network,
one defines
%we will define
$\wgt(\pi) = \wgt(\pi_1) \cdots \wgt(\pi_n)$. 
%We will say that such a family {\em covers} the wiring diagram $G$
%if it contains every edge of $G$ exactly once.
%We will say that $\pi$ has {\em type $w \in \sn$} if
%$\pi_i$ is a path from source $i$ to sink $w_i$ for all $i$.
The {\em (weighted) path matrix} $B = B(G) = (b_{i,j})$ of $G$ is defined by
letting $b_{i,j}$ be the sum of weights of all paths in $G$ from source $i$
to sink $j$.
Thus the product $\permmon bw$ is equal to the sum of weights of all
path families of type $w$ in $G$ (covering $G$ or not).
%It is easy to show that path matrices respect concatenation:
%$B(G_1 \circ G_2) = B(G_1)B(G_2)$.
%When $R$ is commutative, a result known as
%{\em Lindstr\"om's Lemma}~\cite{KMG}, \cite{LinVRep}
%asserts that for row and column sets $I$, $J$ with $|I| = |J|$,
%the minor $\det(B_{I,J})$ is equal to the sum of weights of all nonintersecting
%path families from sources indexed by $I$ to sinks indexed by $J$.

Assigning weights to the edges of $G = G_{J_1} \circ \cdots \circ G_{J_m}$
%$G$ (\ref{eq:wdnetwork})
can aid in
the evaluation of a linear function $\theta_q: \hnq \rightarrow \zqq$ at
%$\theta: \zsn \rightarrow \mathbb Z$ at
%$(1 + s_{i_1}) \cdots (1 + s_{i_m})$
%$\wtc{s_{J_1}\ntnsp}1 \cdots \wtc{s_{J_m}\ntnsp}1$
$\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q$
by relating this evaluation to the generating function
%\begin{equation}\label{eq:imm}
%  \imm{\theta}(x) \defeq \sum_{w \in \sn} \theta(w) \permmon xw
%  \in \zx,
%\end{equation}
\begin{equation}\label{eq:qimm}
\imm{\theta_q}(x) := \sum_{w \in \sn} \qiew \theta_q(T_w) \permmon xw \in \anq,
\end{equation}
%in $\anq$,
which specializes at $\qp12 = 1$ to the
generating function $(\ref{eq:imm})$ in $\zx$.
%called the {\em $\theta$-immanant} in \cite[Sec.\,3]{StanPos}.
In particular, write $G_{J_p} = G_{[i_p,j_p]}$ and
let
$\{ z_{h,p,k} \,|\, 1 \leq p \leq m; i_p \leq h \leq j_p; 1 \leq k \leq 2 \}$
be indeterminate weights satisfying
\begin{equation}\label{eq:quasicommz}
z_{h_2,p_2,k_2} z_{h_1,p_1,k_1} = \begin{cases}
z_{h_1,p_1,k_1} z_{h_2,p_2,k_2} &\text{if $p_1 \neq p_2$, or $k_1 \neq k_2$,}\\
\qp12 z_{h_1,p_1,k_1} z_{h_2,p_2,k_2} &\text{if $p_1 = p_2$, $k_1 = k_2$, and $h_1 < h_2$.}
\end{cases}
\end{equation}
We assign weights to the edges of $G_{J_p}$ as follows.
\begin{enumerate}
	\item Assign weight $1$ to the $n - j_p + i_p - 1$ edges not incident
	upon the central vertex.
        % of $G_{J_p}$.
	%receive weight $1$.
	\item Assign weights $z_{i_p,p,1}, z_{i_p+1,p,1}, \dotsc, z_{j_p,p,1}$,
	to the $j_p - i_p + 1$ edges entering the central vertex,
	%of $G_{[i_p,j_p]}$
	from bottom to top.
	\item Assign weights $z_{i_p,p,2}, z_{i_p+1,p,2}, \dotsc, z_{j_p,p,2}$,
	to the $j_p - i_p + 1$ edges leaving the central vertex,
	%of $G_{[i_p,j_p]}$
	%receive weights $z_{i_p,p,2}, z_{i_p+1,p,2}, \dotsc, z_{j_p,p,2}$,
	from bottom to top.
\end{enumerate}
%for $p = 1,\dotsc, m$ and $k = 1,2$,
%and
%Thus in the star network
%$G = G_{J_1} \circ \cdots \circ G_{J_m}$,
%the $p$th simple star network $G_{J_p} = G_{[i_p,j_p]}$ is weighted as
%in the diagram
%%Omitting weights of $1$, we have
%\begin{equation}\label{eq:onestar}
%\raisebox{1.25cm}{
%  \begin{tikzpicture}[scale=.7,baseline=-25]
%  \draw[-] (0,2.4) -- (2.4,2.4); % node[above,midway,]{1};
%%      \draw[-] (0,2.4) -- (2.4,2.4) node[below,midway, xshift = -4mm]{1};
%  \draw[-,fill] (0,2.4) circle (2pt);  \draw[-,fill] (2.4,2.4) circle (2pt);
%  \draw[-, fill] (1.2,1.2) circle (1pt); \draw[-, fill] (1.2,1.5) circle (1pt);% \draw[-, fill] (1.2,1.8) circle (1pt);    
%  \draw[-] (0,0.6) -- (2.4,0.6); % node[below,midway]{1};
%  \draw[-,fill] (0,0.6) circle (2pt);  \draw[-,fill] (2.4,0.6) circle (2pt);
%%  
%  \draw[-] (0,0) -- (1.2,-1.5) node[left,midway] {$z_{j_p,p,1}$};
%  \draw[-,fill] (0,0) circle (2pt);  \draw[-,fill] (0,-3) circle (2pt);
%  \draw[-, fill] (0.5,-1.2) circle (1pt); \draw[-, fill] (0.5,-1.5) circle (1pt%); \draw[-, fill] (0.5,-1.8) circle (1pt);
%  \draw[-] (0,-3) -- (1.2,-1.5) node[left,midway] {$z_{i_p,p,1}$};
%  \draw[-,fill] (1.2,-1.5) circle (2pt);
%  \draw[-] (2.4,0) -- (1.2,-1.5) node[right,midway] {$z_{j_p,p,2}$};
%  \draw[-, fill] (1.9,-1.2) circle (1pt); \draw[-, fill] (1.9,-1.5) circle (1pt%); \draw[-, fill] (1.9,-1.8) circle (1pt);  
%  \draw[-] (2.4,-3) -- (1.2,-1.5) node[right,midway] {$z_{i_p,p,2}$};
%  \draw[-,fill] (2.4,0) circle (2pt);  \draw[-,fill] (2.4,-3) circle (2pt);
%%
%  \draw[-] (0,-3.6) -- (2.4,-3.6); % node[above,midway]{1};;
%  \draw[-,fill] (0,-3.6) circle (2pt);  \draw[-,fill] (2.4,-3.6) circle (2pt);
%  \draw[-, fill] (1.2,-4.2) circle (1pt); \draw[-, fill] (1.2,-4.5) circle (1pt%); \draw[-, fill] (1.2,-4.8) circle (1pt);    
%  \draw[-] (0,-5.4) -- (2.4,-5.4); % node[below,midway]{1};;
%  \draw[-,fill] (0,-5.4) circle (2pt);  \draw[-,fill] (2.4,-5.4) circle (2pt);
%\end{tikzpicture}\;},
%%\raisebox{-7mm}
%%{\includegraphics[height=15mm]{starabcd.eps}}
%\end{equation} 
%\begin{equation}\label{eq:onestar}
%\begin{tikzpicture}[scale=.7,baseline=-25]
%  \draw[-] (0,0) -- (1.2,-1.5) node[left,midway] {$z_{j_p,p,1}$};
%  \draw[-] (0,-3) -- (1.2,-1.5) node[left,midway] {$z_{i_p,p,1}$};
%  \draw[-] (2.4,0) -- (1.2,-1.5) node[right,midway] {$z_{j_p,p,2}$};
%  \draw[-] (2.4,-3) -- (1.2,-1.5) node[right,midway] {$z_{i_p,p,1}$};
%  \draw[-] (0.5,-1.2) circle (1pt); \draw[-] (0.5,-1.5) circle (1pt); \draw[-] (0.5,-1.8) circle (1pt); 
%  \draw[-] (1.9,-1.2) circle (1pt); \draw[-] (1.9,-1.5) circle (1pt); \draw[-] (1.9,-1.8) circle (1pt); 
%  \draw[-,fill] (0,0) circle (2pt);  \draw[-,fill] (0,-3) circle (2pt);
%  \draw[-,fill] (2.4,0) circle (2pt);  \draw[-,fill] (2.4,-3) circle (2pt);
%  \draw[-,fill] (1.2,-1.5) circle (2pt);
%\end{tikzpicture}\;.
%\end{equation} 
%\begin{equation}\label{eq:onestar}
%\begin{tikzpicture}[scale=.7,baseline=-25]
%  \draw[-] (0,0) -- (1.2,-1) node[above right,midway,xshift=-1mm,yshift=-1mm] {c};
%  \draw[-] (0,-2) -- (1.2,-1) node[below right,midway,xshift=-1mm,yshift=1mm] {a};
%  \draw[-] (2.4,0) -- (1.2,-1) node[above left,midway,xshift=1mm,yshift=-1mm] {d};
%  \draw[-] (2.4,-2) -- (1.2,-1) node[below left,midway,xshift=1mm,yshift=2mm] {b};
%  \draw[-,fill] (0,0) circle (2pt);  \draw[-,fill] (0,-2) circle (2pt);
%  \draw[-,fill] (2.4,0) circle (2pt);  \draw[-,fill] (2.4,-2) circle (2pt);
%  \draw[-,fill] (1.2,-1) circle (2pt);
%\end{tikzpicture}\;,
%\raisebox{-7mm}
%{\includegraphics[height=15mm]{starabcd.eps}}
%\end{equation}
%with edges $a_h$ and $b_h$ labeled $z_{h,p,1}$ and $z_{h,p,2}$,
%respectively, for $h = i_p,\dotsc,j_p$.
%labeled $a,b,c,d$, respectively.
%\begin{equation}\label{eq:onestar}
%\raisebox{-8mm}
%{\includegraphics[height=20mm]{starabcd.eps}}
%\end{equation} 
%labeled $a,b,c,d$, respectively.
%where we omit weights of $1$ for economy. Thus star networks
%corresponding to expressions
%$s_{J_1} s_{J_2} s_{J_3} = s_{[1,4]}s_{[2,5]}s_{[1,3]}$
%and
%$s_{J_4} s_{J_5} s_{J_6} = s_{[1,4]}s_{[2,5]}s_{[1,3]}$
%%$s_{i_4}s_{i_5}s_{i_6} = s_1s_2s_1$
%are weighted differently
%because of the different indexing of the reversal intervals.
Let $Z_G$ be the quotient of the noncommutative ring
\begin{equation*}
\mathbb Z[\qp12, \qm12]
\langle z_{h_p,p,k}
\,|\, p=1,\dotsc, m; h_p = i_p, \dotsc, j_p; k = 1,2 \rangle
\end{equation*}
modulo the ideal generated by the
%above commuting and quasicommuting
relations (\ref{eq:quasicommz}), and assume that $\qp12$, $\qm12$
commute with all other indeterminates.
Let $z_G$
%$\mathbf{z}$ 
be the product of all indeterminates $z_{h_p,p,k}$, in lexicographic order,
%Let $z_G$
%$\mathbf{z}$ 
%be the product of all indeterminates $z_{i_p,p,k}$,
and for $f \in Z_G$,
%$f \in \mathbb{Z}[z_{1,1,1} ,\dotsc, z_{j_m,m,2}]$,
let $[z_G]f$ denote the coefficient of $z_G$ in $f$.
%The following result was essentially observed by Stembridge in
For example,
%corresponding to
the star network $G_{[2,4]} \circ G_{[1,3]}$
%are the
has weighting
%Its edge weights are
%\begin{equation}\label{eq:Gbraid}
%  \begin{aligned}
%  G = G_{[2,4]} \circ G_{[1,3]}.
%  G_{[i_1,i_1+1]} \circ G_{[i_2,i_2+1]} \circ G_{[i_3,i_3+1]}
%  = G_{[1,2]} \circ G_{[2,3]} \circ G_{[1,2]}.
%\end{equation}
%Assigning weights to the edges of $G$ we have
%($\star$ update with new figure)
\begin{equation}\label{eq:Gbraidfig}
\resizebox{7cm}{!}{
	\raisebox{-2.2cm}{
		\begin{tikzpicture}
		\node at (-4,2) {\large $4$};
		\node at (-4,0) {\large $3$};
		\node at (-4,-2) {\large $2$};
		\node at (-4,-4) {\large $1$};
		\node at (6,2) {\large $4$};
		\node at (6,0) {\large $3$};
		\node at (6,-2) {\large $2$};
		\node at (6,-4) {\large $1$};
		\draw[-] (-3,2) -- (-1,0) node[midway,left,xshift = -1mm] {\large $z_{4,1,1}$}
		-- (1,2) node[midway,right,xshift = 1mm] {\large $z_{4,1,2}$}
		-- (5,2);
		\draw[-] (-3,0)  -- (-1,0) node[midway,above,xshift = -3mm]  {\large $z_{3,1,1}$}
		-- (1,0) node[midway,above,xshift = 3mm]  {\large $z_{3,1,2}$}
		-- (3,-2) node[midway,right,xshift = -2mm,yshift = 2mm] {\large $z_{3,2,1}$}
		-- (5,0)  node[midway,right] {\large $z_{3,2,2}$};
		\draw[-] (-3,-2) -- (-1,0) node[midway,left] {\large $z_{2,1,1}$}
		-- (1,-2) node[midway,left,xshift = 2mm, yshift = -2mm] {\large $z_{2,1,2}$}
		-- (3,-2) node[midway,below,xshift = -3mm] {\large $z_{2,2,1}$}
		-- (5,-2) node[midway,below,xshift = 3mm] {\large $z_{2,2,2}$};
		\draw[-] (-3,-4) -- (1,-4) 
		-- (3,-2) node[midway,left,yshift =-1mm] {\large $z_{1,2,1}$}
		-- (5,-4) node[midway,right,yshift = -1mm] {\large $z_{1,2,2}$};
		\filldraw (-3,0) circle (.7mm);
		\filldraw (1,0) circle (.7mm);
		\filldraw (5,0) circle (.7mm);
		\filldraw (-3,2) circle (.7mm);
		\filldraw (1,2) circle (.7mm);
		\filldraw (5,2) circle (.7mm);
		\filldraw (-3,-2) circle (.7mm);
		\filldraw (1,-2) circle (.7mm);
		\filldraw (5,-2) circle (.7mm);
		\filldraw (-3,-4) circle (.7mm);
		\filldraw (1,-4) circle (.7mm);
		\filldraw (5,-4) circle (.7mm);
		\filldraw (-1,0) circle (.7mm);
		\filldraw (3,-2) circle (.7mm);            
		\end{tikzpicture}%.
}}
%\]
\end{equation}
and monomial $z_G = z_{1,2,1} z_{1,2,2} z_{2,1,1} z_{2,1,2} z_{2,2,1} z_{2,2,2} z_{3,1,1} z_{3,1,2} z_{3,2,1} z_{3,2,2} z_{4,1,1} z_{4,1,2}$.

To complete the description of (\ref{eq:evalwithsigma}),
we define a map which allows us to evaluate a linear functional $\theta_q$
on certain $\hnq$ elements via the corresponding immanant
$\imm{\theta_q}(x) \in \anq$.
Given matrix $B \in \mat{n \times n}(Z_G)$, 
%define $\sigma_B$ to be the $\zqq$-linear map
let $\sigma_B$ be the $\zqq$-linear map
\begin{equation}\label{eq:sigmadef}
\begin{aligned}
%  \sigma_B: \anq_{[n],[n]} &\rightarrow \zqq\\
\sigma_B: \annnq &\rightarrow \zqq\\
\permmon xv &\mapsto [z_G] \permmon bv,
\end{aligned}
\end{equation}
where $[z_G] \permmon bv$ denotes the coefficient of $z_G$ in $\permmon bv$,
%and this coefficient is
taken after $\permmon bv$ 
is expanded in the lexicographic basis of $Z_G$.
Note that the ``substitution'' $x_{i,j} \mapsto b_{i,j}$ is performed
only for monomials of the form $x^{e,v}$ in $\Ann$:
we define $\sigma_B(x^{u,w})$ by first expanding $x^{u,w}$
in the basis $\{ x^{e,v} \,|\, v \in \sn \}$, and {\em then} performing
the substitution.
Now we have the following immanant evaluation identity
for star networks (cf.\,\cite[Thm.\,3.7]{KLSBasesQMBIndSgn}).
%(cf.\,\cite[p.\,1081]{StemConj}).
\begin{thm}\label{t:qstem}
	Assign weights to the edges of $G = G_{J_1} \circ \cdots \circ G_{J_m}$
	%  $G$ (\ref{eq:wdnetwork})
	as above
	and let $B$ be the resulting path matrix.
	Then for any linear function $\theta_q: \hnq \rightarrow \zqq$ we have
	\begin{equation}\label{eq:qstem}
	\theta_q(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)
	= [z_G]\sigma_B(\imm{\theta_q}(x)).
	\end{equation}
\end{thm}
\begin{proof}
	Omitted.
\end{proof}
%\begin{prop}\label{p:stem}
%  Assign weights to the edges of $G = G_{J_1} \circ \cdots \circ G_{J_m}$
%  %  $G$ (\ref{eq:wdnetwork})
%  as above
%and let $B$ be the resulting path matrix.
%Then for any linear function $\theta: \zsn \rightarrow \mathbb Z$ we have
%\begin{equation}\label{eq:stem}
%\theta(\wtc{s_{J_1}\ntnsp}1 \cdots \wtc{s_{J_m}\ntnsp}1) = [z_G]\imm{\theta}(B).
%\end{equation}
%\end{prop}

%($\star$ Redo this with $q$.)
To illustrate, we let $n = 4$ and consider the element
\begin{equation}\label{eq:sigmaexample}
%  \wtc{s_{[2,4]}}(1) \wtc{s_{[1,3]}}(1) = 2 \sum_{w \leq s_2s_3s_2s_1s_2} w
\wtc{s_{[2,4]}\ntnsp}q \wtc{s_{[1,3]}\ntnsp}q = (1+q) \sum_{w \leq 3421} T_w
\end{equation}
of $H_4(q)$.  Its star network (\ref{eq:Gbraidfig})
%wiring diagram
%the network
%in (\ref{eq:wdnetwork}) be
%The
has weighted path matrix
%of $G$ is
%has four entries consisting of two terms,
%$(2,2)$-entry equal to
%\begin{equation}\label{eq:B1212}
%  \begin{gathered}
%  b_{1,1} = z_{1,1,1}z_{1,1,2}z_{1,3,1}z_{1,3,2}
%  + z_{1,1,1}z_{2,1,2}z_{2,2,1}z_{2,2,2}z_{2,3,1}z_{1,3,2}, \\
%  b_{1,2} = z_{1,1,1}z_{1,1,2}z_{1,3,1}z_{2,3,2}
%  + z_{1,1,1}z_{2,1,2}z_{2,2,1}z_{2,2,2}z_{2,3,1}z_{2,3,2}, \\
%  b_{2,1} = z_{2,1,1}z_{1,1,2}z_{1,3,1}z_{1,3,2}
%  + z_{2,1,1}z_{2,1,2}z_{2,2,1}z_{2,2,2}z_{2,3,1}z_{1,3,2}, \\
%  b_{2,2} = z_{2,1,1}z_{1,1,2}z_{1,3,1}z_{2,3,2}
%  + z_{2,1,1}z_{2,1,2}z_{2,2,1}z_{2,2,2}z_{2,3,1}z_{2,3,2},
%  \end{gathered}
%\end{equation}
\begin{equation}\label{eq:Bex}
%  \begin{aligned}
%    B &=
%  \begin{bmatrix}
%    1 & 0              & 0              & 0               \\
%    0 & z_{2,1,1}z_{2,1,2} & z_{2,1,1}z_{3,1,2} & z_{2,1,1}z_{4,1,2} \\
%    0 & z_{3,1,1}z_{2,1,2} & z_{3,1,1}z_{3,1,2} & z_{3,1,1}z_{4,1,2} \\
%    0 & z_{2,1,1}z_{2,1,2} & z_{2,1,1}z_{3,1,2} & z_{2,1,1}z_{4,1,2} 
%  \end{bmatrix}\ntksp\ntksp
%  \begin{bmatrix}
%    z_{1,2,1}z_{1,2,2} & z_{1,2,1}z_{2,2,2} & z_{1,2,1}z_{3,2,2} & 0 \\
%    z_{2,2,1}z_{1,2,2} & z_{2,2,1}z_{2,2,2} & z_{2,2,1}z_{3,2,2} & 0 \\
%    z_{3,2,1}z_{1,2,2} & z_{3,2,1}z_{2,2,2} & z_{3,2,1}z_{3,2,2} & 0 \\
%    0              & 0              & 0               & 1 
%  \end{bmatrix}\\
%  &= \begin{bmatrix}
B = \begin{bmatrix}
z_{1,1,1}z_{1,2,2} & z_{1,2,1}z_{2,2,2} & z_{1,2,1}z_{3,2,2} & 0 \\
z_{2,1,1}(z_D + z_U)z_{1,2,2}
& z_{2,1,1}(z_D + z_U)z_{2,2,2}
& z_{2,1,1}(z_D + z_U)z_{3,2,2}
& z_{2,1,1}z_{4,1,2} \\
%    & & & \\
%    \begin{matrix} z_{2,1,1}z_{2,1,2}z_{2,2,1}z_{1,2,2} \\
%      + z_{2,1,1}z_{3,1,2}z_{3,2,1}z_{1,2,2}\end{matrix}
%    &\begin{matrix} z_{2,1,1}z_{2,1,2}z_{2,2,1}z_{1,2,2} \\
%      + z_{2,1,1}z_{3,1,2}z_{3,2,1} z_{1,2,2}\end{matrix}
%    &\begin{matrix} z_{2,1,1}z_{2,1,2}z_{2,2,1}z_{1,2,2} \\
%       + z_{2,1,1}z_{3,1,2}z_{3,2,1}z_{1,2,2}\end{matrix}
%    & z_{2,1,1}z_{4,1,2} \\
%    & & & \\
%    & z_{2,1,1}(z_{2,1,2}z_{2,2,1} + z_{3,1,2}z_{3,2,1})z_{2,2,2}
%    & z_{2,1,1}(z_{2,1,2}z_{2,2,1} + z_{3,1,2}z_{3,2,1})z_{3,2,2}
z_{3,1,1}(z_D + z_U)z_{1,2,2}
& z_{3,1,1}(z_D + z_U)z_{2,2,2}
& z_{3,1,1}(z_D + z_U)z_{3,2,2}
& z_{3,1,1}z_{4,1,2} \\
z_{4,1,1}(z_D + z_U)z_{1,2,2}
& z_{4,1,1}(z_D + z_U)z_{2,2,2}
& z_{4,1,1}(z_D + z_U)z_{3,2,2}
& z_{4,1,1}z_{4,1,2} 
\end{bmatrix},
%  \end{aligned}
\end{equation}
where
$z_D = z_{2,1,2}z_{2,2,1}$,
$z_U = z_{3,1,2}z_{3,2,1}$.
Now consider the linear function
%$\theta: \mathbb Z[\mfs 4] \rightarrow \mathbb Z$
$\theta_q: H_4(q) \rightarrow \zqq$
defined by
%$\theta(s_2s_3s_1s_2) = 1$, {$\theta(s_3s_2s_1s_3s_2) = -1$},
$\theta_q(T_{3412}) = 1$, $\theta_q(T_{4312}) = -1$,
and $\theta_q(T_w) = 0$ otherwise.
%\begin{equation*}
%  \theta(w) = \begin{cases}
%    1 & \text{if $w = e$}\\
%    -1 & \text{if $w = s_1s_2s_1$}\\
%    0 &\text{otherwise}
%  \end{cases}
%\end{equation*}
%This function satisfies
Computing the left-hand side of (\ref{eq:qstem})
%for $\wtc{s_{[2,4]}\ntnsp}q\wtc{s_{[2,4]}\ntnsp}q$ 
we have
\begin{equation}\label{eq:thetaex}
%  \theta(\wtc{s_{[2,4]}\ntnsp}1\wtc{s_{[2,4]}\ntnsp}1 ) = 1(2) - 1(0),
\theta_q(\wtc{s_{[2,4]}\ntnsp}q\wtc{s_{[1,3]}\ntnsp}q ) = (1+q)(1) + (0)(-1) = 1+q,
\end{equation}
since $T_{3412}$
%$s_2s_3s_1s_2$
appears in (\ref{eq:sigmaexample}) with coefficient $1+q$
and $T_{4312}$
%$s_3s_2s_1s_3s_2$
appears with coefficient $0$.
To compute the right-hand side of (\ref{eq:qstem}), we begin by writing
%we first factor the immanant as
\begin{equation*}
\imm{\theta_q}(x) = q^{-2} x_{1,3}x_{2,4}x_{3,1}x_{4,2}
- \qm52 x_{1,4}x_{2,3}x_{3,1}x_{4,2}.
%  = \det(x_{12,34})x_{3,1}x_{4,2}.
\end{equation*}
Substituting $b_{i,j}$
%from (\ref{eq:Bex})
for $x_{i,j}$, we have
$\qm52 b_{1,4}b_{2,3}b_{3,1}b_{4,2} = 0$ and
\begin{equation}\label{eq:zGex}
%\begin{aligned}
q^{-2}b_{1,3}b_{2,4}b_{3,1}b_{4,2} = q^{-2} z_{1,2,1}z_{3,2,2}z_{2,1,1}z_{4,1,2}
z_{3,1,1}(z_D + z_U)z_{1,2,2}z_{4,1,1}(z_D + z_U)z_{2,2,2}.
%\\
%\end{aligned}
\end{equation}
%gnore monomials which are not square-free,
%and extract the coefficient of $z_G$. 
%Substituting, we have
Now since $z_D^2$ and $z_U^2$ are not square-free, we ignore terms
in the expansion containing these. Since 
we have
\begin{equation*}
\begin{aligned}
z_{3,2,2}z_{4,1,2}z_D &= \qp12 z_Dz_{3,2,2}z_{4,1,2},\\
z_{3,2,2}z_{4,1,2}z_U &= \qp12 z_Uz_{3,2,2}z_{4,1,2},
%  z_U z_D &= q z_D z_U,
\end{aligned}
\qquad
\begin{aligned}
z_{3,2,2}z_{1,2,2} &= \qp12 z_{1,2,2}z_{3,2,2},\\
z_{3,2,2}z_{2,2,2} &= \qp12 z_{2,2,2}z_{3,2,2},
%  b_{1,4} &= 0,
\end{aligned}
\qquad
z_U z_D = q z_D z_U,
\end{equation*}
%and that 
we express the nonzero square-free monomials of (\ref{eq:zGex}) in lexicographic
order to obtain 
\begin{equation*}
q^{-2} (q^2 + q^3)
z_{1,2,1}z_{1,2,2}z_{2,1,1}z_D z_{2,2,2}z_{3,1,1}z_U z_{3,2,2}z_{4,1,1}z_{4,1,2} = (1+q)z_G,
\end{equation*}
which matches (\ref{eq:thetaex}).
%The coefficient of $z_G$ in this expression is $1 + q$.

%By Lindstr\"om's Lemma and inspection of
%the wiring diagram (\ref{eq:Gbraidfig}), we have
%to that that the corresponding immanant
%satisfies
%\begin{equation*}
%  [z_G]\imm{\theta}(B) = [z_G]\det(B_{12,34})b_{3,1}b_{4,2} = 2,
%  \end{equation*}
%since exactly two
%families of paths $\pi = (\pi_1, \pi_2, \pi_3, \pi_4)$ from sources
%$1, 2, 3, 4$ to sinks $3, 4, 1, 2$, respectively,
%satisfy
%\begin{enumerate}
%\item $\pi_1$ and $\pi_2$ do not intersect,
%\item $\pi$ covers $G$ and therefore has weight $z_G$.
%\end{enumerate}

%($\star$ Define defective and proper crossings and noncrossing somewhere.
%Use triples $\pi_i$, $\pi_j$, $p$ to do so.)

%that allows us to
%evaluate $\hnq$ traces at $D_w$.
%Let $\pi = (\pi_1, \dotsc, \pi_n)$
%be a path family which covers a star network
%\begin{equation}\label{eq:starnetfactor}
%  G = G_{J_1} \circ \cdots \circ G_{I_m}.
%  \end{equation}
%We assume that for some $v \in \sn$, we have that $\pi_i$ is a path from
%source $i$ to sink $v_i$.  In this case we say that the {\em type} of $\pi$ is
%$v$. 
An important property of the map $\sigma_B$ is that its
%The
evaluation
%of $\sigma_B$
at natural basis elements of $\Ann$ is closely related to
coefficients in the natural expansion of 
%$q_{s_{[i_1,j_1]}} C'_{s_{[i_1,j_1]}}(q) \cdots q_{s_{[i_m,j_m]}} C'_{s_{[i_m,j_m]}}(q)$.
%$\wtc{s_{[i_1,j_1]}\ntnsp}q \cdots \wtc{s_{[i_m,j_m]}\ntnsp}q$.
$\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q$.

\begin{prop}\label{p:qewcoeff}
	%  Let star network $G$ correspond to
	%  reversal sequence $s_{[i_1,j_1]}, \dotsc, s_{[i_m,j_m]}$.
	Let $B$ be the weighted path matrix of star network $G = G_{J_1} \circ \cdots \circ G_{J_m}$, and fix $w \in \sn$.
	Then 
	$\sigma_B(x^{e,w})$ is equal to ${\smash{\qp{\ell(w)}2}}$ times the coefficient of $T_w$ in 
	the product
	%$\wtc{s_{[i_1,j_1]}\ntnsp}q \cdots \wtc{s_{[i_m,j_m]}\ntnsp}q$.
	$\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q$.
\end{prop}
\begin{proof}
	Omitted.
\end{proof}


\section{$G$-tableaux and evaluation of induced sign characters}\label{s:main}

\cref{t:qstem} provides half of the solution
to the problem of evaluating
%$\epsilon_q^\lambda((1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}}))$.
%$\epsilon_q^\lambda(D_w)$.
the left-hand side of (\ref{eq:evalwithsigma}).
The other half is a combinatorial interpretation of the
right-hand-side of (\ref{eq:qstem}), which is a linear
combination of expressions of the form $\sigma_B(x^{u,w}) \in \zqq$.
To combinatorially interpret such evaluations,
%Given a path family $\pi$ covering a wiring diagram $G$,
we arrange the paths of a path family $\pi$ covering a star network $G$
into a (French) Young diagram, using each path exactly once.
We call the resulting structure a {\em $G$-tableau},
or more specifically a {\em $\pi$-tableau}.
If $\type(\pi) = w$, we say that the tableau has type $w$.
For example,
the following path family $\pi$
%of type $213$
covering the star network of
$\wtc{s_{[1,2]}\ntnsp}q \wtc{s_{[2,3]}\ntnsp}q \wtc{s_{[1,2]}\ntnsp}q$
yields six $\pi$-tableaux of shape $21$ and type $213$:
\begin{equation}\label{eq:1stGtableaux}
%\begin{tikzpicture}[scale=.5,baseline=-15]
\raisebox{2mm}{
	\begin{tikzpicture}[scale=.6,baseline=-15]
	\draw[dotted, thick] (0,0) -- (1,0) -- (1.5,-.5) -- (2,0) -- (3,0);
	\draw[dashed] (0,-1) -- (.5,-1.5) -- (1.5,-.5) -- (2.5,-1.5) -- (3,-2);
	\draw[-,thick] (0,-2) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-1);
	\node at (-.5,0) {$\pi_3$};
	\node at (-.5,-1) {$\pi_2$};
	\node at (-.5,-2) {$\pi_1$};
	\end{tikzpicture}
} 
\;, \qquad
\tableau[scY]{\pi_3 | \pi_1,\pi_2} \;, \quad
\tableau[scY]{\pi_2 | \pi_1,\pi_3} \;, \quad
\tableau[scY]{\pi_3 | \pi_2,\pi_1} \;, \quad
\tableau[scY]{\pi_1 | \pi_2,\pi_3} \;, \quad
\tableau[scY]{\pi_2 | \pi_3,\pi_1} \;, \quad
\tableau[scY]{\pi_1 | \pi_3,\pi_2} \;.
\end{equation}
Given a $\pi$-tableau $U$, we define (integer) Young tableaux $L(U)$, $R(U)$
by replacing each path by its source index and sink index, respectively.
For example, if $U$ is the first $\pi$-tableau in (\ref{eq:1stGtableaux}),
then we have
\begin{equation*}
L(U) = \tableau[scY]{3 | 1,2} \;, \quad
R(U) = \tableau[scY]{3 | 2,1} \;.
\end{equation*}
It is easy to see that given two Young tableaux $P$, $Q$ of the same shape,
there is at most one $\pi$-tableau $U$ satisfying $L(U) = P$, $R(U) = Q$.

We also define several statistics on $G$-tableaux.
Suppose that two paths $\pi_a$, $\pi_b$ in a star network
$G = G_{J_1} \circ \cdots \circ G_{J_m}$
pass through the central vertex of
some simple star network $G_{J_p}$.
%in the factorization (\ref{eq:starnetfactor}).
We call the triple $(p, \pi_a, \pi_b)$ a {\em crossing} of $\pi$
if the two paths cross there, and a {\em noncrossing} otherwise.
Let $U$ be any $\pi$-tableau.
Define $\cross(U) = \cross(\pi)$ to be the number of crossings of $\pi$.
%Let $U$ be a $\pi$-tableau of any shape $\lambda \vdash n$.
Define $\incross(U)$, the number of {\em inverted noncrossings} of $U$,
to be the number of noncrossings $(p, \pi_a, \pi_b)$
of $\pi$ such that $\pi_a$, $\pi_b$
intersect at the central vertex of $G_{J_p}$
%(\ref{eq:wdnetwork})
with $\pi_b$ above $\pi_a$,
%and $\pi_b$ appears in an earlier column of $U$,
%occurrences of
\begin{equation}\label{eq:inc}
\begin{tikzpicture}[scale=.7,baseline=-25]
\draw[dashed] (0,0) -- (1,-1) -- (2,0);
\draw[-] (0,-2) -- (1,-1) -- (2,-2);
\node at (-.5,0) {$\pi_b$};
\node at (-.5,-2) {$\pi_a$};
\end{tikzpicture}\;,
% \raisebox{-7mm}
%{\includegraphics[height=15mm]{incross.eps}}\ ,
\end{equation}
% \begin{equation}\label{eq:inc}
% \raisebox{-10mm}
% {\includegraphics[height=20mm]{incross.eps}}\ ,
% \end{equation}
and $\pi_b$ appearing in an earlier column of $U$ than $\pi_a$
(whether or not $b > a$).
%Thus inverted noncrossings may be proper or defective.   
%Define $\cross(U) = \cross(\pi)$ to be the number of crossings of $\pi$, i.e.,
%the number of occurrences of
%\begin{equation}\label{eq:cross}
%\begin{tikzpicture}[scale=.7,baseline=-25]
%  \draw[dashed] (0,0) -- (1,-1) -- (2,-2);
%  \draw[-] (0,-2) -- (1,-1) -- (2,0);
%  \node at (-.5,0) {$\pi_b$};
%  \node at (-.5,-2) {$\pi_a$};
%\end{tikzpicture}
%\qquad
%\mathrm{ or }
%\qquad 
%\begin{tikzpicture}[scale=.7,baseline=-25]
%  \draw[-] (0,0) -- (1,-1) -- (2,-2);
%  \draw[dashed] (0,-2) -- (1,-1) -- (2,0);
%  \node at (-.5,0) {$\pi_a$};
%  \node at (-.5,-2) {$\pi_b$};
%\end{tikzpicture} \;.
%\end{equation}
%\begin{equation*}
%\raisebox{-10mm}
%{\includegraphics[height=20mm]{cr1.eps}}\ 
%\qquad
%\mathrm{ or }
%\qquad 
%\raisebox{-10mm}
%{\includegraphics[height=20mm]{cr2c.eps}}\ .
%\end{equation*}
%This depends only upon $\pi$; not upon the 
%locations of $\pi_a$ and $\pi_b$ in $U$.
%($\star$ postpone $\cdncross$ definition until later? It is only used as a
%stepping stone.)
For example, each tableau $U$ in (\ref{eq:1stGtableaux})
satisfies $\cross(U) = 1$ because $\cross(\pi) = 1$.
The inverted noncrossings in these tableaux are triples
$(1,\pi_1,\pi_2)$ with $\pi_2$ in an earlier column than $\pi_1$,
or
$(2,\pi_2,\pi_3)$ with $\pi_3$ in an earlier column than $\pi_2$.
%appearances of $\pi_3$ in an earlier column
%than $\pi_2$, or $\pi_2$ in an earlier column than $\pi_1$.
The numbers of inverted noncrossings 
for the six tableaux are $1, 0, 0, 0, 1, 1$, respectively.
%Since both cannot happen simultaneously in a tableau having only two columns,
%the

Combining the above tableau statistics, we may combinatorially interpret
$\sigma_B(x^{u,w})$ in terms of tableaux of shape $(n)$ (i.e., consisting of
a single row).
%$\star$ Define path tableau and give examples.
A fixed path family $\pi$ of type $v$ and a permutation
$u \in \sn$ determine a path tableau $U(\pi,u,uv) = \pi_{u_1} \cdots \pi_{u_n}$
which satisfies
$L(U(\pi, u, uv)) = u_1 \cdots u_n$ and
$R(U(\pi, u, uv)) = (uv)_1 \cdots (uv)_n$.
The inclusion of $uv$ in our notation $U(\pi,u,uv)$ is superfluous but makes
clear the ordering of sinks as they appear in the tableau.
%If $U$ is any $\pi$-tableau, then we call a noncrossing of $\pi$ an
%{\em inverted noncrossing of $U$} if the path which enters and leaves
%the central vertex of $G_{J_p}$ on top appears in an earlier column of $U$
%than does the other path.  ($\star$ Say this better and use figures.)
%($\star$ Mention this somewhere.)
%Concatenating a path family $\pi^{G'}$
%which covers $G'$
%to a path family $\pi^H$
%which covers $H$
%we obtain a new path family $\pi^{G'H}$
%which covers $G$
%and satisfies
%$\type(\pi^{G'H}) = \type(\pi^{G'}) \type(\pi^H)$.
%Conversely, every path family which covers $G$
%decomposes this way.

\begin{prop}\label{p:sigmastats}
	Let star network $G$ have weighted path matrix $B$.
	For $u,w \in \sn$ we have 
	\begin{equation}\label{eq:stats}
	\sigma_B (x^{u,w}) = 
	\sum_\pi \qp{\cross(\pi)}2
	%\sum_U
	q^{\incross(U)},
	\end{equation}
	where the sum is over path families $\pi$ of type $u^{-1}w$ covering $G$,
	and $U = U(\pi,u,w)$ is the unique $\pi$-tableau of shape $(n)$
	satisfying $L(U) = u_1 \cdots u_n$, $R(U) = w_1 \cdots w_n$.
\end{prop}
\begin{proof}
	Omitted.
	%  By Lemma~\ref{l:sigmareversalinterp}, the claim is true when $G$ is
	%  the simple star network $G_{[i_1,j_1]}$.
	%    
	%  Now suppose the claim is true for $G$ a concatenation of
	%  $1, \dotsc, m$ simple star networks
	%  and consider $G = G' \circ H$ where $G' = G_{J_1} \circ \cdots \circ G_{I_m}$
	%  and $H = G_{I_{m+1}}$.
	%    Let $B'$, $C$ be the path matrices of $G'$, $H$, respectively
	%    so that $B = B'C$ is the path matrix of $G$.
	%
	%By Corollary~\ref{c:sigmacoprodnew}
	%and induction we have
	%\begin{equation}\label{eq:sigmalong}
	%  \sigma_{B'C}(x^{u,w})
	%  =
	%\sum_{t \in \mfs{I_{m+1}}}
	%  \sum_{\pi^{G'}}
	%  \qp{\cross(\pi^{G'})}2 q^{\incross(U(\pi^{G'}\ntksp,u,wt^{-1}))}
	%  \qp{\cross(\pi^H)}2 q^{\incross(U(\pi^H,wt^{-1},w))},
	%  \end{equation}
	%  where $\pi^{G'}$ varies over all path families of type $u^{-1}wt^{-1}$
	%  which cover $G'$.
	%  Since pairs of path families of types $u^{-1}wt^{-1}$ and $t$ covering
	%  $G'$ and $H$, respectively, correspond to path families of type $u^{-1}w$
	%  covering $G$, we may apply
	%  Proposition~\ref{p:decomposestat} to the right-hand side above to obtain
	%  \begin{equation*}
	%  \sum_\pi
	%  \qp{\cross(\pi)}2 q^{\incross(U(\pi,u,w))},
	%  \end{equation*}
	%  where the sum is over all path families $\pi$ of type $u^{-1}w$ which cover
	%  $G$.
\end{proof}

The special case $u = e$ of \cref{p:sigmastats} yields a proof of
a generalization of Deodhar's defect formula \cite[Prop.\,3.5]{Deodhar90}
for coefficients of the expression
$(1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}})$.
Let $\pi = (\pi_1, \dotsc, \pi_n)$ be a path family covering a star network
$G = G_{J_1} \circ \cdots \circ G_{J_m}$.
If two paths $\pi_i$, $\pi_j$ intersect at the central vertex
of $G_{J_p}$, call the intersection {\em defective} if the paths
have previously crossed an odd number of times (i.e., in
%at the central vertices of
$G_{J_1}, \dotsc, G_{J_{p-1}}$).
Define $\dfct(\pi)$, the number of {\em defects} of $\pi$, to
be the number of triples $(p, \pi_i, \pi_j)$
such that $\pi_i$ and $\pi_j$ intersect
defectively at the central vertex of $G_{J_p}$.
%($\star$ simpler? A defect of a path family is a point of intersection of
%of a path family to be a point of intersection
%two paths which have
%previously crossed an odd number of times.
%)
%(\ref{eq:defectdef2}).
\begin{cor}\label{c:sigmastats}
	%  Let $G$ be the wiring diagram of expression $\sprod im$.
	%  have weighted path matrix $B$.
	%  Then t
	The coefficients in the expansion
	%  \begin{equation*}
	$\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q = \sum_w a_w T_w$
	%  \end{equation*}
	are
	%given by
	\begin{equation*}
	a_w = \sum_\pi q^{\dfct(\pi)},
	\end{equation*}
	where the sum is over all path families of type $w$ which cover the star network
	$G_{J_1} \circ \cdots \circ G_{I_m}$.
	%For $w \in \sn$ we have 
	%\begin{equation}\label{eq:cstats}
	%\sigma_B (x^{e,w}) = 
	%\sum_\pi \qp{\cross(\pi)}2
	%%\sum_U
	%q^{\dfct(\pi_1,\dotsc,\pi_n)},
	%\end{equation}
	%where the sum is over path families $\pi$ of type $w$ covering $G$.
	%and $U = (\pi_1,\dotsc,\pi_n)$
	%is the unique $\pi$-tableaux of shape $(n)$
	%satisfying $L(U) = 1 \cdots n$, $R(U) = w$.
\end{cor}
\begin{proof}
	Omitted.
	%  Substitute $u = e$ in Proposition~\ref{p:sigmastats}.
	%  The right-hand side of (\ref{eq:stats}) is a sum over
	%  path families $\pi$ of type $w$, and
	%  each tableau $U = U(\pi,e,w)$ is simply
	%  the sequence $\pi = (\pi_1,\dotsc,\pi_n)$.
	%  Thus an inverted noncrossing of $U$ is simply a noncrossing
	%  in which the upper path has index less than that of the lower path, i.e.,
	%  a defective noncrossing.  To relate these to all defects,
	%  let us temporarily define
	%  \begin{equation*}
	%    \begin{aligned}
	%      \dnc(\pi) &= \text{ number of defective noncrossings of $\pi$},\\
	%      \dc(\pi) &= \text{ number of defective crossings of $\pi$},
	%    \end{aligned}
	%  \end{equation*}
	%  so that $\dfct(\pi) = \dnc(\pi) + \dc(\pi)$.
	%  Now observe that for any path family of type $w$ we have
	%    \begin{equation*}
	%    \cross(\pi) = \inv(w) + 2\dc(\pi)
	%    \end{equation*}
	%    because if paths $\pi_a$, $\pi_b$ cross $k$ times,
	%    then at most one of those crossings contributes to $\inv(w)$,
	%    while exactly half of the remaining crossings are defective.
	%    Thus the right-hand side of (\ref{eq:stats}) becomes
	%    \begin{equation}\label{eq:rhsstats}
	%      \sum_\pi \qp{\inv(w) + 2\dc(\pi)}2 q^{\dnc(\pi)}
	%      = \sum_\pi \qew q^{\dc(\pi)+\dnc(\pi)} = \qew \sum_\pi q^{\dfct(\pi)},
	%    \end{equation}
	%    where the sum is over all path families of type $w$ which cover the
	%    star network $G_{J_1} \circ \cdots \circ G_{I_m}$.
	%    
	%    By Proposition \ref{p:qewcoeff}, the left-hand side of (\ref{eq:stats})
	%    is $\qew a_w$.
	%  Combining this with (\ref{eq:rhsstats}), we have the desired result.
\end{proof}

%\section{Evaluation of induced sign characters}\label{s:indsgneval}

By \cref{t:qstem}, the map $\sigma_B$ (\ref{eq:sigmadef})
can be used to evaluate
$\epsilon_q^\lambda(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)$
%$\epsilon_q^\lambda(D_w)$
when one has a simple
expression for the generating function
$\imm{\epsilon_q^\lambda}(x)$.
%which allows us to evaluate
%and can evaluate
%$\sigma_B(\imm{\epsilon_q^\lambda}(x))$.
Such an expression was given by Konvalinka
and the second author in \cite[Thm.\,5.4]{KSkanQGJ}:
%for $\lambda = (\lambda_1,\dotsc,\lambda_r)$, we have
\begin{equation}\label{eq:immepsilon}
\imm{\epsilon_q^\lambda}(x) =
\sum_I \qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r}),
%  \sum_{(I_1,\dotsc,I_r)} \qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r}),
\end{equation}
where
$\qdet$ and $x_{L,M}$ are defined as in \cref{s:hnqanq},
and
the sum is over all ordered set partitions
$I = (I_1,\dotsc,I_r)$ of $[n]$
%$\{1,\dotsc,n\}$
satisfying $|I_j| = \lambda_j$.
%$\lambda = (\lambda_1,\dotsc,\lambda_r)$,
%and
We will say that such an ordered set
partition has {\em type} $\lambda$.

To evaluate $\sigma_B(\imm{\epsilon_q^\lambda}(x))$, we expand
each term
%in the sum
on the right-hand side of (\ref{eq:immepsilon})
%and recognize the resulting
%monomials as elements
in a monomial basis $\{x^{u,v} \,|\, v \in \sn \}$ of $\Ann$,
%presented in Section~\ref{s:qmb}.
%These bases are best described
where $u = u(I)$ is the concatenation of the $r$ strictly increasing subwords
\begin{equation}\label{eq:subwordsofu}
u_1 \cdots u_{\lambda_1}, \quad 
u_{\lambda_1 +1} \cdots u_{\lambda_1 + \lambda_2}, \quad
u_{\lambda_1 + \lambda_2 + 1} \cdots u_{\lambda_1 + \lambda_2 + \lambda_3}, \quad
%\dotsc, \quad
%u_{\lambda_1 + \cdots + \lambda_{j-1} + 1} \cdots u_{\lambda_1 + \cdots + \lambda_j}, \quad
\dotsc, \quad
u_{n-\lambda_r + 1} \cdots u_n
\end{equation}
formed by listing the elements of each block $I_1, \dotsc, I_r$
in increasing order.
As $I$ varies over all ordered set partitions of $[n]$ of type $\lambda$,
the
%resulting
%Write $u(I)$ for the permutation corresponding to $I$.
%The
permutations $u(I)$ vary over
%are precisely
the Bruhat-minimal representatives $\slambdamin$ of cosets $\slambda u$, where
$\slambda$ is the {\em Young subgroup} of $\sn$
%is described in terms of
%$\slambda$ of $\sn$,
%Define $\slambda$ to be the subgroup of $\sn$
generated by
\begin{equation*}
%\label{eq:youngsubgen}
%J \defeq 
\{ s_1, \dotsc, s_{n-1} \} \ssm 
\{ s_{\lambda_1}, s_{\lambda_1 + \lambda_2}, 
s_{\lambda_1 + \lambda_2 + \lambda_3}, 
\dotsc, 
s_{n-\lambda_r} \}.
\end{equation*}
%and let
%and
%Let $\slambdamin$ be the set of
%%the Bruhat-minimal representatives $\slambdamin$ of cosets
%Bruhat-minimal representatives of cosets
%of the form $\slambda v$, i.e., the elements $u \in \sn$ for which each
%of the subwords
%is strictly increasing.
%It is clear that elements of
%A bijection between $\slambdamin$ and
%ordered set partitions $(I_1, \dotsc, I_r)$ of $[n]$ of type $\lambda$
%is given by $u \mapsto I$ where
%\begin{equation}\label{eq:uI}
%  I_j = \{ u_{\lambda_1 +\cdots+ \lambda_{j-1} + 1}, \dotsc, u_{\lambda_1 +\cdots+ \lambda_j} \}
%  \text{ for } j = 1,\dotsc,r.
%\end{equation}
%Letting
%$u(I)$ denote the permutation in $\slambdamin$ which corresponds
%to $I$, we have
%, i.e., the permutation whose one-line notation is the increasing
%rearrangement of $I_1$, followed by the increasing rearrangement of $I_2$, etc.
%If $u \in \slambdamin$ corresponds to $(I_1, \dotsc, I_r)$,
%then the path indices in row $j$ of $U(u,\pi,\lambda)$
%are simply the elements of $I_j$, in increasing order. 
%Now we have
%In order to interpret $\epsilon_q^\lambda((1+T_{s_{i_1}}) \cdots (1+T_{s_{i_m}}))$
%we refer to \cite[Eqn.~(6.1)]{CHSSkanEKL}: 
Expanding each term on the right-hand side of (\ref{eq:immepsilon})
and applying $\sigma_B$ we have
\begin{equation}\label{eq:lambepsilon}
%\begin{aligned}
%\epsilon_q^\lambda((1+T_{s_{i_1}}) \cdots (1+T_{s_{i_m}}))
%&= \sigma_B(\imm{\epsilon_q^\lambda}(x))\\
%&= \sum_{(I_1,\dotsc,I_r)}
\sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r}))
%\\
%= \sum_{u \in \slambdamin}
= \sum_{y \in \slambda}
%&= \sum_{u \in \slambdamin} \sum_{y \in u^{-1}\slambda u} 
%(-1)^{\ell(uy)-\ell(u)}q_{u,uy}^{-1} \sigma_B (x^{u,uy}),
(-1)^{\ell(y)}\qiey \sigma_B (x^{u(I),yu(I)}).
%\end{aligned}
\end{equation}
%(See \cite[Eqn.~(6.1)]{CHSSkanEKL}.)
%where the first sum is over all ordered set partitions $(I_1,\dotsc, I_r)$
%of $[n]$ of type $\lambda$.
%Let us therefore consider evaluations of the form 
%\begin{equation*}
%$\sigma_B (\qiuv x^{u,v}) = \qiuv
%$\sigma_B (x^{u,v})$.
%\end{equation*}
%(See, e.g., \cite{CurtisLIT}.)
%satisfying
%\begin{equation*}
%u_1 < \cdots < u_\lambda, \qquad u_{\lambda_1 +1} < \cdots < u_{\lambda_1 + \la%mbda+2}, 
%\dotsc, u_{n-\lambda_r + 1} < \cdots < u_n.
%\end{equation*}
To combinatorially interpret the
%final
sum in (\ref{eq:lambepsilon})
we may apply \cref{p:sigmastats}
%need to
%to expressions $\sigma_B(x^{u,yu})$
%\,|\, u \in \slambdamin, y \in \slambda \}$
%evaluations of the form
%(\ref{eq:stats}) only
%with $u \in \slambdamin$,
%for some partition $\lambda = (\lambda_1,\dotsc,\lambda_r) \vdash n$,
%and $v = uy$ for $y \in u^{-1}\slambdamin u$.
%and
%$v = yu$ for
%$y \in \slambda$,
%Since elements of $\slambdamin$ correspond bijectively to ordered set
%partitions $I$ of $[n]$ of type $\lambda$, and $\sigma_B(x^{u,yu})$ has
%the interpretation given in Proposition~\ref{p:sigmastats},
%it will be convenient to
and compute certain statistics for tableaux belonging to the set
%define
\begin{equation*}
%  \begin{aligned}
\mathcal U_I = \mathcal U_I(G)
%    =\{ U(\pi, u, uy ) \,|\, \pi \text{ covers } G, u = u(I),
\defeq \{ U(\pi, u, yu ) \,|\, \pi \text{ covers } G, u = u(I), 
y \in \slambda \}.
%    \type(\pi) = y \in u^{-1}\slambda u \}.
%  \end{aligned}
\end{equation*}
Note that our restriction on $y$ forces the sink indices of paths in
components
\begin{equation}\label{eq:component}
(\lambda_1 + \cdots + \lambda_{k-1}+1), \dotsc, (\lambda_1 + \cdots + \lambda_k)
\end{equation}
%of $U(\pi,u,uy)$ to be a permutation of the source indices of the same paths.
of $U(\pi,u,yu)$ to be a permutation of the source indices of the same paths.

On the other hand, the
%final
sum in (\ref{eq:lambepsilon}) has both positive and negative signs.
We obtain a subtraction-free expression
for the sum
%$\sigma_B(\imm{\epsilon_q^\lambda}(x))$
%$\epsilon_q^\lambda((1+T_{s_{i_1}}) \cdots (1+T_{s_{i_m}}))$
by applying
%defining
a sign-reversing involution
%$\zeta = \zeta_I$ on
to the tableaux in each set $\mathcal U_I$.
Tableaux which remain after cancellation are in bijection with
$G$-tableaux in which paths in a single column have increasing indices and
do not intersect.  We call such tableaux {\em column-strict}.
%\rightarrow \mathcal U_I$,
%for each ordered set partition $I$ of type $\lambda$.
%This will cancel all negative
%contributions from expressions of the form $\sigma_B(x^{u,uy})$
%with some positive contributions of the same form.

%Finally we can state and justify a
%subtraction-free formula for
%$\epsilon_q^\lambda(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)$.
\begin{thm}\label{t:qepsilon}
	Let $G = G_{J_1} \circ \cdots \circ G_{I_m}$.
	%be the wiring diagram of $\sprod im$.
	%represent $(1 + T_{s_{i_1}}) \cdots (1 + T_{s_{i_m}}) \in \hnq$.
	Then for $\lambda \vdash n$ 
	we have
	\begin{equation}\label{eq:epsilonmain}
	\epsilon_q^\lambda(\wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q)  
	= \sum_W q^{\incross(W)+\cross(W)/2},
	\end{equation}
	where the sum is over all column-strict $G$-tableaux 
	%$U$ 
	of 
	type $e$
	and 
	shape $\lambda^\tr$.
\end{thm}
\begin{proof} (Idea)
	Let $B$ be the path matrix of $G$. Combining the \hyperref[t:qstem]{Theorems~}\ref{t:qstem}
	and \cite[Thm.\,5.4]{KSkanQGJ} (i.e., (\ref{eq:immepsilon}))
	with the identity (\ref{eq:lambepsilon}), we see that
	the left-hand side of (\ref{eq:epsilonmain}) is
	\begin{equation}\label{eq:epsilonsecond}
	\begin{aligned}
	\sigma_B(\imm{\epsilon_q^\lambda}(x)) &=
	%      \sum_{I = (I_1,\dotsc,I_r)}
	\sum_I
	\sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r}))\\
	&=
	\sum_I
	\sum_{\smash{y \in \slambda}} (-1)^{\ell(y)}\qiey
	\sigma_B (x^{u(I),yu(I)}),
	%      \sum_{I = (I_1,\dotsc,I_r)}
	%\sum_{u \in \slambdamin}
	%   \sum_{y \in \slambda} (-1)^{\ell(y)}q_{e,y}^{-1} \sigma_B (x^{u(I),yu(I)}),
	\end{aligned}
	\end{equation}
	where the first two sums are over ordered set partitions
	$I = (I_1,\dotsc,I_r)$ of $[n]$ of type $\lambda$.
	Fixing one such partition $I$ and writing $u = u(I)$, we may use
\cref{p:sigmastats} and other lemmas
	%Lemma~\ref{l:TU}
	to express the sum over elements of $\slambda$ as
	%  and $u = u(I)$.  
	%  (i.e., \cite[Thm.\,5.4]{KSkanQGJ}), 
	%  with the identities
	%  and let $(I_1,\dotsc,I_r)$ be a set
	%partition of $[n]$ of type $\lambda$.
	%By 
	%(\ref{eq:subwordsofu}) -- (\ref{eq:hqlambetaalt}),
	%(\ref{eq:lambepsilon}),
	%there is a permutation $u \in \slambdamin$ 
	%corresponding to $(I_1,\dotsc,I_r)$ such that we have
	%\begin{equation*}
	%\sigma_B(\qdet(x_{I_1,I_1}) \cdots \qdet(x_{I_r,I_r})) = 
	%\sum_{y \in \slambda} (-1)^{\ell(y)}  q_{e,y}^{-1} \sigma_B ( x^{u, yu} ).
	%\end{equation*}
	%By Proposition~\ref{p:sigmastats}
	%this is equal to
	\begin{equation}\label{eq:ypiinvuupi}
	%  \sum_{u \in \slambdamin}
	%  \sum_I
	%  \sum_{(y,\pi)} (-1)^{\ell(y)}
	\sum_{y \in \slambda}
	\sum_\pi 
	(-1)^{\ell(y)} \qiey
	%  q^{\rinv(U(u,\pi,\lambda)^\tr)},
	%\end{equation}
	%\begin{equation}
	%\sigma_B (x^{u,v}) = 
	%\sum_\pi
	\qp{\cross(\pi)}2
	%\sum_U
	%q^{\incross(U(\pi,u(I),yu(I)))},
	q^{\incross(U(\pi,u,yu))}
	%,
	%\end{equation}
	%\begin{equation}\label{eq:sumtocancel}
	%  \sum_{(y,\pi)} (-1)^{\ell(uy)-\ell(y)}  q_{u,uy}^{-1}
	%  \sum_I
	%  \sum_{\smash{u \in \slambdamin}}
	=
	\sum_{y \in \slambda}
	\sum_\pi 
	(-1)^{\ell(y)} \qiey
	%\sum_{(y,\pi)} (-1)^{\ell(y)}  q_{e,y}^{-1}
	\qp{\cross(\pi)}2 q^{\incross(W)+\cdncross(W)},
	%  q^{\incross(\delta(U))+\cdncross(\delta(U))},
	\end{equation}
	where
	the inner
	%$u = u(I)$ and
	sums are over
	%pairs $(y,\pi)$
	%such that 
	%$y \in u^{-1}\slambda u$ and $\pi$ is a path family of type $y$ which covers $G$.
	%with $y \in \slambda$ and $\pi$ is a
	path families $\pi$ of type
	%$u(I)^{-1}yu(I)$
	$u^{-1}yu$
	which cover $G$,
	and where $W = W(\pi,u,yu)$ is a related tableau of shape $\lambda^\tr$,
	and $\cdncross$ is a statistic related to defects and noncrossings.
	
	A sign reversing involution eliminates those tableaux $W$ which are not
	column-strict, and another lemma allows us to interpret the given powers of $q$
	in terms of crossings and inverted noncrossings in the remaining column-strict
	tableaux. Thus the three expressions in (\ref{eq:epsilonsecond})
	are equal to the right-hand side of (\ref{eq:epsilonmain}).
	%which are fixed by $\zeta$.
	%($\star$ Continue.)
\end{proof}

To illustrate the theorem, we compute 
%\begin{equation*}
$\epsilon_q^{211}
(\wtc {s_{[1,2]}\ntnsp}q \wtc{s_{[2,4]}\ntnsp}q \wtc{s_{[1,2]}\ntnsp}q)$
%\epsilon_q^{211}
%((1+T_{s_1})(1+T_{s_2} + T_{s_3} + T_{s_2s_3} + T_{s_3s_2} + T_{s_2s_3s_2})(1+T_{s_1}))
%\end{equation*}
using the star network $G = G_{[1,2]}\circ G_{[2,4]} \circ G_{[1,2]}$
pictured in (\ref{eq:simplestarnets}).
%= 
%\begin{tikzpicture}[scale=.5,baseline=-10]
%  \draw (0,1) -- (1,1) -- (1.5,0) -- (2,1) -- (3,1);
%  \draw (0,0) -- (1,0) -- (1.5,0) -- (2,0) -- (3,0);
%  \draw (0,-1) -- (.5,-1.5) --  (1,-1) -- (1.5,0) -- (2,-1) -- (2.5,-1.5) -- (3,-1);
%  \draw (0,-2) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-2);
%\end{tikzpicture}.
%\end{equation}
There are two path families of type $e$ which cover $G$, and four 
column-strict $G$-tableau of shape $211^\tr = 31$ for each:
\begin{equation*}
%\label{eq:Gtableauxpi}
\begin{tikzpicture}[scale=.5,baseline=-10]
\draw[dashed, ultra thick] (0,1) -- (1,1) -- (1.5,0) -- (2,1) -- (3,1);
\draw[dotted, thick] (0,0) -- (1,0) -- (1.5,0) -- (2,0) -- (3,0);
\draw[dashed] (0,-1) -- (.5,-1.5) --  (1,-1) -- (1.5,0) -- (2,-1) -- (2.5,-1.5) -- (3,-1);
\draw[-,thick] (0,-2) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-2);
\node at (-.5,1) {$\pi_4$};
\node at (-.5,0) {$\pi_3$};
\node at (-.5,-1) {$\pi_2$};
\node at (-.5,-2) {$\pi_1$};
\end{tikzpicture}
\;, \quad
U_\pi^{(1)} = \tableau[scY]{\pi_3 | \pi_1,\pi_2,\pi_4} \;,\ 
U_\pi^{(2)} = \tableau[scY]{\pi_3 | \pi_1,\pi_4,\pi_2}\;,\ 
U_\pi^{(3)} = \tableau[scY]{\pi_4 | \pi_1,\pi_2,\pi_3}\;,\ 
U_\pi^{(4)} = \tableau[scY]{\pi_4 | \pi_1,\pi_3,\pi_2}\;; 
\end{equation*}
\begin{equation*}
%\label{eq:Gtableauxrho}
\begin{tikzpicture}[scale=.5,baseline=-10]
\draw[dashed, ultra thick] (0,1) -- (1,1) -- (1.5,0) -- (2,1) -- (3,1);
\draw[dotted, thick] (0,0) -- (1,0) -- (1.5,0) -- (2,0) -- (3,0);
\draw[dashed] (0,-1) -- (.5,-1.5) -- (1,-2) -- (2,-2) -- (2.5,-1.5) -- (3,-1);
\draw[-, thick] (0,-2) -- (.5,-1.5)  --  (1,-1) -- (1.5,0) -- (2,-1) -- (2.5,-1.5) -- (3,-2);
\node at (-.5,1) {$\rho_4$};
\node at (-.5,0) {$\rho_3$};
\node at (-.5,-1) {$\rho_2$};
\node at (-.5,-2) {$\rho_1$};
\end{tikzpicture}\;, \quad
U_\rho^{(1)} = \tableau[scY]{\rho_3 | \rho_2,\rho_1,\rho_4}\;,\ 
U_\rho^{(2)} = \tableau[scY]{\rho_3 | \rho_2,\rho_4,\rho_1}\;,\ 
U_\rho^{(3)} = \tableau[scY]{\rho_4 | \rho_2,\rho_1,\rho_3}\;,\ 
U_\rho^{(4)} = \tableau[scY]{\rho_4 | \rho_2,\rho_3,\rho_1}\;.
\end{equation*}
The path family $\pi$ has no crossings, so tableau $U_\pi^{(i)}$ contributes 
$q^{\textsc{invnc}(U_\pi^{(i)})}q^{\textsc{c}(U_\pi^{(i)})/2} = q^{\textsc{invnc}(U_\pi^{(i)})}$ for all $i$. 
We have one noncrossing for each of the pairs $(\pi_2,\pi_3)$, $(\pi_2,\pi_4)$ and $(\pi_3,\pi_4)$ 
and two for the pair $(\pi_1,\pi_2)$. Counting only the inverted noncrossings,
%only for pairs where the path which intersects the other from above appears in a column left of the other, for instance
such as $\pi_2$ and $\pi_3$ 
in $U_\pi^{(1)}$, we find the contributions from $U_\pi^{(1)},\dots,U_\pi^{(4)}$ are
$q, q^2, q^2, q^3$, respectively. 
The tableaux for the path family $\rho$ each have two crossings, 
and one noncrossing for 
each of the pairs $(\rho_1,\rho_3)$, $(\rho_1,\rho_4)$ and $(\rho_3,\rho_4)$. 
Adding the contributions together we 
find the contributions for $U_\rho^{(1)},\dots,U_\rho^{(4)}$ 
are $q^1q^{2/2} = q^2$, $q^2q^{2/2} = q^3$, 
$q^2q^{2/2} = q^3$ and $q^3q^{2/2}= q^4$ respectively. 
Hence we have
$\smash{\epsilon_q^{211}}(
\wtc {s_{[1,2]}\ntnsp}q \wtc{s_{[2,4]}\ntnsp}q \wtc {s_{[1,2]}\ntnsp}q)
%(1+T_{s_1})(1+T_{s_2} + T_{s_3} + T_{s_2s_3} + T_{s_3s_2} + T_{s_2s_3s_2}) (1+T_{s_1}))
= q + 3q^2 + 3q^3 + q^4$.


\cref{t:qepsilon} allows one to
%coincides with a special case of 
%\cite[Thm.\,6.4]{CHSSkanEKL}, which
combinatorially interpret evaluations of $\epsilon_q^\lambda$ at
(multiples of) certain elements $\wtc wq$ 
%\begin{equation*}
%  \{ \qew C'_w(q) \,|\, \text{$w$ \avoidsp} \}
%\end{equation*}
of the Kazhdan-Lusztig basis of $\hnq$.
In particular, for some elements $\wtc wq$ there exists a polynomial $g_w(q)$
such that we have
\begin{equation}\label{eq:klfactor}
g_w(q) \wtc wq = \wtc{s_{J_1}\ntnsp}q \cdots \wtc{s_{J_m}\ntnsp}q
\end{equation}
for some sequence $s_{J_1},\dotsc,s_{J_m}$ of reversals.
Such permutations include all \pavoiding permutations,
all of $\mfs 4$ (even $4231$ and $3412$),
all of $\mfs 5$ except $45312$,
and all $321$-{\em hexagon-avoiding} permutations.
(See \cite{BWHex}.)

\begin{cor}\label{c:klfactor}
	Suppose that $\wtc wq$ satisfies
	a factorization of the form (\ref{eq:klfactor})
	and define $G = G_{J_1} \circ \cdots \circ G_{J_m}$.
	Then we have
	\begin{equation}\label{eq:klfactoreval}
	\epsilon_q^\lambda(\wtc wq) = \frac1{g_w(q)}\sum_U q^{\incross(U)+\cross(U)/2},
	\end{equation}
	where the sum is over all column-strict $G$-tableaux 
	%$U$ 
	of type $e$ and shape $\lambda^\tr$.
\end{cor}

It would be interesting to characterize the factorizations
%know which Kazhdan-Lusztig basis elements
%factor as in
(\ref{eq:klfactor})~\cite[Quest.\,4.5]{SkanNNDCB}.


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%\section{A section}
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%\lipsum[3]
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%\begin{equation}\label{eqn:eq1}
%\int_0^\infty \exp(-x)dx = 1.
%\end{equation}
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%\begin{lem}\label{lem:lem1}
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%\centering
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%\caption{A plot of a function.}
%\label{fig:plot}
%\end{figure}
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%\subsection{A subsection}
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%\begin{align*}
%1 + 1 + 1 &= 2 + 1 \\ &= 3.
%\end{align*}
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%\subsubsection{A subsubsection}
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%\lipsum[17]
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%Here is a citation~\cite{greenwade93} with URL. Here is another
%citation~\cite{MR4}, which is the earliest-numbered Math Review with a
%functioning DOI: MR0000004. Finally, a reference to
%equation~\eqref{eqn:eq1}. And a reference to Figure~\ref{fig:plot}.
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%\acknowledgements{\lipsum[20]}

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