%%  This article has been submitted and accepted for publication in
%%  THE FOATA FESTSCHRIFT, 
%%  a special issue of THE ELECTRONIC JOURNAL OF COMBINATORICS
%%  dedicated to DOMINIQUE FOATA at the occasion of his 60th birthday.
%%
%%  Author(s):		ALUN MORRIS AND A. A. ABDEL-AZIZ
%%  Title:		SCHUR $Q$-FUNCTIONS AND SPIN CHARACTERS OF SYMMETRIC GROUPS I
%%  Date of submission:	March 31, 1995
%%  TeX version:	LaTeX
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%%  Number of pages:	12
%%  Author's email:	aom@aber.ac.uk
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\title{SCHUR $Q$-FUNCTIONS AND SPIN CHARACTERS OF SYMMETRIC GROUPS I}
\author{A. O. Morris\thanks{%
Department of Mathematics, University of Wales, Aberystwyth, Dyfed SY23 3BZ}
\ and A. A. Abdel-Aziz\thanks{%
Department of Mathematics, Faculty of Education in Abha, PO Box 9002, Abha,
Saudi Arabia}\\
{\em Dedicated to Dominique Foata on his 60th birthday}
}
\date{\small Submitted: March 31, 1995; Accepted: November 15, 1995}
\begin{document}
\maketitle
\pagestyle{myheadings} 
\markright{\sc the electronic journal of combinatorics 3 (2) (1996), \#R20\hfill} 
\thispagestyle{empty}

\section{Introduction}

In a classic paper, I. Schur [6] introduced a class of symmetric functions,
now called Schur $Q$-functions, in order to determine the irreducible spin
(projective) characters of symmetric groups. In the case of the ordinary
characters of symmetric groups, going back to the early work of D. E.
Littlewood and A. R. Richardson [3], the corresponding Schur functions have
been used to give useful combinatorial formulae for determining explicit
values for these characters (see for example [2]). Our aim in this paper is
to obtain combinatorial formulae for the spin characters. We see that as
always in the spin case, the formulae are considerably more complicated with
a number of new interesting phenomena arising.

\section{Notation and Preliminaries}

\subsection{Compositions and partitions}

If $\ell $ is a positive integer, then $\lambda =(\lambda _1,\ldots ,\lambda
_m)$ is a {\it composition} of $\ell $ if $\lambda _i(i=1,\ldots ,m)$ are
positive integers such that 
\[
\sum_{i=1}^m\lambda _i=\ell . 
\]
If, in addition 
\[
\lambda _1\geq \lambda _2\geq \ldots \lambda _m>0 
\]
then $\lambda $ is a {\it partition} of $\ell $; $\lambda $ is called a {\it %
strict} partition if 
\[
\lambda _1>\lambda _2>\ldots \lambda _m>0. 
\]
When necessary, we write $\ell (\lambda )$ for $m$, the {\it length} of $%
\lambda $ and $|\lambda |$ for $\ell $, the {\it weight }of $\lambda $. Let $%
P(\ell )$ denote the set of partitions of $\ell $, $SP(\ell )$ the set of
strict partitions of $\ell $ and $O(\ell )$ the set of partitions where all
the parts $\lambda _i$ are odd. Also, we write $\lambda \setminus \{\lambda
_i\}$ for the partition $\{\lambda _1,\ldots ,\lambda _{i-1},\lambda
_{i+1},\ldots ,\lambda _m\}$, that is, with the $i$th part deleted with an
obvious extension to the deletion of more parts.

Partitions are also denoted as $\lambda = (1^{m_1}2^{m_2}\ldots)$,
indicating that the part $i$ is repeated $m_i$ times.

If $\lambda\in P(\ell)$, let 
\[U(\lambda) = \{\mu\in P(| \mu | \leq \ell)|\{
\mu_1,\ldots,\mu_{\ell(\mu)}\} \subseteq \{\lambda_1,\ldots,\lambda_m\},\]
that is, $U(\lambda)$ is the set of {\it sub-partitions} of $\lambda$. 
For $%
r \leq \ell$, let 
\[U(\lambda,r) = \{\mu\in U(\lambda)|\mu\in P(r)\},\] $%
u(\lambda) = |U(\lambda)|,\ u(\lambda,r) = |U(\lambda,r)|$. Then, clearly $%
U(\lambda,\ell) = \{\lambda\}$ and $u(\lambda) \leq 2^{\ell(\lambda)}$ and $%
u(\lambda) = 2^{\ell(\lambda)}$ if and only if $\lambda\in SP(\ell)$.

\begin{definition}
A {\it separation} of $\lambda \in P(\ell )$ corresponding to the
composition $w=(w_1,w_2,\ldots ,w_k)$ of $\ell $ is a $k$-set of partitions $%
(\lambda ^{(1)},\lambda ^{(2)},\ldots ,\lambda ^{(k)})$ such that $\lambda
^{(i)}\in P(w_i)\ (i=1,\ldots ,k)$.
\end{definition}

A complete set of separations $(\lambda ^{(1)},\ldots ,\lambda ^{(k)})$ of $%
\lambda $ corresponding to the composition $w=(w_1,\ldots ,w_k)$ can be
obtained by the following algorithm. 
\[
\lambda ^{(1)}\in U(\lambda ,w_1),\ \lambda ^{(2)}\in U(\lambda \setminus
\lambda ^{(1)},w_2),\ldots ,\lambda ^{(k)}\in U(\lambda \setminus (\lambda
^{(1)},\ldots ,\lambda ^{(k-1)}),w_k) 
\]
where 
\[
U(\lambda )\supset U(\lambda \setminus \lambda ^{(1)})\supset U(\lambda
\setminus (\lambda ^{(1)},\lambda ^{(2)}))\supset \ldots \supset U(\lambda
\setminus (\lambda ^{(1)},\ldots ,\lambda ^{(k)}))=\emptyset . 
\]
In this way, we form a tree of partitions of $0,w_1,\ldots ,w_k$ and
corresponding to each branch of the tree is a separation of $\lambda $. For
example, the separation of the partition $\lambda =(1^73^25)$ corresponding
to the composition $w=(7,5,6)$ is represented by the following tree.

%
%-------------------------------------------------- 
%
% THE FIRST PICTURE
%
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\put(130,10){(15)}
\put(155,10){$(1^33)$}
\put(185,10){$(3^2)$}
\put(33,20){\line(0,1){25}}
\put(61,20){\line(0,1){25}}
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\put(61,65){\line(-1,2){14}}
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%
%-------------------------------------------------- 
%
\subsection{Hall-Littlewood Symmetric Functions}

Let ${\bf x}=\{x_1,x_2,\ldots ,x_\ell \}$ be a set of $\ell $ independent
indeterminates and $t$ an independent parameter. If $\alpha =(\alpha
_1,\alpha _2,\ldots ,\alpha _\ell )\in {\bf Z}^\ell $, write $e^\alpha
:=x_1^{\alpha _1}\ldots x_\ell ^{\alpha _\ell }$ and 
\[
\theta _\ell ({\bf {x}},t)=\prod_{1\leq i<j\leq \ell }\frac{x_i-tx_j}{x_i-x_j%
}. 
\]
If $\lambda $ is a partition with $\ell (\lambda )=m\leq \ell $, put $%
\lambda _i=0$ for $i=m+1,\ldots ,\ell $. Then the {\it Hall-Littlewood
symmetric functions} $P_\lambda ({\bf x};t)$ are defined by 
\begin{equation}\label{e21}
P_\lambda ({\bf x};t)=\nu _\lambda (t)^{-1}\sum_{w\in S_\ell }w(e^\lambda
\theta _\ell ({\bf x};t)),
\end{equation}
where $S_\ell $ is the symmetric group acting on ${\bf x}$, 
\[ \nu _\lambda (t)=\prod_{i\geq 0}\nu _{n_i}(t), 
\;\; \nu _r(t)=(1-t)^{-r}\prod_{i=1}^r(1-t^j)\]
and $n_i:=n_i(\lambda )$ is the multiplicity of $i$ in $\lambda $.
Alternatively, as shown by Macdonald [4] 
\begin{equation}\label{e22}
P_\lambda ({\bf x};t)=\sum_{w\in S_\ell /S_\ell ^\lambda }w(e^\lambda \theta
_\ell ({\bf x};t)),
\end{equation}
where $S_\ell ^\lambda $ is the subgroup of $w\in S_\ell $ such that $%
\lambda _{w(i)}=\lambda _i\ (1\leq i\leq \ell )$, and the summation is taken
over left coset representatives of $S_\ell ^\lambda $ in $S_\ell $. In fact,
$S_\ell ^\lambda =\prod_{i\geq 0}S_{n_i}$. We are going to be more concerned
with another version of the Hall-Littlewood symmetric functions, namely 
\begin{equation}\label{e23}
Q_\lambda ({\bf x};t):=b_\lambda (t)P_\lambda ({\bf x};t),
\end{equation}
where $b_\lambda (t):=\prod_{i\geq 1}\prod_{j=1}^{n_i}(1-t^j)$. In
particular, we put $q_k({\bf x};t)=Q_{(k)}({\bf x};t)$ and note that from
(\ref{e21}) we have 
\begin{equation}\label{e24}
q_k({\bf x};t)=(1-t)\sum_{i=1 }^\ell x_i^k\beta _i({\bf x};t),
\end{equation}
where 
\[
\beta _i({\bf x};t)=\prod_{{{j=1}}\atop{{i\neq j}}}\frac{x_i-tx_j}{x_i-x_j}%
\quad (i=1,2,\ldots ,\ell ). 
\]
Macdonald [4] has given the following useful recurrence relation for the $%
Q_\lambda ({\bf x};t)$. 
\begin{equation}\label{e25}
Q_\lambda ({\bf x};t)=(1-t)\sum_{i=1}^\ell x_i^{\lambda _1}\beta _i({\bf x}%
;t)Q_{\lambda \setminus \{\lambda _1\}}({\bf x}\setminus \{x_i\};t).
\end{equation}
We are, in particular, interested in the case corresponding to $t=-1$, as
these are the so-called Schur $Q$-functions which were introduced by I. Schur
[6] to deal with the spin characters of symmetric groups. In this case, we
have
%
%noindent (i) If $\lambda \in P(\ell ),$ then 
\begin{equation}\label{e26}
\left.
 \begin{array}{l}
  \mbox{(i) If } \lambda \in P(\ell ),\mbox{ then }
   b_\lambda (-1)=\left\{ 
   \begin{array}{cc}
     2^{\ell (\lambda )}   & \mbox {if} \lambda \in SP(\ell )\;, \\ 
     0 & \mbox {otherwise,}
   \end{array}
   \right. 
 \\
\\
 \mbox{(ii) } Q_\lambda ({\bf x};-1)=\left\{ 
 \begin{array}{c}
  2^{\ell (\lambda )}P_\lambda ({\bf x};-1)\ \mbox {if}\ \lambda
  \in SP(\ell)\;, \\ 
  0\ \mbox{otherwise,}\;
 \end{array}
 \right. 
 \\
\\
 \mbox{(iii) } q_k({\bf x};-1)=
  2\sum_{i=1}^\ell x_i^k\beta _i({\bf x};-1) \quad (k=1,2,\ldots ).
\end{array}
\right\}
\end{equation}
 From now on, unless otherwise stated, we assume that $\lambda \in SP(\ell)$.
Furthermore, in order to simplify the notation, 
we write $Q_\lambda({\bf x})$
for $Q_\lambda ({\bf x},-1),q_k({\bf x})$ for $q_k({\bf x},-1)$, $\beta_k(%
{\bf x})$ for $\beta_k({\bf x},-1),\ \theta_\ell({\bf x})$ for $\theta_\ell(%
{\bf x},-1)$, etc.

I. Schur [6] defines $Q$-functions by a recursive formula 
and then determines
a closed formula similar to (\ref{e22}) above. 
We reverse the process and show
that the $Q_\lambda ({\bf x})$ as defined above satisfy certain properties,
including the recurrence properties used in I. Schur's definition.

If $\lambda =(\lambda _1,\ldots ,\lambda _m)\in SP(\ell )$, 
then from (\ref{e22})
and (\ref{e26})(ii), we have 
\begin{equation}\label{e27}
Q_\lambda ({\bf x})=2^m\sum_{w\in S_\ell /S_{\ell -m}}w(e^\lambda \theta
_\ell ({\bf x})),
\end{equation}
where $S_{\ell -m}$ is the subgroup of $S_\ell $ which fixes $x_{m+1},\ldots
,x_\ell $ and the summation is over all left coset representatives of $%
S_{\ell -m}$ in $S_\ell $. Thus, we have 
\begin{equation}\label{e28}
Q_\lambda ({\bf x})=2^m\sum_{w\in S_\ell /S_{\ell -m}}(-1)^{\frac{m(m-1)}%
2}w\left( e^\lambda \frac{\theta _m({\bf x})}{\prod^m_{i=1}\beta _i({\bf x})}%
\right) ,
\end{equation}
which is the form of the $Q$-function obtained by I. Schur.

We now proceed to show that the recurrence relations used to define Schur's $%
Q$-functions are satisfied. In order to do this, we need the following which
was indeed proved by I. Schur [6, p.226]. 
\begin{equation}\label{e29}
\theta _m({\bf x})=\sum_{i=2}^m(-1)^i\theta _2(x_1,x_i)\theta _{m-2}({\bf x}%
\setminus \{x_1,x_i\})\qquad \mbox {if}\ m\ \mbox {is even}
\end{equation}
and 
\begin{equation}\label{e210}
\theta _m({\bf x})=\sum_{i=1}^m(-1)^{i-1}\theta _{m-1}({\bf x}\setminus
\{x_i\})\ \mbox {if}\qquad m\ \mbox {is odd}.
\end{equation}

\begin{theorem}
Let $\lambda =(\lambda _1,\ldots ,\lambda _m)\in SP(\ell )$. Then

\begin{enumerate}
\item[(i)]  $Q_{(r,s)}({\bf x})=q_r({\bf x})q_s({\bf x})+2%
\sum_{i=1}^s(-1)^sq_{r+i}({\bf x})q_{s-i}({\bf x})$.

\item[(ii)]  If $m$ is even 
\[
Q_\lambda ({\bf x})=\sum_{i=2}^m(-1)^iQ_{(\lambda _1,\lambda _i)}({\bf x}%
)Q_{\lambda \setminus \{\lambda _1,\lambda _i\}},({\bf x}). 
\]

\item[(iii)]  If $m$ is odd 
\[
Q_\lambda ({\bf x})=\sum_{i=1}^m(-1)^{i-1}q_{\lambda _i}({\bf x})Q_{_\lambda
\setminus \{\lambda _i\}}({\bf x}). 
\]
\end{enumerate}
\end{theorem}

\begin{proof} 
(i) From (2.5) we have
\begin{displaymath}
\begin{array}{ll}
Q_{(r,s)}({\bf x}) &= \sum_{i=1}^\ell x_i^r \beta_i({\bf x})q_s({\bf x}\setminus 
\{x_i\})\\
\\
&= \sum_{i=1}^\ell x_i^r \beta_i({\bf x})\left( \sum^\ell_{j=1} x_j^s 
\beta_j({\bf x})
\frac {x_j-x_i}{x_j+x_i}\right)\\
\\
&= \sum_{i=1}^\ell x_i^r \beta_i({\bf x}) \left(\sum_{j=1}^\ell x_j^s 
\beta_j({\bf x})\left(
1 - \frac {2x_i}{x_j+x_i}\right)\right),
\end{array}
\end{displaymath}
from which we deduce that
\begin{equation}\label{e211}
Q_{(r,s)}({\bf x}) = q_r({\bf x})q_s({\bf x}) - 2 \sum_{i=1}^\ell 
x_i^{r+1}\beta_i({\bf x})
\left(\sum_{j=1}^\ell x_j^s\beta_j({\bf x}) \frac{1}{x_j+x_i}\right).
\end{equation}
A similar calculation, but now writing $\frac {x_j-x_i}{x_j+x_i} = -1 + \frac 
{2x_j}{x_j+x_i}$
gives
\begin{equation}\label{e212}
Q_{(r+1,s-1)}({\bf x}) = -q_{r+1}({\bf x}) q_{s-1}({\bf x}) + 2\sum_{i=1}^\ell
x_i^{r-1} \beta _i({\bf x})
\left(\sum_{j=1}^\ell x_j^s\beta_j({\bf x})
\frac {1}{x_j+x_i}\right).
\end{equation}
Now, adding (\ref{e211}) and (\ref{e212}), we have
\begin{displaymath}
Q_{(r+1,s-1)}({\bf x}) = q_r({\bf x})q_s({\bf x}) - q_{r+1}({\bf x})
q_{s-1}({\bf x}) - Q_{(r,s)}({\bf x}).
\end{displaymath}
A similar calculation to the above shows that
\begin{displaymath}
Q_{(r,1)}({\bf x}) = q_r({\bf x})q_1({\bf x}) - 2q_{r+1}({\bf x})
\end{displaymath}
and the proof is now completed by induction on $s$.

\noindent (ii) If $m$ is even, by substituting (\ref{e29}) in 
(\ref{e28}), we obtain
\begin{displaymath}
\begin{array}{ll}
Q_\lambda({\bf x}) &= 2^m \sum_{w\in S_\ell/S_{\ell-m}} 
(-1)^{m(m-1)/2}w\left(\frac
{e^\lambda \sum_{i=2}^m(-1)^i\theta_2(x_1,x_i)\theta_{m-2}({\bf x}
\setminus
\{x_1,x_i\})}
{\prod_{j=1}^m \beta_j({\bf x})}\right)\\
\\
&= \sum_{i=2}^m(-1)^i\left(2^2\sum_{w\in S_\ell/S_{\ell-2}} - w\left
(e^{(\lambda_1,\lambda_i)}
\frac {\theta_2(x_1,x_i)}{\beta_1({\bf x})\beta_i({\bf x})}\right)\right)\\
\\
&\qquad \left(2^{m-2}\sum_{w\in S_\ell/S_{\ell-m+2}}
(-1)^{(m-2)(m-3)/2} w\left(\frac 
{(e^{(\lambda\setminus\{\lambda_1,\lambda_i\})}\theta_{m-2}
({\bf x}\setminus \{x_1,x_i\})}
{\prod^m_{{j=2}\atop{j\neq i}}\beta_j({\bf x})}\right)\right)\\
\\
&= \sum_{i=2}^m (-1)^iQ_{(\lambda_1,\lambda_i)}({\bf x})Q_{\lambda\setminus 
\{\lambda_1,\lambda_i
\}}({\bf x})
\end{array}
\end{displaymath}
The proof of (iii) is similar, but using (\ref{e210}).
\end{proof}
Macdonald [4, p.109] has shown that the inductive formula 
(\ref{e25}) can be used
to extend the definition of the Schur $Q$-functions to 
any $Q_\mu ({\bf x})$,
where $\mu =(\mu _1,\ldots ,\mu _k)\in {\bf Z}^k$, for any $k$.
Furthermore, Theorem 2.1(i) has been used to reduce such a $Q_\mu ({\bf x})$
to a linear combination of $Q_\lambda ({\bf x})$ such that $\lambda \in
SP(\ell )$ (see also Morris [5]). The reduction rules are as follows.

\begin{theorem}
(i) $Q_{(\ldots ,r,s,\ldots )}({\bf x})=
- -Q_{(\ldots ,s,r,\ldots )}({\bf x})$
with $Q_{(\ldots ,i,\ldots ,i,\ldots )}({\bf x})=0$.\\%\break
\noindent (ii) $Q_{\ldots ,r,-r,\ldots )}({\bf x})=0$,
$Q_{(\mu _1,\ldots
,\mu _{i-1},-r,r,\mu _{i+1},\ldots ,\mu _n)}({\bf x})=2(-1)^rQ_{(\mu
_1,\ldots ,\mu _{i-1},\mu _{i+1},\ldots ,\mu _n)}({\bf x})$;\\
in particular 
$Q_{(s,r)}({\bf x})=-Q_{(r,s)}({\bf x}),\quad r\neq s$ 
and $Q_{(-r,r)}({\bf x})=2(-1)^r$.
\end{theorem}

A consequence of (i) is that if $\lambda =(\lambda _1,\ldots ,\lambda _m)\in
SP(\ell )$, then 
\begin{equation}\label{e213}
Q_\lambda ({\bf x})=\epsilon (\sigma )Q_{(\lambda _{\sigma (1)},\ldots
,\lambda _{\sigma (m)})} 
\end{equation}
for all $\sigma \in S_m$, where $\epsilon (\sigma )$ is the sign of the
permutation $\sigma $.

\section{Further Results on Schur $Q$-functions}

 From now on, it will be essential that it is clear on what sets our
symmetric groups act, thus the notation will be modified. 
We denote by $%
S_{\underline m}$ the symmetric group which acts on the set 
${\underline m}
= \{1,2,\ldots,m\}$ and $S_{\underline m\setminus \underline k}$ the
symmetric group which acts on ${\underline m} \setminus {\underline k} =
\{k+1,\ldots,m\}$. Also, we need to determine certain explicit left coset
representations of certain groups. These are given by the following.

\begin{lemma}\label{l31}
For $k=2,\ldots ,m$, put $w_k=(2k)\in S_{\underline{m}}$ and for $%
k=1,2,\ldots ,m$, put $u_k=(1k)\in S_{\underline{m}}$. Then

\begin{enumerate}
\item[(i)]  $\{w_k|k=2,\ldots ,m\}$ is a set of left coset representatives
of $S_{\underline{m}\setminus \underline{2}}$ in $S_{{\underline{m}}%
\setminus \underline{1}}$.

\item[(ii)]  $\{u_k|k=1,\ldots ,m\}$ is a set of left coset representatives
of $S_{\underline{m}\setminus \underline{1}}$ in $S_{\underline{m}}$.
\end{enumerate}
\end{lemma}

We can now use the following lemma to give an alternative 
form for the $%
Q_\lambda({\bf x})$.

\begin{lemma}\label{l32}
If $\lambda =(\lambda _1,\ldots ,\lambda _m)\in SP(\ell )$, then 
\begin{equation}\label{313}
Q_\lambda ({\bf x})=\sum_{\eta \in S_{\underline{m}\setminus \underline{1}%
}/S_{\underline{m}\setminus \underline{2}}}\epsilon (\eta )Q_{(\lambda
_1,\lambda _{\eta (2)})}({\bf x})Q_{(\lambda _{\eta (3)},\ldots ,\lambda
_{\eta (m)})}({\bf x})
\end{equation}
if $m$ is even, and 
\begin{equation}\label{e314}
Q_\lambda ({\bf x})=\sum_{\eta \in S_{\underline{m}}/S_{\underline{m}%
\setminus \underline{1}}}\epsilon (\eta )q_{\eta (1)}({\bf x})Q_{(\lambda
_{\eta (2)},\ldots ,\lambda _{\eta (m)})}({\bf x})
\end{equation}
if $m$ is odd.
\end{lemma}

\begin{proof}
If $\eta\in S_{\underline m\setminus \underline 1}$, then by Lemma 3.1(i) we 
have
$\eta=w_k\nu_k$, where $w_k = (2k),\ \nu_k\in S_{\underline m\setminus 
\underline 2}$ for some 
$2\leq k\leq m$, and so
$\eta = $
\begin{displaymath}
(2k)
\left(
\begin{array}{ccccccc}
3&\ldots&k&\ldots&s&\ldots&m\\
\eta(3)&\ldots&\eta(k)&\ldots&\eta(2)&\ldots&\eta(m)
\end{array}
\right)
=
\left(
\begin{array}{cccccccc}
2&3&\ldots&k&\ldots&s&\ldots&m\\
k&\eta(3)&\ldots&\eta(k)&\ldots&2&\ldots&\eta(m)
\end{array}
\right)
\end{displaymath}
\begin{displaymath}
= (k,\eta(k))
\left(
\begin{array}{cccccccc}
2&3&\ldots&k&\ldots&s&\ldots&m\\
\eta(k)&\eta(3)&\ldots&\eta(2)&\ldots&\eta(s)&\ldots&\eta(m)
\end{array}
\right),
\end{displaymath}
where $\eta(2)=k$ and $\eta(s) = 2$.  Hence, if $m$ is even
\begin{displaymath}
\sum_{\eta\in S_{{\underline m}\setminus{\underline 1}}/S_{{\underline 
m}\setminus{\underline 2}}}
\epsilon(\eta)Q_{(\lambda_1,\lambda_{\eta(n)})}({\bf x})Q_{(\lambda_{\eta(3)},
\ldots,\lambda_{\eta(m)})}({\bf x})
\end{displaymath}
\begin{displaymath}
= \sum_{k=2}^m \epsilon(k,\eta(k)) \epsilon \left(
\begin{array}{cccccccc}
2&3&\ldots&k&\ldots&s&\ldots&m\\
\eta(k)&\eta(3)&\ldots&\eta(2)&\ldots&\eta(s)&\ldots&\eta(m)
\end{array}
\right)
Q_{(\lambda_1,\lambda_k)}({\bf 
x})Q_{(\lambda_{\eta(3)},\ldots,\lambda_{\eta(m)})}
({\bf x})
\end{displaymath}
\begin{displaymath}
= \sum_{k=2}^m \epsilon(k,\eta(k))Q_{(\lambda_1,\lambda_k)}({\bf 
x})Q_{(\lambda_3,\ldots,
\lambda_2,\ldots,\lambda_m)}({\bf x}),
\end{displaymath}
with $\lambda_2$ in the $k$ position in 
$Q_{(\lambda_3,\ldots,\lambda_2,\ldots,\lambda_m)}
({\bf x})$, where we have used (\ref{e213}).
Now, by using Theorem 2.3(i) repeatedly, we have that the right hand side of the 
above reduces
to 
\begin{displaymath}
\sum_{k=2}^m (-1)^{k-2} Q_{(\lambda_1,\lambda_k)}({\bf x}) Q_{\lambda\setminus 
\{\lambda_1,
\lambda_k\}}({\bf x}) = Q_\lambda({\bf x})
\end{displaymath}
by Theorem 2.2(ii).  A similar argument can be used to prove (ii).
\end{proof}

\begin{lemma}\label{l33}
If $m$ is even and $\lambda \in SP(\ell )$, then 
\[
Q_\lambda ({\bf x})=\frac 1m\sum_{\sigma \in S_{\underline{m}}/S_{{%
\underline{m}}\setminus {\underline{2}}}}\epsilon (\sigma )Q_{(\lambda
_{\sigma (1)},\lambda _{\sigma (2)})}({\bf x})Q_{(\lambda _{\sigma
(3)},\ldots ,\lambda _{\sigma (m)})}({\bf x}) 
\]
\end{lemma}

\begin{proof}
If $m$ is even, then from (\ref{e213}), Theorem 2.2(ii) 
and Lemma 3.2, we have
\begin{displaymath}
\begin{array}{ll}
Q_\lambda({\bf x}) &= \frac {1}{m} \sum_{\sigma\in S_{\underline 
m}/S_{{\underline m}\setminus {\underline 1}}}
\epsilon(\sigma) Q_{(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(m)})}({\bf x})\\
\\
&= \frac {1}{m} \sum_{\sigma\in S_{\underline m}/S_{{\underline m}\setminus 
{\underline 1}}}   \epsilon(\sigma)
\sum_{i=2}^m (-1)^i Q_{(\lambda_{\sigma(1)},\lambda_{\sigma(i)})}({\bf x})
Q_{(\lambda_{\sigma(2)},\ldots,\lambda_{\sigma(m)})\setminus
\{\lambda_{\sigma(1)},\lambda_{\sigma(i)}\}}({\bf x})\\
\\
&= \frac {1}{m} \sum_{\sigma\in S_{\underline m}/
S_{{\underline m}\setminus {\underline 1}}}\epsilon(\sigma)
\sum_{ \eta\in S_{{\underline m}\setminus {\underline 1} }/
S_{ {\underline m} \setminus {\underline 2}}}
%}}
\epsilon(\eta)
Q_{(\lambda_{\sigma\eta(1)},\lambda_{\sigma\eta(2)})} ({\bf x})
Q_{(\lambda_{\sigma\eta(3)},\ldots,\lambda_{\sigma\eta(m)})}
({\bf x})\\
\\
&= \frac {1}{m}\sum_{\sigma\in S_{\underline m}/S_{{\underline m}\setminus 
{\underline 2}}}
 \epsilon(\sigma)
Q_{(\lambda_{\sigma(1)},\lambda_{\sigma(2)})}({\bf x}) 
Q_{(\lambda_{\sigma(3)},\ldots,\lambda_{\sigma(n)})}({\bf x}).
\end{array}
\end{displaymath}
\end{proof}
We can now use this result as the basis for an inductive proof
of the following.

\begin{theorem}
(i) If $m=2\mu $ is even, then 
\[
Q_\lambda ({\bf x})=\frac 1{2^\mu \mu !}\sum_{\sigma \in S_{\underline{m}%
}}\epsilon (\sigma )Q_{(\lambda _{\sigma (1)},\lambda _{\sigma (2)})}({\bf x}%
)Q_{(\lambda _{\sigma (3)},\lambda _{\sigma (4)})}({\bf x})\ldots
Q_{(\lambda _{\sigma (m-1)},\lambda _{\sigma (m)})}({\bf x}) 
\]
(ii) If $m=2\mu +1$ is odd, then 
\[
Q_\lambda ({\bf x})=\frac 1{2^\mu \mu !}\sum_{\sigma \in S_{\underline{m}%
}}\epsilon (\sigma )q_{\lambda _{\sigma (1)}}({\bf x})Q_{(\lambda _{\sigma
(2)},\lambda _{\sigma (3)})}({\bf x})Q_{(\lambda _{\sigma (4)},\lambda
_{\sigma (5)})}({\bf x})\ldots Q_{(\lambda _{\sigma (m-1)},\lambda _{\sigma
(m)})}({\bf x}) 
\]
\end{theorem}

\begin{proof}
(i) The proof is by induction on $m$.  If $m=1$, then clearly
\begin{displaymath}
Q_{(\lambda_1,\lambda_2)}({\bf x}) = \frac {1}{2} \sum_{\sigma\in S_2}
\epsilon(\sigma) Q_{(\lambda_{\sigma(1)},\lambda_{\sigma(2)})}({\bf x}).
\end{displaymath}
If $m>1$, then by Lemma \ref{l33} and the inductive assumption, 
we have
\begin{displaymath}
\begin{array}{ll}
Q_\lambda({\bf x}) &= 
\frac {1}{2\mu}
\sum_{\sigma\in
	S_{\underline m}/S_{{\underline m}
	\setminus {\underline 2}}}
\epsilon(\sigma) Q_{(\lambda_{\sigma(1)},\lambda_{\sigma(2)})}
({\bf x}) \frac {1}{2^{\mu-1}(\mu-1)!}
\sum_{\rho\in S_{({\underline m})\setminus \sigma(2)}}
\epsilon(\rho) Q_{(\lambda_{\rho \sigma(3)},\lambda_{\rho\sigma(4)})}
({\bf x})\\
&\qquad \ldots Q_{(\lambda_{\rho\sigma(m-1)},\lambda_{\rho\sigma(m)}}
({\bf x})\\
\\
&= \frac {1}{2^\mu \mu!} 
\sum_{\sigma\in S_{\underline m}/S_{{\underline m}\setminus {\underline 2}}}
\epsilon(\sigma)
Q_{(\lambda_{\sigma(1)},\lambda_{\sigma(2)})} ({\bf x})
\sum_{\sigma'\in S_{{\underline m}\setminus 2}} 
\epsilon(\sigma') 
Q_{\lambda_{\sigma'(3)},\lambda_{\sigma'(4)})}({\bf x})\\
&\qquad \ldots Q_{\lambda_{(\sigma'(m-1)},\lambda_{\sigma'(m)})}({\bf x})\\
\\
&= \frac {1}{2^{\mu}\mu!} \sum_{\sigma\in S_m} \epsilon(\sigma)
Q_{(\lambda_{\sigma(1)},\lambda_{\sigma(2)})}({\bf x})
Q_{(\lambda_{\sigma(3)},\lambda_{\sigma(4)})} ({\bf x})
\ldots Q_{(\lambda_{\sigma(m-1)},\lambda_{\sigma(m)})} ({\bf x}),
\end{array}
\end{displaymath}
as required.
The proof of (ii) is similar.
\end{proof}
The above result can be interpreted in terms of wreath products of certain
groups; in fact, the hyperoctahedral group.

If $\lambda =(\lambda _1,\ldots ,\lambda _m)$ where $m=2\mu $ is even, let 
\[
M=\{\{\lambda _1,\lambda _2\},\{\lambda _3,\lambda _4\},\ldots ,\{\lambda
_{m-1},\lambda _m\}\} 
\]
and if $\sigma \in S_m$, let 
\[
\sigma (M)=\{\{\lambda _{\sigma (1)},\lambda _{\sigma (2)}\},\{\lambda
_{\sigma (3)},\lambda _{\sigma (4)}\},\ldots ,\{\lambda _{\sigma
(m-1)},\lambda _{\sigma (m)}\}\}. 
\]
If $H=\{\sigma \in S_m|\sigma (M)=M\}$, then $H$ is a subgroup of $S_m$ of
order $2^\mu \mu !$. In fact, if $\pi =(12)(34)\ldots (m-1,m)\in S_m$, then $%
H$ is the centralizer $C_{S_m}(\pi )$ of $\pi $ in $S_m$, which implies that 
$H\cong S_2\wr S_m$ (see [2,p.135]), the wreath product of $S_2$ by $S_m$, which
is the hyperoctahedral group or the Weyl group of type $B_\mu $. Thus, the
above results can be rewritten as

\begin{theorem}
For $m\geq 1$, we have

(i) if $m=2\mu $ is even 
\[
Q_\lambda ({\bf x})=\sum_{\sigma \in S_m\setminus S_2\wr S_\mu }\epsilon
(\sigma )Q_{(\lambda _{\sigma (1)},\lambda _{\sigma (2)})}({\bf x})\ldots
Q_{(\lambda _{\sigma (m-1)},\lambda _{\sigma (m)})}({\bf x}) 
\]

(ii) if $m=2\mu +1$ is odd 
\[
Q_\lambda ({\bf x})=\sum_{\sigma \in S_m\setminus S_2\wr S_\mu }\epsilon
(\sigma )q_{\lambda _{\sigma (1)}}Q_{(\lambda _{\sigma (2)},\lambda _{\sigma
(3)})}({\bf x})\ldots Q_{(\lambda _{\sigma (m-1)},\lambda _{\sigma (m)})}(%
{\bf x}) 
\]
\end{theorem}

\section{Application to the Spin Characters of Symmetric 
\protect{\newline } Groups}

As was mentioned earlier, Schur $Q$-functions were first 
introduced by Schur
[6] in his work on the spin (projective) characters of 
symmetric groups (see
also [1]). He showed that the irreducible spin characters 
of $S_\ell $ were
parameterised by strict partition $\lambda \in SP(\ell )$, 
but that this
correspondence is not one-to-one. In fact, if we say that $\lambda $ is even
(odd) if $\ell -\ell (\lambda )$ is even (odd), there is one irreducible
self-associate spin character $\zeta ^\lambda $ corresponding to $\lambda
\in SP(n)$ if $\lambda $ is even and two irreducible associate spin
characters $\zeta ^\lambda $ and $\zeta ^{\lambda ^{\prime }}$ if $\lambda $
is odd. The connection with Schur $Q$-functions is given by 
\begin{equation}\label{e415}
Q_\lambda ({\bf x})=\sum_{\pi \in O(\ell )}2^{\frac 12(\ell (\lambda )+\ell
(\pi )+\epsilon (\lambda ))}z_\pi ^{-1}\zeta _\pi ^\lambda p_\pi ({\bf x})
\end{equation}
where $z_\pi =\prod_{i\geq 1}i^{m_i}m_i!$, where $m_i$ is the multiplicity
of $i$ in the partition $\pi =(1^{m_1}3^{m_3}\ldots )$, $p_\pi ({\bf x})=p_1(%
{\bf x})^{m_1}p_3({\bf x})^{m_3}\ldots (p_i({\bf x})=\sum_{k=1}^\ell x_k^i)$%
, $\zeta _\pi ^\lambda $ is the value of the character $\zeta ^\lambda $ at
the class $\pi $ and 
\[
\epsilon (\lambda )=\left\{ 
\begin{array}{ll}
0 & \mbox {if $\ell-\ell(\lambda)$ is even} \\ 
1 & \mbox {if $\ell-\ell(\lambda)$ is odd.}
\end{array}
\right. 
\]
This formula gives the value of $\zeta ^\lambda $ on the classes $\pi \in
O(\ell )$ only; however, if $\lambda $ is even $\zeta _\pi ^\lambda =0$ if $%
\pi \in P(\ell )\setminus O(\ell )$ and if $\lambda $ is odd $\zeta _\pi
^\lambda =\zeta _\pi ^{\lambda ^{\prime }}=\zeta _\pi ^{\lambda ^{\prime }}$
if $\pi \in O(\ell )$, while if $\pi \not\in O(\ell )$, $\zeta _\pi
^\lambda \neq 0$ only if $\pi =\lambda $ and then $\zeta _\lambda ^\lambda $
is given explicitly by 
\begin{equation}\label{e416}
\zeta _\lambda ^\lambda =(-1)^{(-\ell (\lambda )+1)}{\sqrt{\frac{\lambda
_1\lambda _2\ldots \lambda _{\ell (\lambda )}}2}}
\end{equation}
and $\zeta _\lambda ^{\lambda ^{\prime }}=-\zeta _\lambda ^\lambda $. Thus,
formula (\ref{e415}) gives all the information required. 
In addition, we require 
\begin{equation}\label{e417}
\zeta _\pi ^{(\ell )}=2^{\frac 12(\ell (\sigma )-1-\epsilon )}
\end{equation}
where 
\[
\epsilon =\left\{ 
\begin{array}{ll}
1 & \mbox {if $\ell$ is odd} \\ 
0 & \mbox {if $\ell$ is even}
\end{array}
\right. 
\]
and 
\begin{equation}\label{e418}
q_\ell ({\bf x})=\sum_{\pi \in O(\ell )}2^{\ell (\pi )}z_\pi ^{-1}p_\pi (%
{\bf x}).
\end{equation}
Both results are originally due to Schur [6]; in fact (\ref{e418}) is 
(\ref{e415}) for
the particular case $\lambda =(\ell )$ using the value given by (\ref{e417}) for
the character $\zeta _\pi ^{(\ell )}$.

Our intention now is to give a recursive formula for the calculation of the $%
\zeta_\pi^\lambda$ based on the preceding work on Schur $Q$-functions.

We shall assume from now on that $\pi = (1^{m_1}3^{m_3}\ldots) \in O(\ell)$.
Let $\{\pi^1$, $\pi^2$, $\ldots$, $\pi^m\}$ be a separation of $\pi$; then
we note that 
\begin{equation}\label{e419}
\frac {z_\pi}{z_{\pi^1}z_{\pi^2}\ldots z_{\pi^m}} = \frac {y_\pi}
{y_{\pi^1}y_{\pi^2}\ldots y_{\pi^m}},
\end{equation}
where $y_\pi = \prod_{i\geq 1} m_i!$ (recall that $z_\pi = \prod_{i\geq 1}
i^{m_i}m_i!$).

We first give an explicit formula for $\zeta_\pi (r,s)\ (r>s)$, that is for
two part partitions. We prove

\begin{theorem}
If $\lambda =(r,s)\in SP(\ell )$ $(r>s)$, $\pi \in O(\ell )$, then 
\[
\zeta _\pi ^{(r,s)}=2^{\frac 12(\ell (\pi )-2-\epsilon )}y_\pi \left\{
\sum_{(\pi _r,\pi _s)=\pi }y_{\pi _r}^{-1}y_{\pi
_s}^{-1}+2\sum_{j=1}^s(-1)^j\left( \sum_{(\pi _{r+j},\pi
_{s-j})}y_{r+j}^{-1}y_{s-j}^{-1}\right) \right\} , 
\]
where the sums are taken over all separations $(\pi _{r+j},\pi _{s-j})$ of $%
\pi $ corresponding to the composition (partition) $(r+j,s-j)$ and $\epsilon
=0(1)$ if $\ell $ is even (odd).
\end{theorem}

\begin{proof}
For the partition $(r,s)\in SP(\ell)$, (\ref{e415}) becomes
\begin{equation}\label{e420}
Q_{(r,s)}({\bf x}) = \sum_{\pi\in O(\ell)} 2^{\frac 
{1}{2}(\ell(\pi)+2+\epsilon)}
z_\pi^{-1}\zeta_\pi^{(r,s)} p_\pi({\bf x}),
\end{equation}
where $\epsilon=0(1)$ if $\ell$ is even (odd).  
Now, by substituting (\ref{e418}) and (\ref{e420}) in Theorem 2.2(i),
we obtain 
\begin{displaymath}
\sum_{\pi\in O(\ell)} 2^{\frac {1}{2}(\ell(\pi)+2+\epsilon)}
z_\pi^{-1}\zeta^{(r,s)}_\pi p_\pi({\bf x}) =
\left( \sum_{\pi_r\in O(r)} z^{-1}_{\pi_r} 2^{\ell(\pi_r)} p_{\pi_r}({\bf 
x})\right)
\left(\sum_{\pi_s\in O(s)} z_{\pi_s}^{-1} 2^{\ell(\pi_s)} p_{\pi_s({\bf 
x})}\right)
\end{displaymath}
\begin{displaymath}
+ 2 \sum_{j=1}^s (-1)^j \left( \sum_{\pi_{r+j}\in O(r+j)} z^{-1}_{\pi_{r+j}} 
2^{\ell(\pi_{r+j})}
p_{\pi_{r+j}}({\bf x})\right)
\end{displaymath}
\begin{displaymath}
= \sum_{(\pi_r,\pi_s)=\pi\in O(\ell)} z_{\pi_r}^{-1} z_{\pi_s}^{-1} 
2^{\ell(\pi)}
p_\pi ({\bf x}) + 2\sum_{j=1}^s (-1)^j
\sum_{(\pi_{r+j},\pi_{s-j})=\pi\in O(\ell)} z^{-1}_{\pi_{r+j}} 
z^{-1}_{\pi_{s-j}} 2^{\ell(\pi)}
p_\pi ({\bf x}),
\end {displaymath}
where the sums are taken over all separations $(\pi_{r+j},\pi_{s-j})$ of $\pi$ 
corresponding to the composition
$(r+j,s-j)$.  Now, by comparing the coefficients of $p_\pi({\bf x})$ on both 
sides of this 
equation and using (4.19), we obtain the required result.
\end{proof}

Thus, the calculation of $\zeta_\pi^{(r,s)}$ for $\pi\in O(\ell)$, is
reduced to a calculation involving the separation of partitions and is given
in combinatorial terms.

\noindent {\bf Example:} We calculate $\zeta^{(6,3)}_{(1^63)}$. The
separations of $(1^63)$ corresponding to the composition 
\\$(6$, $3)$, $(7$, $%
2)$, $(8$, $1)$ and $(9)$ are $(\{(1^6)$, $(3)\}$, $\{(1^33)$, $(1^3)\})$, $%
\{(1^43)$, $(1^2)\}$, $\{(1^53)$, $(1)\}$ and \\
$\{(1^63),\emptyset\}$
respectively. Thus 
\[
\begin{array}{ll}
\chi^{(6,3)}_{(1^63)} & = 2^26!\left\{ \left( \frac {1}{6!} + \frac
{1}{3!3!}\right) + 2\left(- \frac {1}{4!2!} + \frac {1}{5!} - \frac
{1}{6!}\right)\right\} \\ 
& = 4.
\end{array}
\]
We now use the expansion of the Schur $Q$-functions $Q_\lambda({\bf x})$ in
terms of products of Schur $Q$-functions corresponding to two-part
partitions to give a combinatorial formula for $\zeta_\pi^\lambda$ in the
general case. We use Theorem 3.5 (or Theorem 3.4). We prove

\begin{theorem}
(i) If $m=2\mu $ is even and $\lambda =(\lambda _1,\ldots ,\lambda _m)\in
SP(\ell )$, $\pi =(1^{m_1}3^{m_3}\ldots )\in O(\ell )$, $I=\{1,3,5,\ldots
,m-1\}$, then 
\[
\zeta _\pi ^\lambda =\sum_{\sigma \in S_m/S_2\wr S_\mu }\epsilon (\sigma
)\left\{ \sum_{(\pi _1^\sigma ,\pi _3^\sigma ,\ldots )=\pi }y_\pi \left(
\prod_{i\in I}y_{\pi_i^\sigma}^{-1}\right) 2^{\frac 12(-\epsilon +\sum_{i\in 
I}\epsilon
_i^\sigma )}\prod_{i\in I}\zeta_{{\pi _i^\sigma }}^{(\lambda _{\sigma
(i)},\lambda _{\sigma (i+1)})}\right\} , 
\]
where for each $i\in I$, $\pi _i^\sigma =\pi _{(\lambda _{\sigma
(i)},\lambda _{\sigma (i+1)})}\in O(|\lambda _{\sigma (i)}|+|\lambda
_{\sigma (i+1)}|)$, $\epsilon _i^\sigma =0(1)$ if $|\lambda _{\sigma
(i)}|+|\lambda _{\sigma (i+1)}|$ is even (odd), $\epsilon =0(1)$ if $\ell $
is even (odd), and the sum is taken over all separations $(\pi _1^\sigma
,\pi _3^\sigma ,\ldots ,\pi _{m-1}^\sigma )$ of $\pi $.

\noindent (ii) If $m=2\mu +1$ is odd and $\lambda =(\lambda _1,\ldots ,\lambda 
_m)\in
SP(\ell )$, $\pi =(1^{m_1}3^{m_3}\ldots )\in O(\ell )$,\\
 $J=\{2,4,\ldots
,m-1\}$, then 
\[
\zeta _\pi ^\lambda =\sum_
{\sigma \in S_m/S_2\wr S_\mu }
\epsilon(\sigma)
\left\{ 
\sum_{(\pi
_{\lambda _{\sigma (1)}},\pi _2^\sigma ,\pi _4^\sigma,\ldots  ,\pi
_{m-1}^\sigma )=\pi} 
y_\pi y_{\pi _{\lambda _{\sigma (1)}}}^{-1}\prod_{j\in
J}y_{\pi _j^\sigma }^{-1}\right. 
\]
\[
2^{\frac 12(-\epsilon +\epsilon _{\sigma (1)}+\sum_{j\in J}\epsilon
_j^\sigma ))}\zeta _{\pi _{\lambda _{\sigma (1)}}}^{(\lambda _{\sigma
(1)})}\prod_{j\in J}\zeta _{\pi _j^\sigma }^{(\lambda _{\sigma (j)},\lambda
_{\sigma (j+1)}}), 
\]
where for each $j\in J$, $\pi _j^\sigma =\pi _{(\lambda _{\sigma
(j)},\lambda _{\sigma (j+1)})}\in O(|\lambda _{\sigma (j)}|+|\lambda
_{\sigma (j+1)}|),$ $\epsilon _j^\sigma =0(1)$ if $|\lambda _{\sigma
(j)}|+|\lambda _{\sigma (j+1)}|$ is even (odd), $\epsilon _{\sigma (1)}=0(1)$
if $|\lambda _{\sigma (1)}|$ is odd (even), $\epsilon =0(1)$ if $\ell $ is
odd (even) and the summation is over all separations $(\pi _{\lambda
_{\sigma (1)}},\pi _2^\sigma ,\pi _4^\sigma ,\ldots ,\pi _{m-1}^\sigma )$ of 
$\pi $ corresponding to the composition.
\end{theorem}

\begin{proof}
We prove (i) only; the proof of (ii) is similar.
 From Theorem 3.5(i) and (\ref{e415}) we obtain
\begin{displaymath}
\sum_{\pi\in O(\ell)} 2^{\frac{1}{2}(m+\ell(\pi)+\epsilon)}
z_\pi^{-1} \zeta^{\lambda}_\pi p_\pi({\bf x}) 
\end{displaymath}
\begin{displaymath}
=\sum_{\sigma\in S_m/S_2\wr S_\mu} \epsilon(\mu)
\prod_{i\in I} \sum_{\pi_i^\sigma} 
2^{\frac{1}{2}(2+\ell(\pi_i^\sigma) + \epsilon_i^\sigma)}
z_{\pi_i^\sigma}^{-1} 
\zeta_{\pi_i^\sigma}^{(\lambda_{\sigma(i)},\lambda_{\sigma(i+1)}} 
p_{\pi_i^\sigma}
({\bf x}).
\end{displaymath}
Since, if $\{\pi^1,\pi^2,\pi^3,\ldots,\pi^m\}$ is a separation of $\pi\in 
O(\ell)$, we
have $\ell(\pi) = \sum_{i=1}^m \ell(\pi^i)$, $p_\pi({\bf x}) =
%%%% \Prod_{i=1}^m p_{\pi_i} ({\bf x})$,
\prod_{i=1}^m p_{\pi_i} ({\bf x})$,
and using (\ref{e419}) and comparing the coefficients of
$p_\pi({\bf x})$ on both sides of the above, we obtain the desired formula.
\end{proof}

\noindent {\bf Remark 1:} We note that simplifications of the above formula
can be obtained in most cases; that is, in case (i) 
\[
2^{\frac{1}{2}(-\epsilon + \sum_{i\in I}\epsilon_i^\sigma)} = 1 
\]
in the following cases:

\begin{enumerate}
\item[(a)]  if $\ell $ is even and the parts of $\lambda $ are all even or
are all odd,

\item[(b)]  if $\ell $ is odd and all the parts of $\lambda $ except one are
even or are odd,
\end{enumerate}

and in case (ii) 
\[
2^{\frac{1}{2}(\epsilon+\epsilon_{\sigma(1)} + \sum_{j\in J}
\epsilon_j^\sigma)} = 1 
\]
in the following cases

\begin{enumerate}
\item[(a)]  if $\ell $ is even and all the parts of $\lambda $ are either
even or are odd except $\lambda _{\sigma (1)}$,

\item[(b)]  if $\ell $ is odd and all the parts of $\lambda $ are either odd
or are even except for $\lambda _{\sigma (1)}$.
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Furthermore, in order to apply these formulae, we require to determine left
coset representatives of $S_2\wr S_\mu$ in $S_m$ for both $m=2\mu$ and $%
2\mu+1$. We note that 
\[
[S_m:S_2\wr S_\mu] = \left\{ 
\begin{array}{cc}
1\cdot 3\cdot \ldots .m-1 & \mbox {if}\ m=2\mu\ \mbox {is even,} \\ 
1\cdot 3\cdot \ldots \cdot m & \mbox {if}\ m=2\mu+1\ \mbox {is odd.}
\end{array}
\right. 
\]
If $m=2\mu$ is even, left coset representatives of $S_2\wr S_\mu$ in $S_m$
are of the form $\psi = x_1x_2\ldots x_{m-1}$, where $x_i = (x_{i1},x_{i2})$
are transpositions determined inductively as follows:

\begin{enumerate}
\item[(i)]  Put $I_1=\{2,\ldots ,m\}$, then $x_{11}=2,\ x_{12}=k_1,\ k_1\in
I_1$.

\item[(ii)]  Put $I_2=I\setminus \{x_{11},x_{12}\}$, $d_2=\min I_2$, then $%
x_{21}=d_2,\ x_{22}=k_2,\ k_2\in I_2$ unless $k_1=2$, in which case $%
x_{21}=d_2+1$.

\item[(iii)]  Put $I_r=I_{r-1}\setminus\{x_{r-1,1},x_{r-1,2}\},\ d_r=\min I_r$,
then $x_{r1}=d_r,\ x_{r2}=k_r,\ k_r\in I_r$, unless $k_{r-1}=d_{r-1}$, in
which case $x_{r2}=d_{r-1}+1$.
\end{enumerate}

For example, the left coset representatives of $S_2\wr S_3$ in $S_6$ are
given as follows 
%
%-------------------------------------------------- 
%
\[
%%% SECOND PICTURE TO GO IN HERE
%%% \input pict2.tex
\begin{picture}(250,117)
\put(0,10){(44)}\put(15,10){(45)}\put(30,10){(46)}
\put(50,10){(44)}\put(65,10){(45)}\put(80,10){(46)}
\put(100,10){(33)}\put(115,10){(35)}\put(130,10){(36)}
\put(150,10){(33)}\put(165,10){(34)}\put(180,10){(36)}
\put(200,10){(33)}\put(215,10){(34)}\put(230,10){(36)}
\multiput(7,20)(50,0){5}{\line(1,2){15}}
\multiput(22,20)(50,0){5}{\line(0,1){30}}
\multiput(37,20)(50,0){5}{\line(-1,2){15}}
\put(15,57){(22)}\put(65,57){(23)}\put(115,57){(24)}
\put(165,57){(25)}\put(215,57){(26)}
\put(22,67){\line(5,1){100}}
\put(72,67){\line(5,2){50}}
\put(122,67){\line(0,1){20}}
\put(172,67){\line(-5,2){50}}
\put(222,67){\line(-5,1){100}}
\put(120,92){$e$}
\end{picture}
\]
%
%-------------------------------------------------- 
%
that is, $\{e$, $(45)$, $(46)$, $(23)$, $(23)(45)$, $(23)(46)$, $(24)$, $%
(24)(35)$, $(24)(36)$, $(25)$,\break 
\noindent $(25)(34)$, $(25)(36)$, $(26)$, $(26)(34)$, $(26)(35)\}$. 
%THIS IS ADDED ON

Similarly, if $m=2\mu+1$ is odd, the left coset representatives of $S_2\wr
S_\mu$ in $S_m$ are of the form $\phi = \phi_1\psi$, where $\phi_1$ can be
any of the $m$ transpositions $(1,k)$ $(1\leq k\leq m)$ and $\psi$ is
obtained in exactly the same way as the above but now applied to the set $%
J=\{3,4,\ldots,m=2\mu+1\}$ in place of $I = \{2,\ldots,2\mu\}$. Thus, for
example, the left coset representatives of $S_2\wr S_2$ in $S_5$ 
are \\ $\{e$, $%
(12)$, $(13)$, $(14)$, $(15)\}\times \{e$, $(34)$, $(35)\} = \{e$, $(34)$, $%
(35)$, $12)$, $(12)(34)$, $(12)(35)$, $(13)$, $(13)(34)$, $(13)(35)$,%
$(14)$, $(14)(34)$, $(14)(35)$, $(15)$, $(15)(34)$, $(15)(35)\}$.

\noindent {\bf Example 1:} The values of the character $\zeta^{(6,4,3,2)}$
of $S_{15}$ are calculated at the classes $(1^23^27)$ and $(3,5,7)$. In this
case, as $(\lambda_1,\lambda_2,\lambda_3,\lambda_4) = (6,4,3,2)$, the
relevant left coset representatives are $e,(23),(24)$; thus, by Theorem
4.2(i) and Remark 1(i), we have 
\[
\zeta^{(6,4,3,2)}_{(1^23^27)} = \frac {2!2!}{2!} \zeta^{(6,4)}_{(3,7)}
\zeta^{(3,2)}_{(1^23)} - \frac {2!2!}{2!2!} \zeta^{(6,3)}_{(1^27)}
\zeta^{(4,2)}_{(3^2)} 
\]
\[
+ \frac{2!2!}{2!} \zeta^{(6,2)}_{(1,7)} \zeta^{(4,3)}_{(1,3^2)} + \frac
{2!2!}{2!2!} \zeta^{(6,2)}_{(1^23^2)} \zeta^{(4,3)}_{(7)}, 
\]
noting that $((3,7),(1^23,\phi))$ and $((1^27,\phi),(3^2,\phi))$ are the
only separations of $(1^23^27)$ corresponding to $((6,4),(3,2))$ and $%
((6,3),(4,2))$ respectively, while there are two separations $%
((1,7),(13^2,\phi))$ and $((1^23^2),\phi),((7),\phi)$ of $(1^23^27)$
corresponding to $((6,2),(4,3,1))$.

Now by means of Theorem 4.1, we find that 
\[
\zeta^{(6,4)}_{(3,7)} = 0,\quad \zeta^{(3,2)}_{(1^2,3)} = -1,\quad
\zeta^{(6,3)}_{(1^27)} = 0,\quad \zeta^{(4,2)}_{(3^2)} = 2,\quad
\zeta^{(6,2)}_{(17)} = 0, 
\]
\[
\zeta^{(4,3)}_{(13^2)} = 2,\quad \zeta^{(6,2)}_{(1^23^2)} = -2,\quad
\zeta^{(4,3)}_{(7)} = -1. 
\]
Thus 
\[
\zeta^{(6,4,3,2)}_{(1^23^27)} = \zeta^{(6,2)}_{(1^23^2)}\zeta^{(4,3)}_{(7)}
= 2. 
\]
Since the partition $(3,5,7)$ has no separations corresponding to $\pi_9\in
O(9)$, $\pi_6 \in O(6)$, we have 
\[
\zeta_{(3,5,7)}^{(6,4,3,2)} = \zeta^{(6,4)}_{(3,7)} \zeta_{(5)}^{(3,2)} +
\zeta^{(6,2)}_{(3,5)} \zeta^{(4,3)}_{(7)} 
\]
\[
=0\cdot 1 - 2\cdot (-1) = +2. 
\]

\noindent {\bf Example 2:} The values of the character $\zeta^{(5,4,3,2,1)}$
of $S_{15}$ are calculated at the classes $(1^23^27)$ and $(3,5,7)$. In this
case, there are terms corresponding to the 15 left coset representatives of $%
S_2\wr S_2$ in $S_5$ listed above. However, since $\zeta^{(5,3)}_{(1,7)} =
\zeta^{(5,4)}_{(1^27)} = \zeta^{(4,1)}_{(1^23)} = \zeta^{(5,2)}_{(13^2} = 0$%
, then by Theorem 4.2(ii) and Remark 1(ii) and taking the only possible
separations of $(1^23^27)$ we obtain 
\[
\zeta^{(5,4,3,2,1)}_{(1^23^27)} = \frac{2!2!}{2!} \cdot 
 2\zeta^{(5)}_{(1^23)}
\zeta^{(4,3)}_{(7)} \zeta^{(2,1)}_{(3)} + \frac
{2!2!}{1}\cdot 2\cdot \zeta^{(4)}_{(1,3)} \zeta^{(5,2)}_{(7)} 
\zeta^{(3,1)}_{(1,3)} 
\]
\[
 - (-\frac{2!2!}{2!2!} \cdot  2\zeta^{(2)}_{(1^2)} 
\zeta^{(2)}_{(1^2)}
\zeta^{(5,1)}_{(3^2)} \zeta^{(4,3)}_{(7)}) + \frac{2!2!}{2!}%
\cdot 2\cdot \zeta^{(1)}_{(1)} \zeta_{(7)}^{(5,2)} 
\zeta^{(4,3)}_{(13^2)}. 
\]
Again, by Theorem 4.1, we have $\zeta^{(4,3)}_{(7)} = -1,\
\zeta_{(3)}^{(2,1)} = -1, \zeta^{(5,2)}_{(7)} = 1, \zeta_{(1,3)}^{(3,1)} =
 - (-1), \zeta^{(5,1)}_{(3^2)} = -2$ and $\zeta^{(4,3)}_{(13^2)} = 2$, and by
(4.17), we have $\zeta^{(5)}_{(1^23)} = 2, \zeta^{(4)}_{(13)} =
1,\zeta^{(2)}_{(1^2)} = 1$ and $\zeta^{(1)}_{(1)} = 1$. Thus, we have 
\[
\zeta^{(5,4,3,2,1)}_{(1^23^27)} = 4. 
\]

Due to the separation process, there are only two terms to be considered for
the class $(3,5,7)$ and we obtain 
\[
\begin{array}{ll}
\zeta^{(5,4,3,2,1)}_{(3,5,7)} & = 2\zeta^{(5)}_{(5)} \zeta_{(7)}^{(4,3)}
\zeta^{(2,1)}_{(3)} - 2\zeta^{(3)}_{(3)} \zeta_{(7)}^{(5,2)}
\zeta_{(5)}^{(4,1)} \\ 
& = 2\cdot 1\cdot (-1)\cdot (-1) -2\cdot 1\cdot 1\cdot (-1 )= 4.
\end{array}
\]

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\end{document}

 




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