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\title[Representations of covering groups]{Representations of the covering groups of the symmetric groups 
and their combinatorics}

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\author[C.\ Bessenrodt]{%
	     C.\ Bessenrodt%
             \thanks{supported by a grant from 
             the EC (Network on Algebraic Combinatorics)}\\
             Fakult\"at f\"ur Mathematik,\\ 
             Otto-von-Guericke-Universit\"at Magdeburg\\
             PSF 4120, 39016 Magdeburg, Germany} 
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\date{}

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The theory of the covering groups 
$\tilde S_n$ of the symmetric groups $S_n$ has its origins in 
early papers by Schur [S1,S2,S3] 
who investigated the projective re\-pre\-sen\-tations of the groups $S_n$. 
While the re\-pre\-sen\-tation theory of the symmetric groups made continuous
progress, with the appropriate combinatorial concepts of partitions 
and tableaux developed early on by Young, the representation 
theory of the covering groups saw a long period of stagnation. 
Only 50 years after Schur's fundamental paper on the group $\tilde S_n$, new 
light into the subject was brought by Morris [Mo1, Mo2]. 
He introduced the combinatorial notions which turned out to be the right 
analogues of the notions in the $S_n$ case: shifted tableaux 
instead of Young tableaux and bars instead of hooks. 
Using these notions he obtained new results on the $Q$-functions 
introduced by Schur, and he found a recursive formula for the 
irreducible spin characters (i.e.\ the characters belonging to the 
proper projective re\-pre\-sen\-tations) in analogy to the Murnaghan-Nakayama 
formula, thus opening up a deeper 
investigation of the characters of~$\tilde S_n$.

In the meantime, $p$-modular re\-pre\-sen\-tation theory of finite groups had been 
developed, mainly by Richard Brauer, who had defined the fundamental 
concept of $p$-blocks of finite groups. 
While the $p$-blocks of the groups $S_n$ had been determined already in the 
1940's by a combinatorial algorithm -- still known as the Nakayama 
conjecture -- it was only in 1965 that Morris conjectured how to determine 
the $p$-blocks of~$\tilde S_n$ for odd primes $p$, again by a combinatorial 
procedure. 
It took more than 20 years before this conjecture was settled by 
Humphreys [H] and then, using different methods, by Cabanes [C]. 
In the meantime, there was very little progress on the modular 
re\-pre\-sen\-tation theory of the groups $\tilde S_n$. 
But in the past decade the area has seen many significant contributions, 
not only from re\-pre\-sen\-tation theory, but now also from combinatorialists
interested in tableaux (see [St1, St2]). 
At about the same time in the late 1980's several survey articles were 
published that took a new look at Schur's papers and proceeded with new 
results. 
Jozefiak [Jo] chose the superalgebra approach to Schur's work, 
while Stembridge ([St1, St2]) put more emphasis on the combinatorial side, 
in particular shifted tableaux, and viewed $Q$-functions as 
generating functions for special tableaux. 
A little later, Hoffman and Humphreys [HH] gave a full account of 
the projective re\-pre\-sen\-tations of $S_n$ in their book on this subject. 
Certainly, all these have had an impact on the increasing interest in this 
area where algebra interacts with combinatorics and the theory of 
symmetric functions. 

In the following sections, first a brief introduction into 
the re\-pre\-sen\-tation 
theory of the groups $\tilde S_n$ will be given, along with its 
combinatorics, that corresponds to the time up to 1986. 
This will be incomplete in many respects, e.g.\ there will be little 
included about symmetric functions (for this we refer the reader 
to the monograph by Macdonald [M] and to the survey article by 
Morris [Mo3]) or Clifford algebras (see [Jo]). 
It will focus on those results that are needed for the later sections 
in which we report on the more recent results in $p$-modular re\-pre\-sen\-tation 
theory of~$\tilde S_n$. 
Based on the determination of the $p$-blocks of~$\tilde S_n$ mentioned above, 
Olsson [O1, O2, O3, O4], Morris and Yaseen [MY1] 
and Morris and Olsson [MO] studied the 
$p$-block invariants and developed the appropriate combinatorial tools 
further. Motivated by the 
computations of decomposition matrices for small~$n$ 
by Morris and Yaseen [MY2, Y], general results 
on the decomposition matrices in characteristic 3 [BMO] 
and~5 have been obtained 
and a surprising connection with a deep conjecture by Andrews [A1, A2]
from 1974 was revealed in this context -- motivating a new attack on the 
problem which could then be settled [ABO]. 
Computations of the decomposition matrix at characteristic~2 
by Benson [B1] led to a conjecture of Kn\"orr and Olsson 
on the 2-block structure of~$\tilde S_n$ in 1987 [O1]. 
This could recently be proved [BO1], and in fact very good information 
on the 2-decomposition matrix could be obtained. On the basis of these 
results, now also the heights of spin characters in 2-blocks have been 
studied, and a number of central re\-pre\-sen\-tation theoretical 
{\samepage conjectures have been verified for the groups~$\tilde S_n$ [BO2].} 

\section{Spin characters: from Schur to Morris}

Let $G$ be a finite group, $K=\C$ the field of complex numbers. 
The assertion on the field can be weakened for some of the results 
below; we refer the reader to [HH] and [Jo] for more details. 
A {\it projective re\-pre\-sen\-tation} of~$G$ on a $K$-vector space~$V$  
is a map  $T: G \rarrow GL(V)$, 
satisfying $T(1_G)=id_V$ and for any $x,y \in G$ there is 
a suitable scalar $\alpha (x,y) \in K^*$ with 
$T(x)T(y) = \alpha (x,y) T(xy)$. 
The map $\alpha: G \times G \rarrow K^*$ is then a factor set, i.e.\ 
it satisfies the 2-cocycle condition 
$$\alpha(x,y) \alpha(xy,z) = \alpha(x,yz)\alpha(y,z) \; 
\mbox{ for all } x,y,z \in G \, .$$
For a trivial cocycle $\alpha \equiv 1$, 
the corresponding re\-pre\-sen\-tations are the linear re\-pre\-sen\-tations 
of~$G$. 
The map $T$ induces a map 
$\bar T : G \rarrow PGL(V)$ %=GL(V)/{K^* \cdot \mbox{id}}$, 
which is a homomorphism, often also called a projective re\-pre\-sen\-tation 
of~$G$. 
So from a geometric point of view, a projective re\-pre\-sen\-tation of $G$ is 
equivalent to considering $G$ as a transformation group on projective space, 
which certainly was a natural point of view at the beginning of the century. 
But also from the linear re\-pre\-sen\-tation theoretic point of view, projective 
re\-pre\-sen\-tations come up naturally when one wants to study the connection 
between linear re\-pre\-sen\-tations of a group $G$ and of a normal subgroup $H$ 
of $G$; this is fundamental in the so-called Clifford theory (see [CR], [NT]). 

Two projective re\-pre\-sen\-tations $T_1$ and $T_2$ of $G$ on $K$-vector spaces 
$V_1$ and $V_2$ respectively, are projectively 
equivalent if there is an isomorphism 
$A \in Hom_K(V_1,V_2)$ and a map $\delta : G \rarrow K^*$ such that 
$$\delta(x) A T_1(x) A^{-1} = T_2(x) \; \mbox{ for all } x \in G \, .$$
The equivalence of projective re\-pre\-sen\-tations 
induces an  equivalence of factor sets; 
these equivalence classes  form an abelian group $M(G)$, called the 
{\it Schur multiplier} of $G$, which is isomorphic to the cohomology group 
$H^2(G,\C^*)$. 

Now Schur realized that projective re\-pre\-sen\-tations can be `linearized' 
by enlarging the group. More precisely, there is a central 
extension $\tilde G$ of $G$ such that the 
projective re\-pre\-sen\-tations of $G$ can be lifted to linear 
re\-pre\-sen\-tations of~$\tilde G$. 
A minimal such group is called a {\it re\-pre\-sen\-tation group} of $G$. 
In fact, one obtains a re\-pre\-sen\-tation group by taking the central kernel 
to be the Schur multiplier.  
So we have a non-split extension with central kernel contained in the 
commutator subgroup of~$\tilde G$ 
in the first row of the following 
commutative diagram: 

\[ \begin{array}{ccccccccc}
1 & \longrightarrow & M(G) & \longrightarrow & \widetilde G & 
\: { \stackrel{\mbox{$\pi$}}{ \longrightarrow}} & \: G & \longrightarrow & 1\\
 &&&& \Big\downarrow\rlap{$\vcenter{\hbox{$\rho$}}$} &&
 \Big\downarrow\rlap{$\vcenter{\hbox{$\bar T$}}$}&& \\
1 & \longrightarrow & \C^* & \longrightarrow & GL(V) & \longrightarrow & PGL(V) 
     & \longrightarrow & 1\\
\end{array} \]
 
and $T$ as above is equivalent to the projective 
re\-pre\-sen\-tation $\rho \circ u : G \rarrow GL(V)$, where $u$ is a fixed 
section of $\pi$ and $\rho$ is a suitable linear re\-pre\-sen\-tation 
of~$\tilde G$. \\

We now turn to the specific situation $G=S_n$ 
which Schur studied in his 1911 paper. 

\begin{theorem}  
(Schur [S3])  
\begin{displaymath}
H^2(S_n,\C^*) \simeq  
\left\{ \begin{array}{cl}
0 & \mbox{for } n \leq 3\\
\Z_2 & \mbox{for  } n>3\\
\end{array} \right. \; .
\end{displaymath}
\end{theorem}
\vspace{1ex}

\begin{theorem}  
(Schur [S3]) 
For $n \geq 4$, there are two re\-pre\-sen\-tation groups of $S_n$, which 
are  isomorphic only for $n=6$.
\end{theorem}
\vspace{1ex}

Both re\-pre\-sen\-tation groups can be given explicitly in terms of generators 
and relations. Since the re\-pre\-sen\-tation theory for these two groups 
is virtually the same, i.e.\ one easily transforms results for one into such 
for the other, we will only deal with one of them and take Schur's choice: 
\[ \begin{array}{rcll}
\tilde S_n &=& <\! t_1, \dots , t_{n-1}, z \mid 
                    & z^2=1, t_i^2 =z, 1 \leq i \leq n-1; \\
	  &&&	    t_{i+1}t_it_{i+1}=t_it_{i+1}t_i,  1 \leq i \leq n-2; \\
          &&&       t_it_j=zt_jt_i \mbox{ for }  |i-j|>1, 1 \leq i,j \leq n-1 \!>
\end{array}\]
for $n \in \N$. So $\tilde S_n$ is a central non-split extension of $S_n$ 
by $\gen{z}$, a double cover of~$S_n$, and it is a representation 
group for $S_n$, for $n \geq 4$. 

Classification of irreducible projective $S_n$-re\-pre\-sen\-tations 
now means classification of the irreducible linear $\tilde S_n$-re\-pre\-sen\-tations. 
The irreducible linear $\tilde S_n$-re\-pre\-sen\-tations with $z$ in their kernel 
correspond to the well-known irreducible linear $S_n$-re\-pre\-sen\-tations. 
So we are only interested in the linear $\tilde S_n$-re\-pre\-sen\-tations 
which map $z$ to $-$id${}_V$, and we call these {\it spin  re\-pre\-sen\-tations}. 

Now Schur succeeded in classifying the irreducible complex spin re\-pre\-sen\-tations
by giving their characters; he also produced the {\it basic spin re\-pre\-sen\-tations} 
explicitly. Only recently Nazarov [N] has constructed all the irreducible 
spin re\-pre\-sen\-tations explicitly by presenting suitable orthogonal 
matrices, in analogy to Young's orthonormal representations in the case 
of~$S_n$.

The first step towards the computation of the irreducible spin character table 
is the determination of the conjugacy classes in $\tilde S_n$. 
Let $\lambda$ be a partition of $n$, i.e.\ a sequence 
$\lambda = (\ell_1, \ell_2, \dots, \ell_m)$ of positive integers where 
$\ell_1 \geq \ell_2 \geq \dots \geq \ell_m > 0$ and 
$\sum_{i=1}^m \ell_i =n$; we call $m = \ell(\lambda)$ the {\it length} 
of~$\lambda$.
We then set 
$$C_{\lambda} = \{ \sigma \in \tilde S_n \mid \pi(\sigma) \in S_n 
   \mbox{ is of cycle type } \lambda \} \, .$$
Furthermore, we let ${\cal P}(n)$ be the set of all partitions of $n$, 
${\cal O}(n)$ the set of partitions of $n$ 
with odd parts only, and we let ${\cal D}(n)$ be the set of partitions of $n$ 
into distinct parts. Then ${\cal D}^-(n)$ resp.\ ${\cal D}^+(n)$ denote 
the sets of those partitions in ${\cal D}(n)$ with an odd resp.\ an even 
number of even parts; we call the corresponding partitions 
{\it odd} resp.\ {\it even partitions}. 

For the following results of Schur see [S3], [Jo], [HH], [St1, St2]. 

\begin{theorem} (Schur) 
Let $\lambda \in {\cal P}(n)$. 
Then $C_{\lambda}$ splits into two $\tilde S_n$-conjugacy classes 
if and only if $\lambda \in {\cal O}(n) \cup {\cal D}^-(n)$, 
otherwise $C_{\lambda}$ does not split. 
\end{theorem}

In the case that $C_{\lambda}$ splits, a specific labelling 
for the two $\tilde S_n$-conjugacy classes in $C_{\lambda}$ is needed. 
If 
$\lambda = (\ell_1 , \dots , \ell_m) \in {\cal P}(n)$ we set 
$$\sigma_{\lambda} = v_1 \cdots v_m \: , \mbox{ where } 
v_j = t_{i+1}t_{i+2} \cdots t_{i+\ell_j-1} \: , 
i = \sum_{k=1}^{j-1} \ell_k \; ; j=1,\dots , m \: .$$
Then we let $C_{\lambda}^+$ denote the conjugacy class of $\sigma_{\lambda}$, 
and $C_{\lambda}^-$ the conjugacy class of $z \sigma_{\lambda}$. 
Note that any spin character vanishes on the non-split classes 
so we only have to consider the values on the split classes in the following. 

Since the number of non-equivalent 
irreducible complex representations of a finite 
group $G$ is equal to the number of $G$-conjugacy classes, 
we can already conclude that the number of non-equivalent 
irreducible complex spin re\-pre\-sen\-tations equals 
$$|{\cal O}(n)| + |{\cal D}^-(n)| = |{\cal D}(n)| + |{\cal D}^-(n)|
                                  = |{\cal D}^+(n)| + 2|{\cal D}^-(n)| \: .$$

In fact the last expression corresponds nicely to the classification of 
irreducible spin characters. Before giving this, we need one further 
definition. 
Let $\sgn$ denote the sign character of~$\tilde S_n$ (induced from the sign 
character of $S_n$). 
A character $\chi$ of~$\tilde S_n$ is called {\it self-associate} if 
$\sgn \cdot \chi = \chi$, otherwise we have a pair $\chi$, 
$\chi'= \sgn \cdot \chi$ 
of {\it associate} characters. 				  

\begin{theorem} (Schur) 
A complete list of irreducible complex spin characters of~$\tilde S_n$ 
is given as follows. \\
For each $\lambda \in {\cal D}^+(n)$ there is a self-associate spin character 
$\spin{\lambda}$, and for each $\lambda \in {\cal D}^-(n)$ there is a pair 
of associate spin characters $\spin{\lambda}$, $\spin{\lambda}'$ which take 
the following values on $\sigma_{\alpha} \in C_{\alpha}^+$: 
\[ \begin{array}{rcll}
\spin{\lambda} (\sigma_{\alpha}) &=& \spin{\lambda}'(\sigma_{\alpha})
      & \mbox{for } \alpha \in {\cal O}(n), \lambda \in {\cal D}^-(n)\\
\spin{\lambda} (\sigma_{\alpha}) &=& 0 
      & \mbox{for } \alpha \in {\cal D}^-(n), \lambda \neq \alpha\\
\spin{\lambda} (\sigma_{\lambda}) &=& - \spin{\lambda}'(\sigma_{\lambda})
      = i^{(n-m+1)/2} \sqrt{\prod_j \ell_j/2} 
      & \mbox{for } \lambda = (\ell_1 , \dots , \ell_m) \in {\cal D}^-(n)\\
\end{array}\]
and $\spin{\lambda} (\sigma_{\alpha})$ for $\alpha \in {\cal O}(n)$ 
is determined by the following expansion of the Schur $Q$-function into 
power sum functions: 
$$Q_{\lambda} = 
\sum_{\alpha \in {\cal O}(n)} 2^{(\ell(\lambda)+\ell(\alpha)+\epps(\lambda))/2}
  \frac{1}{z_{\alpha}} \spin{\lambda} (\sigma_{\alpha}) p_{\alpha} \: , $$
where 
$z_{\alpha} = |C_{S_n}(\pi(\sigma_{\alpha}))| = \prod_{j\geq 1} j^{m_j} (j!)$ 
if $\alpha = (1^{m_1}, 2^{m_2}, \ldots )$ and 
\begin{displaymath}
\epps(\lambda) = 
\left\{ \begin{array}{cl}
0 & \mbox{if } \lambda \mbox{ is even}\\
1 & \mbox{if } \lambda \mbox{ is odd}\
\end{array} \right.
\end{displaymath}  
\end{theorem}

The $Q$-functions appearing above are special instances of Hall-Littlewood 
functions, more precisely: 
$$Q_{\lambda} (x_1 , \dots , x_n) = 2^m \sum_{w\in S_n/S_{n-m}} 
     w\left(\prod_{i=1}^m x_i^{\ell_i}\prod_{i=1}^m \prod_{j=i+1}^n 
       \frac{x_i+x_j}{x_i-x_j}\right)$$
for $\lambda = (\ell_1 , \dots , \ell_m)$, $m \leq n$; 
if $\ell(\lambda) > n$, then $Q_{\lambda} (x_1 , \dots , x_n) = 0$. 

Their combinatorial significance comes from the fact that they are tableaux 
generating functions, indeed, $Q_{\lambda}$ corresponds to a weight 
enumeration of marked 
shifted $\lambda$-tableaux (see [St1, St2]). 
The shifted $\lambda$-diagrams and shifted $\lambda$-tableaux are the 
analogues of the Young diagrams and Young tableaux in the ordinary $S_n$-case, 
the shifting meaning that the rows are indented along the diagonal. 
For example, for $\lambda = (5,3,2)$ the shifted Young diagram $S(\lambda)$ 
is depicted by 
\begin{center} 
\setlength{\unitlength}{.5mm}
\begin{picture}(50,30)
\put(0,30){\line(1,0){50}}
\put(0,20){\line(1,0){50}}
\put(10,10){\line(1,0){30}}
\put(20,0){\line(1,0){20}}
\put(0,20){\line(0,1){10}}
\put(10,10){\line(0,1){20}}
\put(20,0){\line(0,1){30}}
\put(30,0){\line(0,1){30}}
\put(40,0){\line(0,1){30}}
\put(50,20){\line(0,1){10}}
\end{picture}
\end{center}
The standard shifted $\lambda$-tableaux are obtained by filling the 
shifted $\lambda$-diagram $S(\lambda)$ with the integers $\{ 1, \ldots , n\}$ 
such that entries increase along rows and down the columns; for example 
the following is a standard shifted $(5,3,2)$-diagram: 
$$
\begin{array}{ccccc}
1 & 2 & 3 & 6 & 7\\
& 4 & 5 & 9\\
& & 8 & 10
\end{array}
 $$

That these are the right combinatorial concepts is seen 
for example in the degree formula  which explicitly computes the spin 
character values at 1 ([S3], [Mo1], [St1, St2]). 
Before giving this, we need one further central notion due to Morris:
the {\it bar lengths} in $\lambda$. 
These are the hook lengths 
of the $\lambda$-nodes in the shift-symmetric diagram $SS(\lambda)$ associated 
with $\lambda$. Instead of giving a formal definition, we illustrate 
this again with our example $\lambda=(5,3,2)\in {\cal D}(10)$. 
We have to adjoin the parts of $\lambda$ as columns to the shifted 
diagram and then get the following shift-symmetric diagram $SS(\lambda)$, 
where we have already written the bar lengths into the $\lambda$-nodes
of $SS(\lambda)$: 

$$
\begin{array}{cccccc}
\Box & 8 & 7 & 5 & 4 & 1\\
\Box & \Box & 5 & 3 & 2\\
\Box & \Box & \Box & 2 & 1\\
\Box & \Box & \Box \\
\Box
\end{array}
 $$


\begin{theorem}
Let $\lambda = (\ell_1 , \dots , \ell_m) \in {\cal D}(n)$. Then 
$$ 
\spin{\lambda}(1)=2^{\left[(n-m)/2 \right]}
\frac{n!}{\prod(\ell_i!)}\prod_{i<j}
\frac{\ell_i-\ell_j}{\ell_i+\ell_j}
 $$
Denoting by $H_{\lambda}$ the product of the bar lengths in $\lambda$ 
and by $g_{\lambda}$ the number of standard shifted $\lambda$-tableaux, 
we also have: 
$$\spin{\lambda}(1) = 2^{\left[(n-m)/2 \right]} \frac{n!}{H_{\lambda}}
                    = 2^{\left[(n-m)/2 \right]} g_{\lambda} $$
\end{theorem}


Schur explicitly constructed the basic spin re\-pre\-sen\-tation which is 
the one labelled 
by $\lambda = (n)$ and calculated the values of the basic spin character 
$\spin{n}$. 
The problem of actually computing all the spin character values by a recursive 
formula analogous to the Murnaghan-Nakayama formula for the characters 
of the symmetric groups was solved by Morris [Mo1, Mo2]. 
First we have to explain the process of $\ell$-bar removal. 
Given a partition $\lambda \in {\cal D}(n)$, we may subtract $\ell$ from 
a part of~$\lambda$ (if the resulting partition is in ${\cal D}(n-\ell)$), 
or remove a part $\ell$ from $\lambda$ (if there is such a part), 
or we may remove two parts $m$ and $\ell -m$ from $\lambda$ (if possible). 
Any of these operations is called removal of an $\ell$-bar; an $\ell$-bar 
corresponds to an $\ell$-hook in $SS(\lambda)$ belonging to one of the 
$\lambda$-nodes. 
The leg length $L(b)$ of the $\ell$-bar $b$ is then defined to be the 
leg length of the corresponding $\ell$-hook in $SS(\lambda)$. 
The partition resulting in removing $b$ from $\lambda$ is denoted by 
$\lambda \setminus b$.

\begin{theorem} (Morris' recursion formula) 
Let $\lambda \in {\cal D}(n)$, and let $\alpha \in {\cal O}(n)$ 
be a partition with~$\ell$ as a part. 
Then 
$$
\spin{\lambda}(\sigma_{\alpha}) = 
        \sum_{b \: \ell\mbox{\footnotesize -bar}}  (-1)^{L(b)} 	2^{m(b)}
	\spin{\lambda \setminus b}(\sigma_{\alpha \setminus \ell})
$$
where 
\[
m(b)=\left\{ \begin{array}{cl} 
           1 & \mbox{if }\; \epps(\lambda\setminus b)-\epps (\lambda)=1\\
           0 & \mbox{otherwise}
	   \end{array}
      \right.	   \; .
\]
\end{theorem}

As a consequence, one easily deduces the Branching Theorem which describes 
$\spin{\lambda}$ restricted to $\tilde S_{n-1}$, where 
$\tilde S_{n-1} = \gen{t_1, \dots , t_{n-2}} \subseteq \tilde S_n$, 
and dually induction of $\spin{\lambda}$ to $\tilde S_{n+1}$. 
Up to the modification coming from associate spin characters this is again 
very similar to the ordinary Branching Theorem for $S_n$. 
For later purposes we state only the induction version; 
first we need to set up some further notation. 

For $\lambda \in {\cal D}(n)$ we put 
$$\hat{\spin{\lambda}} = 
  \left\{ \begin{array}{ll}
  \spin{\lambda} & \mbox{if } \lambda \in {\cal D}^+(n)\\
  \spin{\lambda}+\spin{\lambda}' & \mbox{if } \lambda \in {\cal D}^-(n)
  \end{array} \right.$$
and set 
\begin{eqnarray*}
N(\lambda) & =&  \{ \mu \in {\cal D}(n+1) \mid \lambda \mbox{ is obtained from } 
     \mu \mbox{ by removing a 1-bar} \} \: ,\\
N(\lambda)' &= &\{ \mu \in N(\lambda) \mid \ell(\mu) = \ell(\lambda) \} \: .\\
\end{eqnarray*}

\begin{theorem} (Branching Theorem  [Mo2])
If $\lambda \in {\cal D}^+(n)$, then 
$$\spin{\lambda}\!\uparrow^{\tilde S_{n+1}} = 
\sum_{\mu \in N(\lambda)} \hat{\spin{\mu}} \: .$$
If $\lambda = (\ell_1 , \dots , \ell_m) \in {\cal D}^-(n)$, then 
\[ \begin{array}{rll}
\spin{\lambda}\!\uparrow^{\tilde S_{n+1}} & =  \spin{\lambda}'\!\uparrow^{\tilde S_{n+1}} 
= \sum_{\mu \in N(\lambda)} \spin{\mu}  & \: \mbox{ if } \ell_m =1 \: , \\
\spin{\lambda}\!\uparrow^{\tilde S_{n+1}} & =
\sum_{\mu \in N(\lambda)'} \spin{\mu} + \spin{\ell_1, \ldots , \ell_m, 1} & \: \mbox{ if } 
\ell_m >1 \: , 
\end{array}\]
(and similarly for the associate character).
\end{theorem}     
  
\section{Generalities from modular re\-pre\-sen\-tation theory}

Let $p$ be a prime dividing the order of the finite group $G$, 
and let $(R,K,F)$ be a $p$-modular splitting system, 
i.e.\ $R$ is a  complete discrete valuation ring with quotient field $K$ of 
characteristic~0 and residue field $F$ of characteristic $p$, 
and $F$ and $K$ are splitting fields for $G$ (for more details on this and 
any unexplained notation we refer to [CR] or [NT]). 
Typically, $K$ is an extension of the $p$-adic field $\Q_p$, 
containing a primitive $|G|$-th root of unity, $R$ is the standard 
valuation ring of $K$, and $F$ is the residue field of $R$.

Given a $KG$-module $V$ (finite-dimensional over $K$) 
there is an $RG$-lattice $U$ with 
$V=U \otimes_R K$, called an $R$-form of $V$. 
Then $U$ and $\bar U = U \otimes_R F$ are not uniquely determined  
up to isomorphism, but the following important result 
due to Brauer and Nesbitt holds 
(see [CR], [NT]): 

\begin{theorem} 
The $FG$-composition factors of $\bar U = U \otimes_R F$ 
only depend on $V$ and not on the choice of $U$. 
\end{theorem}

This allows to define the decomposition matrix of $G$ as follows. 
Let $V_1 , \dots , V_t$ be the (pairwise nonisomorphic) irreducible 
$KG$-modules, $\chi_i$ the character belonging to $V_i$, 
let $U_1 , \ldots , U_t$ be $R$-forms of $V_1 , \ldots , V_t$ resprectively, 
and let $S_1 , \dots , S_r$ be the (pairwise nonisomorphic) simple 
$FG$-modules. 
Then we define the $p$-decomposition matrix $D = (d_{ij})_{i,j}$ 
by 
$$d_{ij} = \mbox{multiplicity of } S_j \mbox{ as a composition factor of } 
           \bar U_i$$
If $P_1, \dots , P_r$ are the projective indecomposable $RG$-lattices, ordered 
such that $\bar P_j = P_j \otimes_R F$ is a projective cover of $S_j$, 
$j = 1, \dots , r$, and $\chi_{P_j}$ are their characters, then we also have: 

$$d_{ij} = \gen{\chi_i , \chi_{P_j}} \: , \mbox{ for } i= 1, \dots ,t; 
         j= 1, \dots ,r \, , $$
where we take the usual inner product $< .,.>$ on characters. 

Since we have chosen our fields to be sufficiently large, we know the size 
of $D$:

$t$ is the number of conjugacy classes of $G$, and 
$r$ is the number of $p$-regular conjugacy classes of $G$ 
(which are the ones corresponding to elements whose order is not divisible 
by~$p$). 

Brauer associated certain complex-valued class functions 
(now called Brauer characters) 
also with $FG$-modules, but these take values only on the $p$-regular classes. 
The Brauer character uniquely determines the composition factors of the 
$p$-modular representation to which it is associated. 
Denoting by $\varphi_j$ the irreducible Brauer character belonging to $S_j$, 
we have 
$$\chi_i= \sum_j d_{ij} \varphi_j \quad \mbox{on $p$-regular classes}$$
So the $p$-decomposition matrix is an important link between ordinary 
(i.e.\ characteristic 0) and $p$-modular re\-pre\-sen\-tation theory of $G$, 
and any information on its entries is valuable for both areas. 

Let us first consider the decomposition of the decomposition matrix 
into indecomposable diagonal matrix blocks! That is, sort the modules 
$V_i$ and $S_j$ in such a way that 
\[ 
D = \left( 
\begin{tabular}{cccccc}
\cline{1-1}
\multicolumn{1}{|c|}{*} \\ \cline{1-2} 
& \multicolumn{1}{|c|}{}  \\ 
& \multicolumn{1}{|c|}{*} \\ 
& \multicolumn{1}{|c|}{}  && 0\\ \cline{2-2}
&& $\ddots$ \\
& 0 && $\ddots$ \\ \cline{5-6}
&&&& \multicolumn{2}{|c|}{} \\
&&&& \multicolumn{2}{|c|}{\hbox{$\; * \;$}} \\
&&&& \multicolumn{2}{|c|}{} \\ \cline{5-6}
\end{tabular}
   \right)
\]   
and no further decomposition of this type is possible. 

Then these matrix blocks correspond to the $p$-blocks of the group algebra 
which may be defined resp.\ viewed in various ways. 
In algebra terms, we have 
$$RG=B_1 \oplus \dots \oplus B_s$$
with indecomposable 2-sided ideals $B_1 , \dots , B_s$, these are the 
$p$-{\it blocks} of $RG$. 
They are of the form $B_i = e_iRG$, $e_i$ a primitive central idempotent, 
the {\it block idempotent} belonging to $B_i$. 
Reduction modulo the maximal ideal of $R$ sends $p$-blocks of $RG$ to 
$p$-blocks of $FG$, thus giving a corresponding decomposition for $FG$. 

Let $A \in \{ R,F \}$. 
For any indecomposable $AG$-module $V$ there is an $i \in \{1, \dots s\}$ such 
that $Ve_i=V$ and $Ve_j=0$ for all $j \neq i$. We then say that $V$ 
(and its character) 
belongs to the block $B_i$ and write $V \in B_i$. 
Thus in particular the simple and the projective indecomposable modules 
are sorted into blocks. Also, sometimes a block is viewed as a set of 
irreducible characters; indeed, there is a direct criterion to test whether 
two irreducible characters belong to the same $p$-block 
which can be read off the character table [CR, NT]. 
We write $\chi \in B$ if $\chi$ belongs to the block $B$, and denote by 
Irr$(B)$ the set of irreducible characters belonging to~$B$. 

The $p$-blocks play a fundamental r\^ole in modular re\-pre\-sen\-tation theory, 
and it is of central importance to compute their invariants. 
First, there are the obvious arithmetical invariants for a $p$-block $B$ 
of $G$:
\begin{eqnarray*}
k(B) &=& |\mbox{Irr}(B)| \\   	  
l(B) &=& |\{ 1 \leq j \leq r \mid S_j \in B \}| 
\end{eqnarray*}
As mentioned before, the $p$-block decomposition of the group algebra $AG$ 
corresponds to the matrix decomposition of the $p$-decomposition matrix $D$ 
of $G$, so the invariants $k(B)$ and $l(B)$ determine the size of the 
matrix block corresponding to~$B$ in $D$. 

A structural invariant is the {\it defect group} 
$\delta(B)$ of~$B$ which is a 
$p$-subgroup of~$G$ (unique up to $G$-conjugacy). It can be computed from 
the block idempotent~$e_B$, or from considering the indecomposable $AG$-modules 
in~$B$ (see [CR]). 
The {\it defect} $d(B)$ of~$B$ is then defined by $|\delta(B)|= p^{d(B)}$. 
Blocks of defect 0 are just (isomorphic to) full matrix rings; 
their decomposition matrix is $D= (1)$. 

For any $n \in \N$, let $\nu_p(n)=r$ if $p^r$ is the exact $p$-power 
dividing $n$. 
Let  $a=\nu_p(|G|)$, then it is well-known that for any $\chi \in $ Irr$(B)$ 
the power $p^{a-d(B)}$ divides the degree~$\chi(1)$. 
The {\it height} $h(\chi)$ of~$\chi$ is then defined by 
$$a - d(B) + h(\chi) = \nu_p(\chi(1)) \, .$$
Since $\chi(1)$ divides $|G|$, the height $h(\chi)$ is at most~$d(B)$. 
The invariant $k(B)$ is now refined to 
$$k_i(B) = | \{\chi \in \mbox{Irr}(B) \mid  h(\chi) = i \}| \; , 
\mbox{ for } 0 \leq i \leq d(B) \, .$$
The central conjectures in modular re\-pre\-sen\-tation theory are about these 
invariants. Here we recall some long standing conjectures dealing with 
ordinary characters: 

\begin{conj} (Brauer's Height 0 Conjecture) 
The defect group $\delta(B)$ of a  $p$-block~$B$ is abelian if and only if 
$k(B) = k_0(B)$.
\end{conj}

\begin{conj} (Brauer) 
Any $p$-block~$B$ satisfies  $k(B) \leq |\delta(B)|$.
\end{conj}


\begin{conj} (Olsson) 
Let~$B$ be a $p$-block with  defect group $\Delta$.  
Then  $k_0(B) \leq |\Delta: \Delta'|$.
\end{conj}

Finally we would like to mention the following conjecture on heights 
of characters which was recently put forward by Robinson [R]: 

\begin{conj} (Robinson)
Let~$B$ be a $p$-block with  non-abelian defect group $\Delta$. 
Then for any $\chi \in$ Irr$(B)$ one has 
$h(\chi) < |\Delta:Z(\Delta)|$. 
\end{conj}

By now, quite a bit of evidence has been collected for these conjectures; 
in particular, they are known to be true for the symmetric groups~$S_n$. 
In coming back to $\tilde S_n$, we will keep in mind to check the conjectures 
also for this family of groups. 

\section{Modular re\-pre\-sen\-tation theory of~$\tilde S_n$ 
at odd characteristic~$p$}

The guide for work on the spin re\-pre\-sen\-tations of the double covers 
$\tilde S_n$ of~$S_n$ 
is always the theory of the symmetric groups~$S_n$ itself.
In characteristic~0 we have seen 
that the r\^ole of the partitions as labels of 
the irreducible characters of~$S_n$ is played 
in the $\tilde S_n$-case by the partitions into distinct 
parts (sometimes called bar partitions) as labels of self-associate resp.\ 
pairs of associate irreducible spin characters.

The $p$-modular irreducible $S_n$-re\-pre\-sen\-tations are labelled by the 
$p$-regular partitions, i.e.\ the partitions where all parts have 
mulitiplicity $< p$. 
The main point about these combinatorial labels is that re\-pre\-sen\-tation 
theoretical invariants like the $p$-blocks~$B$ and their invariants 
$k(B)$, $l(B)$, $k_i(B)$ and $d(B)$ can be computed via combinatorial 
algorithms on partitions, and one obtains some general information on the 
decomposition matrix by combinatorial considerations. 
In fact, there is more behind this, namely the Specht modules for $S_n$ 
which have a characteristic-free definition.
We refer the reader to [JK], [J], [O4]
for details on this. 
Note that there is as yet no general analogue for the $p$-modular 
labels on the $\tilde S_n$-side, and there are no analogues for the 
Specht modules so far. 

\iffalse{Also, recently there was new progress on the question of the existence of 
$p$-blocks of defect 0 for $S_n$ and $\tilde S_n$; here number theoretical 
methods have been important [Erdmann-Michler; Granville-Ono]. 

verschieben!!??
}\fi 

First, we recall the $p$-block distribution of irreducible spin characters 
of~$\tilde S_n$; we assume for the rest of this section 
that {\bf $p \neq 2$}. The situation for $p=2$ 
is completely different and will be discussed in the next section. 

In the ordinary $S_n$ case one has to remove $p$-hooks from a labelling 
partition until the $p$-core is reached; the $p$-blocks are then determined 
by the $p$-core and the weight (this is the content of the Nakayama Conjecture 
which was proved by Brauer and G.\ de B.\ Robinson in 1947). 
In the spin case, hooks are replaced by bars as we have already seen in the 
degree formula before: 
given $\lambda \in {\cal D}(n)$, remove $p$-bars as long as possible; 
we then obtain a (uniquely determined) bar partition $\lambda_{(\bar p)}$, 
called the $\bar p$-core. 
There is also a description by a suitable $\bar p$-abacus. 
This abacus has runners labelled $0, \dots p-1$, and we place the numbers 
$0,1,2,\dots$ on the runners as follows: 
\[
\begin{array}{ccccc}
0 & 1 & 2 & \dots & p-1\\
\\
\noalign{\hrule}\\
0 & 1 & 2 & \dots & p-1\\
p & p+1 & p+2 & \dots & 2p-1\\
2p & 2p+1 & 2p+2 & \dots & 3p-1\\
3p & \vdots & \vdots & \dots & \vdots\\
\vdots & \vdots & \vdots & \dots & \vdots
\end{array}
\]
The runners labelled $i$ and $p-i$ are called {\it conjugate runners}, 
for $i=1, \dots , \frac{p-1}{2}$.

For a given bar partition $\lambda$ we place its parts as beads on this abacus. 
For example, let $p=5$ and take 
$\lambda=(14,9,7,6,5,3)$ This has the bead configuration:
\[
\begin{array}{ccccc}
0 & 1 & 2 & 3 & 4\\
\\
\noalign{\hrule}\\
0 & 1 & 2 & \framebox{3} & 4\\
\framebox{5} & \framebox{6} & \framebox{7} & 8 & \framebox{9}\\
10 & 11 & 12 & 13 & \framebox{14}\\
15 & \vdots & \vdots & \dots & \vdots\\
\vdots & \vdots & \vdots & \dots & \vdots
\end{array}
\]
The removal of~$p$-bars now corresponds to the following operations 
on a configuration as above on the abacus. 
\begin{enumerate}
 \item[(i)] Sliding a bead one position up the runner (if the new
position is not yet occupied); beads in position $0$ are dumped.
 \item[(ii)] Removing the beads on two conjugate runners in the top row.
\end{enumerate}
Operating according to these rules as long as possible we finally
obtain the configuration of the $\bar p$-core $\lambda_{(\bar p)}$ 
of~$\lambda$; 
the process is recorded by the $\bar p$-{\it quotient}  
$\lambda^{(\bar p)} = \rho = (\rho_0;\rho_1, \dots , \rho_t)$, 
$t = \frac{p-1}{2}$, where $\rho_0$ is a bar partition describing the 
0-runner of~$\lambda$ while $\rho_i$ is a partition describing  
simultaneously the $i$-th and $(p-i)$-th runner of~$\lambda$ 
(for details, see  [O4]).
The {\it weight} of this $\bar p$-quotient is 
$ w=|\rho_0|+ |\rho_1| + \ldots + |\rho_t|$ and its {\it sign} is 
$(-1)^{w-\ell(\rho_0)}$. 
In the example above one easily checks that $\lambda_{(\bar 5)} = (4)$. 

We now have the following result on the $p$-block distribution of the 
characters of~$\tilde S_n$: 

\begin{theorem}\label{pblocks} (Morris' Conjecture [Mo2]; Humphreys [H], Cabanes [C])
\begin{enumerate}
\item[(i)] Let $\lambda$, $\mu \in {\cal D}(n)$. 
Then the spin characters $\spin{\lambda}$ and $\spin{\mu}$ are in the 
same $p$-block of~$\tilde S_n$ if and only if $\lambda_{(\bar p)} = 
\mu_{(\bar p)}$.
\item[(ii)] Let $\lambda \in {\cal D}^-(n)$. 
If $\lambda \neq \lambda_{(\bar p)}$, then $\spin{\lambda}$ and 
$\spin{\lambda}'$ belong to the same $p$-block of~$\tilde S_n$. 
If $\lambda = \lambda_{(\bar p)}$, then $\spin{\lambda}$ and 
$\spin{\lambda}'$ each form a $p$-block of defect 0 on their own. 
\end{enumerate}
\end{theorem}

Based on this, Olsson has determined the block invariants 
in [O2, O3] (see also [O4]). We now collect some of the results 
on the $p$-blocks of~$\tilde S_n$: 

\begin{theorem}\label{datapblocks}
Let~$B$ be a $p$-block of~$\tilde S_n$, $p  \neq 2$. 
\begin{enumerate}
\item[(i)] Irr$(B)$ consists only of spin characters or only of ordinary 
characters. 
\end{enumerate}
Assume now that~$B$ is a spin $p$-block of~$\tilde S_n$, i.e.\ all characters 
in~$B$ are spin characters. Let $\mu$ be the $\bar p$-core of the spin 
characters in~$B$. Then we have: 
\begin{enumerate}
\item[(ii)] $d(B)=\nu_p((pw)!)$, where $w= \frac{n-|\mu|}{p}$ is the 
{\em weight} of~$B$.
\item[(iii)] Set $\epps = (-1)^{\epps(\mu)}$. Then 
$$k(B) = %k^{\epps}(\bar p,w) = 
q^{\epps}(\bar p,w) + 2q^{-\epps}(\bar p,w)$$
where $q^{\epps}(\bar p,w)$ is the number of~$\bar p$-quotients of weight $w$ 
and sign $\epps$.
\item[(iv)] Let $l^s(B)$ resp.\ $l^{ns}(B)$ denote the number of self-associate 
resp.\ pairs of associate modular irreducible spin re\-pre\-sen\-tations in~$B$. 
Then 
$$l(B) = l^s(B)+2 \, l^{ns}(B)$$
and 
\[
l^s(B)=\left\{ \begin{array}{cl} 
        k(t,w) & \mbox{if }\; w \equiv \epps(\mu) \: (\mbox{mod } 2) \\
           0 & \mbox{otherwise}
	   \end{array}
      \right.	         
\]
\[ 
l^{ns}(B)=\left\{ \begin{array}{cl} 
        0 & \mbox{if }\; w \equiv \epps(\mu) \: (\mbox{mod } 2) \\
        k(t,w) & \mbox{otherwise}
	   \end{array}
      \right.	         
\]
where $k(t,w)$ is the number of~$t$-tuples of partitions with total sum~$w$. 
\end{enumerate}
\end{theorem}

Also, an explicit formula for the height $h(\spin{\lambda})$ in terms of the 
so-called $\bar p$-{\it core tower} of~$\lambda$ is known, 
and the refined invariants 
$k_i(B)$ have been computed by Olsson in analogy to the formula for $k(B)$ 
(see [O2]). 
As consequences of these results, the Brauer Conjectures and Olsson's Conjecture 
stated above hold for the $p$-blocks of~$\tilde S_n$, $p \neq 2$. \\

What do we know about the $p$-decomposition matrix $D$ of~$\tilde S_n$ 
at this point? 

We set 
${\cal D}_p(n) = 
\{ \lambda = (\ell_1 , \dots , \ell_m) \in {\cal D}(n) \mid 
\ell_i \not\equiv 0 \: (\mbox{mod } p), \, i=1, \ldots , m \}$, 
and then for a sign $\epps$ we let  
${\cal D}^{\epps}_p(n) = {\cal D}_p(n) \cap {\cal D}^{\epps}(n)$.  
Then the decomposition matrix is of size $t \times r$ with 
$t=|{\cal D}^+(n)| + 2 |{\cal D}^-(n)|$, 
$r= |{\cal D}^+_p(n)| + 2 |{\cal D}^-_p(n)|$. 

We can also calculate combinatorially the block decomposition of~$D$ 
by Theorem~\ref{pblocks}, and we 
have good formulae for the size of the block matrices by 
Theorem~\ref{datapblocks}. 

By a result of Wales [W] 
the rows of~$D$ corresponding to $\spin{n}$ and 
$\spin{n-1}$ (and their associates) are known. 
For `small' $n$ the decomposition matrices of~$\tilde S_n$ have been computed 
by Morris and Yaseen; 
more precisely, the decomposition
matrices for $n\le 11$ for $p=3$ are given in [MY2] 
and for $n\le 13$, $p=5, 7$ and $n\le 14$ for $p=11$ in [Y]. 
Yaseen has also obtained tables for $n=12, 13$ for $p=3$ 
but with a few ambiguities unresolved. 
More recently, the Brauer character tables for $\tilde S_n$ for $n \le 13$ 
have also been calculated using computers. 
In the work of Morris and Yaseen, 
the question of finding the `right' labels was not dealt with, 
the labels were chosen arbitrarily to a certain extent. 

In the case of~$S_n$, there is so far no algorithm available that 
automatically computes the $p$-decomposition matrix, but there is 
at least the following general shape result (see [JK]) due 
to Farahat, M\"uller and Peel [FMP] and James [J]. 

Order the partitions labelling the rows of the $p$-decomposition matrix 
of~$S_n$ in the following way: first the $p$-regular partitions in decreasing 
lexicographic ordering, then the other partitions. 
The $p$-regular partitions labelling the columns are also ordered 
lexicographically decreasing. 
Then the $p$-decomposition matrix $D$ of~$S_n$ has the shape: 
\[
\begin{array}{l}
\\
\\
p\mbox{-regular}\\
\mbox{partitions}\\
\\
\\
\\
p\mbox{-singular}\\
\mbox{partitions}
\end{array}
\stackrel{\mbox{$p$-regular partitions}}{ 
\left( {
\begin{array}{cccccc}
1\\
& 1 & & & 0\\
& & \ddots \\
 &   *\\
& & & & \ddots \\
& & & & & 1\\
\hline
\\
& & & *\\
\\
\end{array} }
\right)
}
\] 
Furthermore, the final non-zero entry in the row labelled by a partition 
$\alpha$ is a 1 in the column labelled by the $p$-regular partition 
$\alpha^R$, where $\alpha^R$ is obtained by a combinatorial `regularization' 
from $\alpha$. The only non-zero entries in this row are in columns labelled 
by $p$-regular partitions $\beta \unrhd \alpha^R$, where $\unrhd$ denotes the 
usual dominance order on partitions. 

From the data obtained by Morris and Yaseen in the spin case 
it seemed likely that a similar result might also hold for the 
$p$-decomposition matrix of~$\tilde S_n$, $p \neq 2$, except that one has to take 
the complication arising from the associate pairs into account and perhaps with 
suitable 2-powers instead of the 1's above. 
The early proof of the result in the $S_n$ case was by combinatorial 
arguments on the $p$-residue diagram together with suitable inductions 
from Young subgroups; later this was proved using the fundamental 
Specht modules. 
As mentioned before, there is no $\tilde S_n$ analogue in sight for the Specht 
modules, so the approach followed in the work described below 
is similar in spirit to the early proof in the ordinary case. 

First we have to introduce some further combinatorial concepts 
(see [MY1, MY2]); remember that we still assume $p \neq 2$. 

The $\bar p$-{\it residue diagram} of a partition $\lambda \in {\cal D}(n)$ 
is the $\lambda$-part of the {\it shifted $\bar p$-residue diagram}  

\[ 
\matrix{
1 & 2 & \dots & \frac{p-1}2 & \frac{p+1}2 & \frac{p-1}2 & \dots & 2 & 1 &1 &2 & \dots \cr
& 1 & 2 & \dots & \frac{p-1}2 & \frac{p+1}2 & \frac{p-1}2 & \dots & 2 & 1 & \dots \cr
& & 1 & 2 & \dots & \dots \cr
& & & \ddots }
\]

The $\bar p$-{\it content} of~$\lambda$ is given by 
$(1^{c_1} 2^{c_2} \dots \frac{p+1}{2}^{c_{(p+1)/2}})$, 
where $c_i$ is the multiplicity of~$i$ in the $\bar p$-residue diagram of 
$\lambda$. 

For example, take $\lambda = (6,5,3,1)$ 
and $p=5$, then the 
$\bar 5$-residue diagram of~$\lambda$ is 
\[
\matrix{
1 & 2 & 3 & 2 & 1 & 1 \cr 
& 1 & 2 & 3 & 2 & 1 \cr 
& & 1 & 2 & 3 \cr 
& & & 1 }
\iffalse{
1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1\cr
& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 \cr
& & 1 & 2 & 3 & 2 & 1 & 1 \cr
& & & 1 & 2 & 3 & 2 & 1 \cr
& & & & 1 & 2 & 3 }}\fi 
\]
and the $\bar 5$-content of~$\lambda$ is $(1^{7} 2^{5} 3^3)$. 

\begin{theorem} (Morris and Yaseen [MY1])
Let $\lambda \in {\cal D}(n)$. Then the $\bar p$-content of~$\lambda$ 
determines the $\bar p$-core $\lambda_{(\bar p)}$. 
\end{theorem}

For the {\bf proof} one uses the analogous result on the $p$-content 
of the partition corresponding to the shift-symmetric diagram $SS(\lambda)$.\\ 

Consequently one can control the distribution of summands of 
$\spin{\lambda}\!\uparrow^{\tilde S_{n+1}}$ into $p$-blocks by only 
adding nodes of a specified $\bar p$-residue to the shifted diagram 
of~$\lambda$ at a time. 
This is the principle of~$(r,\bar r)$-induction as it was called by Morris 
and Yaseen [MY2]. For a given $\bar p$-residue $r \in \{1, \dots \frac{p+1}2 \}$
we will denote by $\spin{\lambda}\!\uparrow^r$ the sum of the constituents 
of~$\spin{\lambda}\!\uparrow^{\tilde S_{n+1}}$ which are labelled by a 
$\mu \in {\cal D}(n+1)$ reached from $\lambda$ by adjoining an $r$-node. 
By the above, 
this is one block component of~$\spin{\lambda}\!\uparrow^{\tilde S_{n+1}}$.

Let us consider again an example for $p=5$. Take $\lambda = (5,3,2)$ 
and calculate the block components of the induced character 
$\spin{\lambda}\!\uparrow^{\tilde S_{11}}$. We have indicated in bold the 
nodes to be added. 

\[
\matrix{
1 & 2 & 3 & 2 & 1 & {\bf 1} \cr 
& 1 & 2 & 3 & {\bf 2} \cr 
& & 1 & 2 \cr 
& & & {\bf 1} } \qquad 
{\begin{array}{rcl}
\spin{5,3,2}\!\uparrow^1 &=& \hat{\spin{6,3,2}} + \spin{5,3,2,1} \\
\spin{5,3,2}\!\uparrow^2 &=& \hat{\spin{5,4,2}}\\
\spin{5,3,2}\!\uparrow^3 &=& 0
\end{array} } 
\] 

Now it is our aim to find column labels for the decomposition matrix that 
are of `high' type, i.e.\ such that with respect to a suitable ordering 
of the labels the decomposition matrix has the shape 
\[
\begin{array}{l}
\\
\\
\mbox{high type}\\
\mbox{partitions}\\
\\
\\
\\
\mbox{other}\\
\mbox{partitions}
\end{array}
\stackrel{\raisebox{1ex}{high type partitions}}{ 
\left( {
\begin{tabular}{cccccccc}
$\ddots$ \\ \cline{2-3}
&\multicolumn{2}{|c|}{}  \\
&\multicolumn{2}{|c|}{}  &&& 0\\ \cline{2-5}
&&& \multicolumn{2}{|c|}{}  \\ \cline{4-6}
&&&&& \multicolumn{1}{|c|}{}  \\ 
&& * &&& \multicolumn{1}{|c|}{}  \\ \cline{6-6}
&&&&&& $\ddots$ \\ \cline{8-8}
&&&&&&& \multicolumn{1}{|c|}{}\\
\hline
\\
& & & *\\
\\
\end{tabular} }
\right)
}
\] 
Here we have indicated that 
we may have $2 \times 1$, $1 \times 2$ or $2 \times 2$ matrix blocks 
instead of just an entry along the `diagonal' in the upper part of the matrix 
in the case of associated rows resp.\ columns occurring. 

For achieving this, we define a set of partitions of~$n$ by the following 
algorithm which we call {\it the top node algorithm}. 
We keep a prime $p \neq 2$ fixed. 

We set ${\cal C}_p(1) = \{ (1) \}$. \\
Assume that ${\cal C}_p(n-1)$ has already been constructed. Then the partition 
$\lambda$ belongs to ${\cal C}_p(n)$ if it can be constructed from some 
$\mu \in {\cal C}_p(n-1)$ by adding a node to the $\bar p$-residue diagram of 
$\mu$ which is the highest among the nodes with the same $\bar p$-residue 
that could be adjoined to $\mu$. 

Let us illustrate this by an example for $p=3$. 
From $(1)$ we can only construct~$(2)$, by adding a 2-node. 
Now consider the $\bar 3$-residue diagram of~$(2)$ and the nodes that 
could be adjoined (these are marked bold below): 
\[ \matrix{
1&2&{\bf 1}\cr
&{\bf 1}}\]
We are only allowed to add the highest 1-node to the diagram of~$(2)$, hence 
${\cal C}_3(3)=\{(3)\}$. Similarly, ${\cal C}_3(4)=\{(4)\}$. 
Now we are looking at 
\[ \matrix{
1&2&1&1&{\bf 2}\cr
&{\bf 1}}\]
and we have the option of adding either the 1-node or the 2-node indicated above, 
so ${\cal C}_3(5)=\{(5), (4,1)\}$. 

We now face the following problems: 
\begin{enumerate}
\item[(I)] What is the `internal' description of~${\cal C}_p(n)$?
\item[(II)] What is $|{\cal C}_p(n)|$?
\end{enumerate}
In (II), we would like to have $|{\cal C}_p(n)| = |{\cal D}_p(n)|$, 
since then ${\cal C}_p(n)$ could serve as a set of column labels for 
the decomposition matrix of~$\tilde S_n$.  

For $p=3$, both of these problems turned out to find satisfying answers: 

\begin{theorem}
Let $n \in \N$. Then 
\[  {\begin{array}{rl} 
 {\cal C}_3(n) = 
\{ \lambda  = (\ell_1 , \dots , \ell_m)  & \in {\cal D}(n) \mid 
\ell_i - \ell_{i+1}  \geq 3 \: , i=1, \dots , m-1;  \\
 & \ell_i - \ell_{i+1} >3  \mbox{ if } \ell_i \equiv 0 (\mbox{mod } 3)\, , 
 i=1, \dots , m-1 \}
  \end{array}} \]
and $ |{\cal C}_3(n)| = |{\cal D}_3(n)| .$
\end{theorem}

{\bf Proof. } The internal description 
follows easily with the combinatorial description 
of the set on the right hand side given 
via `ladders' in the ${\bar 3}$-residue diagram, 
see below. 

The enumerative identity 
is a special case of a partition identity due to Schur~[S4].
\\

We call the partitions in ${\cal C}_3(n)$ {\it Schur regular} 
partitions of~$n$. 
For any such partition of~$n$, say $\lambda$,  
there are usually quite different construction 
paths in the top node algorithm; we need one for which we have good control 
over the constituents in the corresponding induction of~$\spin{1}$ 
to $\tilde S_n$. 
We will then use this to obtain an approximation to the column of the 
decomposition matrix which we want to label by $\lambda$. 
This will be achieved in this induction process since an induced 
projective character is again projective (where here projective is again 
used in the sense of ordinary re\-pre\-sen\-tation theory, i.e.\ meaning a
character belonging to a projective lattice). 
Since by Theorem~\ref{pblocks} above $\bar 3$-cores label irreducible 
projective characters, we will often use such Schur regular partitions 
as a starting point for the induction. 
For describing an induction  path good for our purposes, we need the 
concept of ladders in the $\bar 3$-residue diagram. 

Let $(i,j)$ denote the $j$-th node in the $i$-th row of the
$\bar p$-residue diagram. For $i \in \N$, the $i$-th ladder $L_{i,r}$ 
in the $\bar 3$-residue diagram joins the following $r$-nodes (from
bottom to top): 
\[ \begin{array}{cl}
L_{i,1}{:} & (i,1) \rarrow  (i-1,4) \rarrow (i-1,3) \rarrow (i-2,7) \rarrow
(i-2,6) \rarrow (i-3,10) \rarrow \cdots \\
& \qquad \cdots  \rarrow (1,3i-2) \rarrow (1,3i-3) \\[6pt]
L_{i,2}{:} & (i,2)  \rarrow  (i-1,5) \rarrow (i-2,8) \rarrow \cdots
\rarrow (2,3(i-2)+2) \rarrow (1,3(i-1)+2)\,.
\end{array}
\]
The top parts of these ladders are their highest nodes in the $\bar 3$-residue 
diagram. 

If $\lambda$ is a bar partition, the {\it ladders in} $\lambda$ are
the non-empty intersections $L_{i,j}(\lambda)$ of the above ladders
$L_{i,j}$ with (the $\bar 3$-residue diagram of) $\lambda$. 
It is not hard to see that the Schur regular partitions are exactly those 
bar partitions for which their nodes on each ladder form a top interval on that 
ladder (see [BMO]). So the idea is be to construct a Schur regular partition 
along its ladders, i.e.\ adding the nodes along the ladders 
$L_{1,1}$, $L_{1,2}$, $L_{2,1}$, $L_{2,2}$, $L_{3,1}$, \dots , 
and each time from top down. 

The next proposition is a useful ingredient for having better control 
over the coefficients    in the induction process; 
it is used to obtain the 2-powers appearing in the theorem below. 
Note that if we only add a sequence of nodes of the same $\bar 3$-residue, 
then the assumption on the components below is satisfied. 

\begin{prop} 
Let $\bar\beta \in {\cal D}(n-k)$ and $\beta\in{\cal D}(n)$
such that the skew diagram $S(\beta)\setminus S(\bar\beta)$
has connected components of type 
\hbox{\setlength{\unitlength}{.5mm}
\begin{picture}(5,5)
\put(0,0){\line(0,1){5}}
\put(5,0){\line(0,1){5}}
\put(0,0){\line(1,0){5}}
\put(0,5){\line(1,0){5}}
\end{picture}} , 
\hbox{\setlength{\unitlength}{.5mm}
\begin{picture}(10,5)
\put(0,0){\line(0,1){5}}
\put(5,0){\line(0,1){5}}
\put(10,0){\line(0,1){5}}
\put(0,0){\line(1,0){10}}
\put(0,5){\line(1,0){10}}
\end{picture}} , 
and 
\hbox{\setlength{\unitlength}{.5mm}
\begin{picture}(5,10)
\put(0,0){\line(0,1){10}}
\put(5,0){\line(0,1){10}}
\put(0,0){\line(1,0){5}}
\put(0,5){\line(1,0){5}}
\put(0,10){\line(1,0){5}}
\end{picture}}, 
only. 
Then 
 $$
< \spin{\hat{\bar\beta}} \!\uparrow^{\tilde S_n}\,,\,\spin{\beta} > \: =
2^{a(\beta,\bar\beta)}\,k\,! $$
 where
 $a(\beta,\bar\beta)=(i(\beta,\bar\beta)-\epps(\beta)+\epps(\bar\beta))/2$
 and $i(\beta,\bar\beta)$ is the number of isolated nodes in 
 \hbox{$S(\beta)\setminus S(\bar\beta)$} that are not on the diagonal.  
 \end{prop}

For the {\bf proof}, count fillings of the boxes in $S(\beta)\setminus S(\bar\beta)$ 
by $1 , \dots ,k$, keep track of
parity changes from odd to even for the partitions arising and apply 
the Branching Theorem (see [BMO]). \\


Inducing along ladders and using this proposition now allows to obtain `good'
projective characters. Before we can formulate this, we have to introduce the 
notion of {\it Schur regularization} of a bar partition $\alpha \in {\cal D}(n)$ 
in the $\bar 3$-residue diagram. 
For this, take the $\bar 3$-residue diagram of~$\alpha$ and consider its 
nodes as beads on the ladders defined above. 
To obtain the Schur regularization $\alpha^S \in {\cal C}_3(n)$, push the 
beads to the top of their ladders, i.e.\ if $\alpha$ has $j$ nodes on a 
particular ladder $L$, then $\alpha^S$ has the top $j$ nodes on this ladder. 
This is a well-defined process that indeed produces a Schur regular partition. 

For example, for $\alpha = (4,3,2)$ the Schur regularization gives

\[ 
\matrix{
1 & 2 & 1 & 1 &                     & 1 & 2 & 1 &1 & 2 & 1 & 1 & 2  \cr
  & 1 & 2 & 1 & \quad \rarrow \quad & & 1 \cr
  & & 1 & 2  \cr
} 
\]

Hence $\alpha^S = (8,1)$. 

A detailed analysis of the induction process and its combinatorics, in 
particular the compatibility of dominance and regularization, then leads to: 


\begin{theorem}\label{approx3} ([BMO]) 
For any Schur regular partition $\beta\in {\cal C}_3(n)$, 
there is a projective character $\Phi_{\beta}$ satisfying 
 $$
\Phi_{\beta} =
\sum_{\stackrel{\mbox{\scriptsize $\alpha\in {\cal D}(n)$}}
 {\mbox{\scriptsize $\alpha^S \triangleleft \beta$}}} 
   t_\alpha \hat{\spin{\alpha}}
+  
\sum_{\stackrel{\mbox{\scriptsize $\alpha\in {\cal D}(n)$}}
 {\mbox{\scriptsize $\alpha^S = \beta$}}} 2^{a(\alpha)} \hat{\spin{\alpha}}
 $$
with $t_\alpha=0$ 
if $\alpha_{(\bar 3)}\ne\beta_{(\bar 3)}$, 
and 
$a(\alpha)=\left\lceil\frac 12(m_0(\alpha)+\varepsilon
 (\alpha_c)-\varepsilon (\alpha)+o(\alpha))\right\rceil$, 
where $m_0(\alpha) =$ \hbox{$|\{ i \mid \alpha_i \equiv 0 (\mbox{mod } 3)\}|$},
$\alpha_c$~denotes the bar partition corresponding to the union of
the complete ladders in~$\alpha$, 
and 
$o(\alpha)=|\{2\mbox{-ladders in } \,\alpha\setminus\alpha'\, 
\mbox{ with an odd number of nodes}\}|$, 
where $\alpha'$ corresponds to the union of the complete
ladders together with the next two ladders in~$\alpha$.
\end{theorem}

From this we deduce the desired result on the shape of the decomposition 
matrix at characteristic 3: 

\begin{theorem}\label{3decomp}
Let~$B$ be a $3$-spin block of~$\tilde S_n$. 
Order the spin characters in~$B$ by first taking the ones with Schur regular 
label in lexicographic order, and then the others; 
take as column labels for the decomposition matrix $D_B$ the Schur regular 
partitions in lexicographic order (doubling the columns if all irreducible 
modular spin re\-pre\-sen\-tations in~$B$ are non-selfassociate). 
Then we have
\[
D_B = 
\begin{array}{r}
\\
\\
\mbox{Schur regular}\\
\mbox{partitions}\\
\\
\\
\\
\mbox{other}\\
\mbox{partitions}
\end{array}
\stackrel{\raisebox{1ex}{Schur regular partitions}}{ 
\left( {
\begin{tabular}{cccccccc}
$\ddots$ \\ \cline{2-3}
&\multicolumn{2}{|c|}{}  \\
&\multicolumn{2}{|c|}{}  &&& 0\\ \cline{2-5}
&&& \multicolumn{2}{|c|}{}  \\ \cline{4-6}
&&&&& \multicolumn{1}{|c|}{}  \\ 
&& * &&& \multicolumn{1}{|c|}{}  \\ \cline{6-6}
&&&&&& $\ddots$ \\ \cline{8-8}
&&&&&&& \multicolumn{1}{|c|}{}\\
\hline
\\
& & & *\\
\\
\end{tabular} }
\right)
}
\] 

Moreover, the last non-zero entry in the row labelled by 
$\alpha \in {\cal D}(n)$ 
is in the \hbox{(double-)}column labelled by $\alpha^S$, 
and it is at most $2^{a(\alpha)}$ with $a(\alpha)$ as in Theorem~\ref{approx3}. 
\end{theorem}

Furthermore, 
the final (double-)column in $D_B$ (labelled by the minimal 
Schur regular partition $\lambda$ in~$B$) 
is determined quite precisely (see [BMO]).\\

Now we turn to the case $p=5$. It turns out that already the first 
combinatorial step is much more complicated than for $p=3$.

\begin{theorem}\label{partid5} ([ABO])
Let $n \in \N$. Then 
\[
\begin{array}{rl}
{\cal  C}_5(n)=
\lbrace  \lambda= & (\ell_1,\ell_2,\dots,\ell_m)\in {\cal D}(n) 
\mid \ell_i-\ell_{i+2}\ge 5\; \mbox{ for all } i\leq m-1;\, \\
& \ell_i-\ell_{i+2}>5\;\mbox{ if }\;\ell_i\equiv 0\pmod 5\;\mbox{ or if }
\ell_i+\ell_{i+1}\equiv 0\pmod 5\,,\;\\
& \mbox{and there are no subsequences of the following types:}\\
& (5j+3,5j+2),\;(5j+11,5j+9,5j+5),\; 
(5j+10,5j+6,5j+4),\\
& (5j+11,5j+10,5j+5,5j+4),\; j\ge 0 \rbrace
\end{array}
\]
and 
\[ 
|{\cal  C}_5(n)|=|{\cal D}_5(n)| \, .
\]
\end{theorem}

A set $\tilde {\cal  C}_5(n)$ very similar to the set ${\cal  C}_5(n)$ 
was defined by Andrews in the context of generalizing the Rogers-Ramanujan 
identities [A1, A2]; 
in fact, there is an easy bijection between these two sets. 
Hence, the second statement in the Theorem above is equivalent to 
showing $|\tilde{\cal  C}_5(n)|=|{\cal D}_5(n)|$. 
This equality was conjectured by Andrews in 1974, and indeed, for the 
proof of the second part of Theorem~\ref{partid5}   
suitable generating functions for partitions 
in  $\tilde{\cal  C}_5(n)$ are considered in [ABO] 
and shown to satisfy the correct  fourth order recurrence relation. 

The proof of the inclusion `$\subseteq$' for 
the internal description of~${\cal  C}_5(n)$ above 
is by induction on $n$, while the other inclusion is proved by 
actually providing a specific top node construction for the partitions 
under consideration. 
The basic observation is that the partitions described in the Theorem 
have a similar description as in the case $p=3$, this time their nodes are 
`almost' at the top of their ladders, the only possible exception being 
a `hole' in $(j,5k+2)$ at the penultimate position of a 2-ladder. 

For example, take $\alpha = (16,14,6,4) \in {\cal C}_5(40)$
and look at its $\bar 5$-residue diagram (where the critical 
2-holes are indicated as boxes):
\[
\matrix{
1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & \Box  \cr
& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 \cr
& & 1 & 2 & 3 & 2 & 1 & 1 &  \Box  \cr
& & & 1 & 2 & 3 & 2 }
\]


The ladders in the $\bar 5$-residue diagram are defined similarly as in 
the $3$-case, i.e.\ by giving the nodes that they are joining, from 
bottom to top: 
\[ \begin{array}{cl}
L_{i,1}{:} & 
(i,1) \rarrow (i-2,6) \rarrow (i-2,5) \rarrow (i-4,11) \rarrow (i-4,10) 
\rarrow \cdots \\
L_{i,2}{:} & 
(i,2)  \rarrow (i-1,4) \rarrow (i-2,7) \rarrow (i-3,9) \rarrow (i-4,12) 
\rarrow \cdots \\
L_{i,3}{:} & 
(i,3)  \rarrow (i-2,8) \rarrow (i-4,13) \rarrow (i-6,18) \rarrow (i-8,23) 
\rarrow \cdots \\
\end{array}
\]

So in the example above the ladders $L_{5,2}(\alpha)$ and 
$L_{7,2}(\alpha)$ have holes in the penultimate position (indicated by a box), 
and all other ladders have their nodes at the top. 

Now again we construct $\alpha \in {\cal C}_5(n)$ by going along the ladders 
from top down except that we take detours caused by `accessible' holes 
in 2-ladders, i.e.\ holes in position $(j,5k+2)$ where $\alpha_j=5k+1$. 

Let us consider a big example for illustrating the path chosen. 
We first show the $\bar 5$-residue diagram of the partition 
$$\lambda = (36,31,30,26,21,20,16,11,10,6,3,1)$$ 
(with the holes indicated by boxes) and then give the path in tableau 
notation, starting with the 1-ladder before the first 2-ladder with an 
accessible hole. \\

\centerline{$\scriptsize{ 
\matrix{
1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 
   & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 
   & 2 & 3 & 2 & 1 & 1 \cr
& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1
   & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 &  \Box  \cr
&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1
   & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 \cr   
&&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1
   & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 \cr      
&&&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1
   & 1 & 2 & 3 & 2 & 1 & 1 &  \Box  \cr      
&&&&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1
   & 1 & 2 & 3 & 2 & 1 \cr      
&&&&&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 \cr
&&&&&&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 & 1 &  \Box  \cr   
&&&&&&&& 1 & 2 & 3 & 2 & 1 & 1 & 2 & 3 & 2 & 1 \cr   
&&&&&&&&& 1 & 2 & 3 & 2 & 1 & 1  \cr
&&&&&&&&&& 1 & 2 & 3 \cr 
&&&&&&&&&&& 1 \cr 
}}
$}
\vspace{4ex}

\[\scriptsize{ 
\matrix{
&&&&&&&&&&&&&&&& .   & 8 & 19 & 20  & 24 & 29 & 32 & 34 & 35 \cr
&&&&&&&&&&&&& .  & 1 & 2 & 9 & 16 & 25 & 31  & 36 &  \Box  \cr
&&&&&&&&&&&&& .  &  10 & 21 & 22 & 26 & 30 & 33  & 37 \cr
&&&&&&&&&& .  &  3 & 4 & 11 & 17 & 27 & 38 & 39 \cr
&&&&&&&&&& .  & 12 & 23 & 40 &  \Box  \cr
&&&&&&& .  & 5 & 6 & 13 & 18 & 28 & 41 \cr
&&&&&&& .  & 14 & 42 & 43 \cr
&&&& .  & 7 & 44 &  \Box  \cr 
&&&& .  & 15 & 45  \cr 
& .  & 46 & 47 & \cr 
. & .  \cr 
48 \cr 
}}
\]

That is, in going down a 1-ladder one has to watch out for neighbours of 
2-holes; if such a node would be added in the next step one goes to the top of 
the next ladder.
In going down a later 1-ladder, $L_{i,1}(\alpha)$ say, one then may be required 
to add the left-out neighbour of a 2-hole on $L_{i-1,1}(\alpha)$ before 
adjoining the final node of~$L_{i,1}(\alpha)$. Then one jumps back to 
$L_{i-1,1}(\alpha)$ to add the tail end of this, again using the same 
procedure. \\

For $\beta \in {\cal C}_5(n)$ let $s(\beta)$ be the sequence of 
$\bar 5$-residues of its nodes in the order of the construction path 
described above. We say that $\alpha \in {\cal C}_5(n)$ can be constructed 
along $s(\beta)$ if $\alpha$ has a construction path (not necessarily of 
the top node type) with the same $\bar 5$-residue sequence. 

Now we have the following crucial combinatorial result:

\begin{prop}
Let $\beta \in {\cal C}_5(n)$ and suppose $\alpha \in {\cal C}_5(n)$ 
can be constructed along~$s(\beta)$. 
Then $\alpha \unlhd \beta$.
\end{prop}


As a consequence we obtain the desired approximation to the decomposition 
matrix for $\tilde S_n$ at characteristic~5: 

\begin{theorem}
For any $\beta \in {\cal C}_5(n)$ there is a projective character 
$\Phi_\beta$ of~$\tilde S_n$ of the form 
$$\Phi_\beta 
= \sum_{\stackrel{\mbox{\scriptsize $\alpha \in {\cal C}_5(n)$}}
       {\mbox{\scriptsize $\alpha \unlhd \beta$}}}
    c_{\alpha\beta}\hat{\spin{\alpha}} 
  + \sum_{\mbox{\scriptsize $\alpha \not\in {\cal C}_5(n)$}} 
  c_{\alpha\beta}\hat{\spin{\alpha}} \: , $$
with $c_{\beta\beta} \neq 0$. 
In particular, with respect to decreasing lexicographical ordering 
of the ${\cal C}_5(n)$-partitions we obtain an approximation to the decomposition 
matrix of~$\tilde S_n$ at characteristic~5 of the form 
\[
\begin{array}{r}
\\
\\
{\cal C}_5(n)\\
\\
\\
\\
\\
{\cal D}(n)\setminus{\cal C}_5(n)
\end{array}
\stackrel{\raisebox{1ex}{${\cal C}_5(n)$}}{ 
\left( {
\begin{array}{cccccc}
*\\
& * & & & 0\\
& & \ddots \\
 &   *\\
& & & & \ddots \\
& & & & & *\\
\hline
\\
& & & *\\
\\
\end{array} }
\right)
}
\quad \mbox{ (up to splitting of rows)}
\] 
and hence the 5-decomposition matrix of~$\tilde S_n$ 
also has the form above (up to splitting of rows and columns). 
\end{theorem}


For $p=5$ we have as yet no suitable regularization process, 
so this result is weaker than Theorem \ref{3decomp} for $p=3$ where 
we had determined the final non-zero entry for each row. \\

In trying to extend our approach to $p>5$, a bad surprise came up: 
$$|{\cal C}_7(21)| = 52 < |{\cal D}_7(21)|=53$$
so the top node algorithm no longer produces enough labels for $p=7$. 
In fact,  $|{\cal C}_p(3p)|  < |{\cal D}_p(3p)|$
for $p=7$, $11$ and~$13$, 
and the difference between  $|{\cal C}_p(n)|$ and $|{\cal D}_p(n)|$ 
gets worse for larger~$n$. 

On the other hand, more important than the choice of the labelling 
partitions $\beta$ is the right choice of the induction path $s(\beta)$, 
as we have seen above. 
It turns out that indeed for $p=7$, $n=21$, it is possible to choose two 
`independent' induction paths for the partition $(9,7,5) \in {\cal C}_7(21)$
and thus still obtain a full approximation matrix -- which is of the desired 
shape. 
The problem now is: how to get control over the induction paths? 
One should keep in mind though that this approach was led by the hope 
that one can find enough projective characters just via inducing 
-- and it is still an open question whether this holds. 


\section{Spin representations at characteristic 2}

The situation for $p=2$ is completely different from the odd 
characteristic case. 
Whereas the conjecture on the $p$-block distribution of the spin characters 
had been formulated by Morris for odd $p$ already in 1965, 
such a conjecture on the 2-block distribution was only suggested in 1987 
by Kn\"orr and Olsson [O1], based on work by Benson. 
In [B1], Benson had calculated the 2-decomposition matrices up to $n=15$ 
(with slight ambiguities for $n \geq 14$). He had also obtained  information 
on special rows of the 2-decomposition matrix for $\tilde S_n$. 

In contrast to odd characteristic, the 2-blocks of~$\tilde S_n$ are mixed, 
i.e.\ contain ordinary as well as spin characters. 
The simple $\tilde S_n$-modules in characteristic 2 all have $\gen{z}$ in their 
kernel, so they may be identified with the simple $S_n$-modules $D^{\lambda}$ 
which are labelled by partitions $\lambda \in {\cal D}(n)$. 
So we do not have to worry about the labelling of the columns of the 
decomposition matrix. This knowledge of the simple modules 
was exploited in the 2-block determination achieved in [BO1]. 

First we have to introduce some more notation. 
For a partition 
$\lambda = (\ell_1 , \dots , \ell_m) \in {\cal D}(n)$ we set 
$$ \mbox{dbl } (\lambda) =  
\left(\left[\frac{\ell_1+1}{2}\right], \, \left[\frac{\ell_1}{2}\right], 
\, \left[\frac{\ell_2+1}{2}\right], \, \left[\frac{\ell_2}{2}\right], 
\ldots, \left[\frac{\ell_m+1}{2}\right], \, 
\left[\frac{\ell_m}{2}\right]\right) \in {\cal P}(n)\, , $$ 

the {\it doubling} of~$\lambda$. 
Furthermore, for $\alpha \in {\cal P}(n)$ we denote by 
$[\alpha]$ the corresponding ordinary character of~$S_n$. 

Using this terminology, Kn\"orr and Olsson conjectured the following 
2-block distribution [O1] which was recently proved in [BO1]: 

\begin{theorem}  
([BO1]) Let $\lambda \in {\cal D}(n)$. Then 
$\spin{\lambda}$ and $[\mbox{dbl}(\lambda)]$ belong to the same 2-block 
of~$\tilde S_n$.
\end{theorem}

The strategy of the proof of this Theorem is quite surprising: 
we first determined the number of spin characters in a fixed 2-block of 
$\tilde S_n$ (and similarly for the 2-blocks of~$\tilde A_n$, the double 
covers of the alternating groups), 
and the result on the 2-block distribution was then proved by an 
intricate induction on $n$ and the weight of relevant blocks, 
using the outcome of the spin character count (see [BO1]).
The key to the counting of spin characters in a 2-block was that as a 
consequence of Brauer's Second Main Theorem the number of ordinary characters 
in a $p$-block is `locally determined' by the number of 
$p$-modular characters, and in our case 2-modular $\tilde S_n$-representations 
are essentially 2-modular $S_n$-representations. 
Further ingredients were a group theoretic analysis of certain centralizer 
subgroups in $S_n$, Clifford theory, the usage of specific spin character 
values and some combinatorial identities. \\

Before we state the result of this spin character enumeration, 
we recall some more definitions and results from the $S_n$ case. 
Given a partition $\alpha \in {\cal P}(n)$, we obtain its 2-core $\alpha_{(2)}$
by removing as many 2-hooks (i.e.\ dominoes) from the Young diagram 
of~$\alpha$ as possible; $\alpha_{(2)}$ determines the 2-block of 
$[\alpha]$. The number of 
removed 2-hooks is then the weight of the 2-block of~$S_n$ to which  
the character $[\alpha]$ belongs. Any 2-block~$B$ of a symmetric group 
is uniquely determined by its 2-core  and its weight~$w$. 
It is well-known that $k(B)=k(2,w)$ 
where $k(2,w)$ is the number of 2-quotients of weight~$w$, 
i.e.\ the number of pairs of partitions $(\lambda_0, \lambda_1)$ 
with $|\lambda_0|+|\lambda_1| =w$. 
Note also that $l(B) = p(w)$. 


\begin{theorem}
Let~$B$ be a 2-block of~$S_n$ of weight $w$, and let $\tilde B$ be the 
2-block of~$\tilde S_n$ containing~$B$ (as sets of characters). Then 
$$k(\tilde B) - k(B) = p(w) + \tilde p^- (w) \: .$$
Furthermore, let~$B'$ be the block
of~$A_n$ covered by~$B$ and let $\tilde B'$ be the block of~$\tilde A_n$
covered by $\tilde B$. Then
 $$
k(\tilde B')-k(B')=p(w)+\widetilde p^+(w)\;.
 $$
Here, $\tilde p^{\epps}(w)$ counts the partitions $\alpha$ of~$w$ with 
$(-1)^{\ell(\alpha)} = \epps$. 
\end{theorem}


With these results at hand, a combinatorial toolkit for dealing with 
spin characters was developed that was similar to the one encountered 
before in odd characteristic. 

Let us first recall that for studying the representations at characteristic~2 
the partitions labelling the ordinary characters 
of~$S_n$ resp.\ $\tilde S_n$ are handled by the 2-residue diagram, 
i.e.\ by considering ladders and regularization along these ladders 
in the 2-residue diagram: 

\[ 
\matrix{ 
0&1&0&1&0&1&0&1&0& \ldots \cr
1&0&1&0&1&0&1&0& \ldots  \cr
0&1&0&1&0&1&0& \ldots \cr
1&0&1&0&1&0&1& \ldots \cr
\vdots}
\]

Here the ladders connect the nodes (from bottom to top):
\begin{eqnarray*} 
L_{i,0} &:& (2i-1,1) \rarrow (2i-2,2) \rarrow (2i-3,3) \rarrow \ldots \rarrow (1,2i-1)\\
L_{i,1} &:& (2i,1) \rarrow (2i-1,2) \rarrow (2i-2,3) \rarrow \ldots \rarrow (1,2i)
\end{eqnarray*}
Again, the ladders in a partition $\alpha$ are just the intersections of
the $L_{i,j}$ with (the 2-residue diagram of) $\alpha$. It is clear
that the 2-regular partitions are exactly those partitions $\alpha$
where all the nodes on the ladders of~$\alpha$ form top parts of
these ladders. Given an arbitrary partition $\alpha$, we ``regularize''
$\alpha$ by replacing the nodes in each ladder $L_{i,j}(\alpha)$ by
the same number of nodes at the top of~$L_{i,j}$; it is easy to check
that this gives a 2-regular partition which we call $\alpha^R$ (see [JK, 
p.\ 282]).\
 
For example, take $\alpha=(4^2,3,1^2)$.
Then the regularization of the 2-residue diagram of~$\alpha$ 
is as follows: 
\[
\matrix{
0&1&0&1 &                     &0&1&0&1&0\cr
1&0&1&0 & \quad \rarrow \quad &1&0&1&0\cr
0&1&0&  &                     &0&1&0\cr
1& & &  &                     &1 \cr
0} 
\]
so $\alpha^R=(5,4,3,1)$. 

For the spin characters  we consider 
instead of the 2-residue diagram the $\bar 4$-{\it residue diagram}. 
For example,  $\lambda=(13,11,8,5,2) \in {\cal D}(39)$ has
$\bar 4$-residue diagram
\[
\matrix{
0&1&1&0&0&1&1&0&0&1&1&0&0\cr
& 0&1&1&0&0&1&1&0&0&1&1\cr
& & 0&1&1&0&0&1&1&0\cr
&&& 0&1&1&0&0\cr
&&&& 0&1}
\]

In the $\bar 4$-residue diagram the ladders $L_{i,j}$ 
join the 0- resp.\ 1-nodes as follows (from bottom to top): 
\begin{eqnarray*} 
L_{i,0} &:& (i,1) \to (i-1,5) \to (i-1,4) \to (i-2,9) \to
(i-2,8) \to \ldots \\
        & & \qquad \ldots \to (1,4(i-1)+1) \to (1,4(i-1))\\[10pt]
L_{i,1} &:&  (i,3) \to (i,2) \to (i-1,7) \to (i-1,6) \to \dots \\
        & & \qquad \ldots \to (1,4(i-1)+3)\to (1,4(i-1)+2)
\end{eqnarray*} 
Note that with these definitions the $\bar 4$-content of~$\lambda$ 
equals the 2-content of dbl($\lambda$), and this content 
determines the common 2-block of~$\spin{\lambda}$ and [dbl($\lambda$)].  

Now induction along the ladders in the 2-residue diagram of a 
given bar partition $\beta \in {\cal D}(n)$ gives the well-known 
shape result for the part of the 2-decomposition matrix 
corresponding to the ordinary characters of~$S_n$ resp.\ $\tilde S_n$, 
and at the same time, by careful consideration of 
the ladders in the $\bar 4$-diagram 
of the bar partitions labelling the spin characters, gives 
the following result for rows corresponding to the spin characters: 
 
\begin{theorem} ([BO1])
Let $\lambda \in {\cal D}(n)$. Set 
$\mbox{\rm dbl}^2(\lambda)=\mbox{\rm dbl}(\lambda)^R$, 
and let $m_0(\lambda)$ denote the number of even parts of~$\lambda$. 
Furthermore, for $\beta \in {\cal D}(n)$ 
let $D^{\beta}$ denote the corresponding 
2-modular simple representation of~$S_n$.\\
Then the 2-modular composition factors 
of the spin representation labelled by $\lambda$ are given by:
 $$
\overline{\spin{\lambda}}\sim
2^{\left[m_0(\lambda)/2\right]}D^{\mbox{\scriptsize \rm dbl}^2(\lambda)} +
\sum_{\beta\triangleright\mbox{\scriptsize \rm dbl}^2(\lambda)} 
c_\beta D^\beta \, .
 $$
\end{theorem}

For special bar partitions $\lambda$ this result 
was proved by different methods by Benson [B2].\\

Based on the knowledge of the 2-block distribution of spin characters, 
results about the heights of spin characters in 2-blocks 
have recently been proved in [BO2].
Remember that by the degree formula the height of the spin character 
$\spin{\lambda}$ is intimately connected with 
the number $g_{\lambda}$ of standard shifted $\lambda$-tableaux. 

We have already had a glimpse of the significance of the 
$\bar 4$-combinatorics for spin characters before; the $\bar 4$-quotient of 
a bar partition $\lambda \in {\cal D}(n)$ describes the bead positions 
of the $\lambda$-parts on the runners of a suitable $\bar 4$-abacus 
(see [BO1]). We denote this $\bar 4$-quotient of~$\lambda$ by 
$\rho = \lambda^{(\bar 4)}$, 
say $\rho = (i^{2m_i+\epps_i})$ with $\epps_i \in \{0, 
1\},$ and we set $\rho_o = (i^{m_i})$ and $\rho_e = (i^{\epps_i})$.
Let $\lambda_o$ resp.\ $\lambda_e$ denote the partition consisting of all odd 
resp.\ even parts of~$\lambda$. Then $\lambda_e = 2\rho_e$ and $\rho_o = 
\mu(\lambda_o)$ in the notation of [O4]. Furthermore, the spin 
character $\spin{\lambda}$ belongs to a 2-block of weight $w = 
w(\lambda) = 2|\rho_o| + |\rho_e|$. 
Finally, we define 
$\bar h (\lambda) = h (\spin{\lambda})$ 
to be the height of~$\spin{\lambda}$ in its 2-block of~$\tilde S_n$.\\
With these notations we have:

\begin{theorem}
Let $\lambda \in {\cal D}(n)$, $w=w(\lambda)$, $\rho_o, \rho_e$ 
as defined above. Then
$$\bar h(\lambda) = \nu_2 (\left[\rho_o\right] (1)) + 
\nu_2 (\spin{\rho_e}(1)) + \nu_2 
{w\choose |\rho_e|} + 2 |\rho_o| + \left[\frac{|\rho_e|}{2}\right] + 
\gamma(\rho_e)$$
where 
\[
\gamma(\rho_e) = \left\{ \begin{array}{cl}
1 & \mbox{if } \, |\rho_e| \, \mbox{ odd and } \, \rho_e \in {\cal D}^-\\
0 & \mbox{otherwise} 
\end{array} \right. \; .
\]
\end{theorem}

Note that by this formula the height of~$\spin{\lambda}$ does not depend 
on the 2-core of its 2-block but only on the $\bar 4$-quotient of~$\lambda$. 

With this formula and a detailed study of the minimal 2-powers dividing 
spin character degrees we then obtained a sharp lower bound for the 
heights; also  an upper bound was given in [BO2]. 

\begin{theorem}
Let $n \in {\N}$, $\lambda \in {\cal D}(n)$, $w = w(\lambda)$ and let 
$s = s(w)$ be the number of summands in the $2$-adic decomposition
of~$w$. Then 
$$\left[\frac{2w-s}{2}\right] \leq \bar h(\lambda) \leq 
          \left[\frac{3 w - 2s}{2}\right] \: .$$
\end{theorem}

In fact, an explicit description of the bar partitions $\lambda$ 
attaining the lower resp.\ upper bound is given in [BO2]. 

As an application of our results on the 2-block distribution of spin 
characters and their heights, we have shown in [BO2] 
that the conjectures 
by Brauer, Olsson and Robinson stated before  all hold for the 
covering groups $\tilde S_n$ also at characteristic~$p=2$.  


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