\section{Introduction}
Neural networks and especially deep learning architectures have become more and more popular recently \cite{lecun2015deep}. We believe that deep neural networks are the most powerful tools in a majority of classification problems (as in the case of image classification \cite{resnet}). Unfortunately, the use of neural networks in regression tasks is limited and it has been recently showed that a softmax distribution of clustered values tends to work better, even when the target is continuous \cite{wavenet}. In some cases seemingly continuous values may be understood as categorical ones (e.g. image pixel intensities) and the transformation between the types is straightforward \cite{Oord16}. However, sometimes this transformation cannot be simply incorporated (as in the case when targets span a huge set of possible values). Furthermore, forcing a neural network to predict multiple targets instead of just a single one makes the evaluation slower.

We want to present a method which fulfils the following requirements:
\begin{itemize}
\item gains advantage from the categorical distribution which makes a prediction more accurate,
\item outputs a single value which is a solution to a given regression task,
\item may be evaluated as quickly as in the case of the original regression neural network.
\end{itemize}

The method proposed, called \emph{drawering}, bases on temporarily extending a given neural network that solves a regression task. That modified neural network has properties which improve learning. Once training is done, the original neural network is used standalone. The knowledge from the extended neural network seems to be transferred and the original neural network achieves better results on the regression task.

The method presented is general and may be used to enhance any given neural network which is trained to solve any regression task. It also affects only the learning procedure.

\section{Main idea}
\subsection{Assumptions}
The method presented may be applied for a regression task, hence we assume:
\begin{itemize}
\item the data $D$ consists of pairs $(x_i,y_i)$ where the input $x_i$ is a fixed size real valued vector and target $y_i$ has a continuous value,
\item the neural network architecture $f(\cdot)$ is trained to find a relation between input $x$ and target $y$, for \mbox{$(x,y) \in D$},
\item a loss function $\mathcal{L}_f$ is used to asses performance of $f(\cdot)$ by scoring $\sum_{(x,y)\in D} \mathcal{L}_f(f(x), y)$, the lower the better.
\end{itemize}

\subsection{Neural network modification} \label{firtsMentionOfPercentiles}
In this setup, any given neural network $f(\cdot)$ may be understood as a composition $f(\cdot) = g(h(\cdot))$, where $g(\cdot)$ is the last part of the neural network $f(\cdot)$ i.e. $g(\cdot)$ applies one matrix multiplication and optionally a non-linearity. In other words, a vector $z = h(x)$ is the value of last hidden layer for an input $x$ and the value $g(z)$ may be written as $g(z) = \sigma (Gh(x))$ for a matrix $G$ and some function $\sigma$ (one can notice that $G$ is just a vector). The job which is done by $g(\cdot)$ is just to squeeze all information from the last hidden layer into one value.

In simple words, the neural network $f(\cdot)$ may be divided in two parts, the first, core $h(\cdot)$, which performs majority of calculations and the second, tiny one $g(\cdot)$ which calculates a single value, a prediction, based on the output of $h(\cdot)$.

Our main idea is to extend the neural network $f(\cdot)$. For every input $x$ the value of the last hidden layer $z = h(x)$ is duplicated and processed by two independent, parameterized functions. The first of them is $g(\cdot)$ as before and the second one is called $s(\cdot)$. The original neural network $g(h(\cdot))$ is trained to minimize the given loss function $\mathcal{L}_f$ and the neural network $s(h(\cdot))$ is trained with a new loss function $\mathcal{L}_s$.
An example of the extension described is presended in the Figure \ref{drawering_example}.
For the sake of consistency the loss function $\mathcal{L}_f$ will be called $\mathcal{L}_g$.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{drawering_example}
\caption{The sample extension of the function $f(\cdot)$. The function $g(\cdot)$ always squeezes the last hidden layer into one value. On the other hand the function $s(\cdot)$ may have hidden layers, but the simplest architecture is presented.}
\label{drawering_example}
\end{figure}

Note that functions $g(h(\cdot))$ and $s(h(\cdot))$ share parameters, because $g(h(\cdot))$ and $s(h(\cdot))$ are compositions having the same inner function. Since the parameters of $h(\cdot)$ are shared. It means that learning $s(h(\cdot))$ influences $g(h(\cdot))$ (and the other way around). We want to train all these functions jointly which may be hard in general, but the function $s(\cdot)$ and the loss function $\mathcal{L}_s$ are constructed in a special way presented below.

All real values are clustered into $n$ consecutive intervals i.e. disjoint sets $e_1, e_2, ..., e_n$ such that
\begin{itemize}
\item $\cup_{i=1}^n e_i$ covers all real numbers,
\item $r_j < r_k$ for $r_j \in e_j, r_k \in e_k$, when $j < k$.
\end{itemize}

The function $s(h(\cdot))$ (evaluated for an input $x$) is trained to predict which of the sets $(e_i)_{i=1}^n$ contains $y$ for a pair $(x,y) \in D$. The loss function $\mathcal{L}_s$ may be defined as a non-binary cross-entropy loss which is typically used in classifiation problems. In the simplest form the function $s(\cdot)$ may be just a multiplication by a matrix $S$ (whose first dimension is $n$).

To sum up, \emph{drawering} in its basic form is to add additional, parallel layer which takes as the input the value of the last hidden layer of the original neural network $f(\cdot)$. A modified (\emph{drawered}) neural network is trained to predict not only the original target, but also additional one which depicts order of magnitude of the original target. As a result extended neural network simultaneously solves the regression task and a related classification problem.

One possibility to define sets $e_i$, called \emph{drawers}, is to take suitable percentiles of target values to make each $e_i$ contain roughly the same number of them.

\subsection{Training functions $g(h(\cdot))$ and $s(h(\cdot))$ jointly} \label{howToTrainFunctions}
Training is done using gradient descent, hence it is sufficient to obtain gradients of all functions defined i.e. $h(\cdot)$, $g(\cdot)$ and $s(\cdot)$. For a given pair $(x,y) \in D$ the forward pass for $g(h(x))$ and $s(h(x))$ is calculated (note that a majority of calculations is shared). Afterwards two backpropagations are processed.

The backpropagation for the composition $g(h(x))$ using loss function $\mathcal{L}_g$ returns a vector which is a concatenation of two vectors  $grad_g$ and $grad_{h,g}$, such that $grad_g$ is the gradient of function $g(\cdot)$ at the point $h(x)$ and $grad_{h,g}$ is the gradient of function $h(\cdot)$ at the point $x$. Similarly, the backpropagation for $s(h(x))$ using loss function $\mathcal{L}_s$ gives two gradients $grad_s$ and $grad_{h,s}$ for functions $s(\cdot)$ and $h(\cdot)$, respectively.

The computed gradients of $g(\cdot)$ and $s(\cdot)$ parameters (i.e. $grad_g$ and $grad_s$) can be applied as in the normal case -- each one of those functions takes part in only one of the backpropagations.

Updating the parameters belonging to the $h(\cdot)$ part is more complex, because we are obtaining two different gradients $grad_{h,g}$ and $grad_{h,s}$. It is worth noting that $h(\cdot)$ parameters are the only common parameters of the compositions $g(h(x))$ and $s(h(x))$. We want to take an average of the gradients $grad_{h,g}$ and $grad_{h,s}$ and apply (update $h(\cdot)$ parameters). Unfortunately, the orders of magnitute of them may be different. Therefore, taking an unweighted average may result in minimalizing only one of the loss functions $\mathcal{L}_g$ or $\mathcal{L}_s$. To address this problem, the averages $a_g$ and $a_s$ of absolute values of both gradients are calculated.

Formally, the norm $L^1$ is used to define:
\begin{equation}
a_g = \left\lVert grad_{h,g} \right\rVert_1,
\end{equation}
\begin{equation*}
a_s = \left\lVert grad_{h,s} \right\rVert_1.
\end{equation*}
The values $a_g$ and $a_s$ aproximately describe the impacts of the loss functions $\mathcal{L}_g$ and $\mathcal{L}_s$, respectively. The final vector $grad_h$ which will be used as a gradient of $h(\cdot)$ parameters in the gradient descent procedure equals:
\begin{equation}
grad_h = \alpha grad_{h,g} + (1 - \alpha) \frac{a_g}{a_s} grad_{h,s}
\end{equation}
for a hyperparameter $\alpha \in (0,1)$, typically $\alpha = 0.5$. This strategy makes updates of $h(\cdot)$ parameters be of the same order of magnitude as in the procces of learning the original neural network $f(\cdot)$ (without \emph{drawering}).

One can also normalize the gradient $grad_{h,g}$ instead of the gradient $grad_{h,s}$, but it may need more adjustments in the hyperparameters of the learning procedure (e.g. learning rate alteration may be required).

Note that for $\alpha = 1$ the learning procedure will be identical as in the original case where the function $f$ is trained using loss function $\mathcal{L}_g$ only.

It is useful to bear in mind that both backpropagations also share a lot of calculations. In the extreme case when the ratio $\frac{a_c}{a_d}$ is known in advance one backpropagation may be performed simultaneously for loss function $\mathcal{L}_g$ and the weighted loss function $\mathcal{L}_s$. We noticed that the ratio needed is roughly constant between batches iterations therefore may be calculated in the initial phase of learning. Afterwards may be checked and updated from time to time.

\textit{In this section we slightly abused the notation -- a value of gradient at a given point is called just a gradient since it is obvious what point is considered.}
\subsection{Defining \emph{drawers}}
\subsubsection{Regular and uneven}\label{evenUneven}
We mentioned in the subsection \ref{firtsMentionOfPercentiles} that the simplest way of defining \emph{drawers} is to take intervals whose endings are suitable percentiles that distribute target values uniformly. In this case $n$ \emph{regular drawers} are defined in the following way:
\begin{equation}
e_i = (q_{i-1,n}, q_{i,n}]
\end{equation}
 where $q_{i,n}$ is $\frac{i}{n}$-quantile of targets $y$ from training set (the values $q_{0,n}$ and $q_{n,n}$ are defined as \emph{minus infinity} and \emph{plus inifinity}, respectively).
This way of defining \emph{drawers} makes each interval $e_i$ contain approximately the same number of target values.

However, we noticed that an alternative way of defining $e_i$'s, which tends to support classical mean square error (MSE) loss better, may be proposed. The MSE loss penalizes more when the difference between the given target and the prediction is larger. To address this problem $drawers$ may be defined in a way which encourages the learning procedure to focus on extreme values. \emph{Drawers} should group the middle values in bigger clusters while placing extreme values in smaller ones. The definition of $2n$ \emph{uneven drawers} is as follows:
\begin{equation}
e_i = (q_{1,2^{n-i+2}}, q_{2,2^{n-i+2}}], \text{ for } i \leq n,
\end{equation}
\begin{equation*}
e_i = (q_{2^{i-n+1}-2,2^{i-n+1}}, q_{2^{i-n+1}-1,2^{i-n+1}}], \text{ for } i>n.
\end{equation*}
In this case every \emph{drawer} $e_{i+1}$ contains approximately two times more target values as compared to \emph{drawer} $e_i$ for $i<n$. Finally, both $e_n$ and $e_{n+1}$ contain the maximum of $25\%$ of all target values. Similarly to the asceding intervals in the first half, $e_i$ are desceding for $i>n$ i.e. contain less and less target values.

The number of \emph{drawers} $n$ is a hyperparameter. The bigger $n$ the more complex distribution may be modeled. On the other hand each \emph{drawers} has to contain enough representants among targets from training set. In our experiments each \emph{drawer} contained at least 500 target values.

\subsubsection{Disjoint and nested} \label{disjointNested}
We observed that sometimes it may be better to train $s(h(\cdot))$ to predict whether target is in a set $f_j$, where $f_j = \cup_{i=j}^n e_i$. In this case $s(h(\cdot))$ has to answer a simpler question: \textit{"Is a target higher than a given value?"} instead of bounding the target value from both sides. Of course in this case $s(h(x))$ no longer solves a one-class classification problem, but every value of $s(h(x))$ may be assesed independently by binary cross-entropy loss.\\

Therefore, \emph{drawers} may be:
\begin{itemize}
\item \emph{regular} or \emph{uneven},
\item \emph{nested} or \emph{disjoint}.
\end{itemize}
These divisions are orthogonal. In all experiments described in this paper (the section \ref{experiments}) \emph{uneven drawers} were used.

\section{Logic behind our idea}
We believe that \emph{drawering} improves learning by providing the following properties.
\begin{itemize}
\item The extension $s(\cdot)$ gives additional expressive power to a given neural network. It is used to predict additional target, but since this target is closely related with the original one, it is believed that gained knowledge is transferred to the core of the given neural network $h(\cdot)$.
\item Since categorical distributions do not assume their shape, they can model arbitrary distribution -- they are more flexible.
\item We argue that classification loss functions provide better behaved gradients than regression ones. As a result evolution of classification neural network is more smooth during learning.
\item Additional target (even closely related) works as a regularization as typically in multitask learning \cite{thrun1996learning}.
\end{itemize}

\section{Model comparison}
Effectiveness of the method presented was established with the comparison. The original and \emph{drawered} neural network were trained on the same dataset and once trainings were completed the neural networks performances on a given test set were measured.  Since \emph{drawering} affects just a learning procedure the comparision is fair.

All learning procedures depend on random initialization hence to obtain reliable results a few learning procedures in both setups were performed. Adam \cite{adam} was chosen for stochastic optimization.

The comparison was done on two datasets described in the following section. The results are described in the section \ref{experiments}.
\section{Data}
The method presented were tested on two datasets.

\subsection{Rossmann Store Sales}
The first dataset is public and was used during \emph{Rossmann Store Sales} competition on well-known platform \emph{kaggle.com}. The official description starts as follows:

\begin{quote}
Rossmann operates over 3,000 drug stores in 7 European countries. Currently, Rossmann store managers are tasked with predicting their daily sales for up to six weeks in advance. Store sales are influenced by many factors, including promotions, competition, school and state holidays, seasonality, and locality. With thousands of individual managers predicting sales based on their unique circumstances, the accuracy of results can be quite varied.
\end{quote}

The dataset contains mainly categorical features like information about state holidays, an indicator whether a store is running a promotion on a given day etc.

Since we needed ground truth labels, only the train part of the dataset was used (in \emph{kaggle.com} notation). We split this data into new training set, validation set and test set by time. The training set ($648k$ records) is consisted of all observations before year 2015. The validation set ($112k$ records) contain all observations from January, February, March and April 2015. Finally, the test set ($84k$ records) covers the rest of the observations from year 2015.

In our version of this task target $y$ is normalized logarithm of the turnover for a given day. Logarithm was used since the turnovers are exponentially distributed. An input $x$ is consisted of all information provided in the original dataset except for \emph{Promo2} related information. A day and a month was extracted from a given date (a year was ignored).

The biggest challenge linked with this dataset is not to overfit trained model, because dataset size is relatively small and encoding layers have to be used to cope with categorical variables. Differences between scores on train, validation and test sets were significant and seemed to grow during learning. We believe that \emph{drawering} prevents overfitting -- works as a regularization in this case.

\subsection{Conversion value task}
This private dataset depicts conversion value task i.e. a regression problem where one wants to predict the value of the next item bought for a given customer who clicked a displayed ad.

The dataset describes states of customers at the time of impressions. The state (input $x$) is a vector of mainly continuous features like a price of last item seen, a value of the previous purchase, a number of items in the basket etc. Target $y$ is the price of the next item bought by the given user. The price is always positive since only users who clicked an ad and converted are incorporated into the dataset.

The dataset was split into training set ($2,1$ million records) and validation set ($0,9$ million observations). Initially there was also a test set extracted from validation set, but it turned out that scores on validation and test sets are almost identical.

We believe that the biggest challenge while working on the conversion value task is to tame gradients which vary a lot. That is to say, for two pairs $(x_1, y_1)$ and $(x_2, y_2)$ from the dataset, the inputs $x_1$ and $x_2$ may be close to each other or even identical, but the targets $y_1$ and $y_2$ may even not have the same order of magnitude. As a result gradients may remain relatively high even during the last phase of learning and the model may tend to predict the last encountered target ($y_1$ or $y_2$) instead of predicting an average of them. We argue that \emph{drawering} helps to find general patterns by providing better behaved gradients.

\section{Experiments}\label{experiments}
In this section the results of the comparisons described in the previous section are presented.

\subsection{Rossmann Store Sales}
In this case the original neural network $f(\cdot)$ takes an input which is produced from 14 values -- 12 categorical and 2 continuous ones. Each categorical value is encoded into a vector of size $min(k,10)$, where $k$ is the number of all possible values of the given categorical variable. The minimum is applied to avoid incorporating a redundancy. Both continuous features are normalized. The concatenation of all encoded features and two continuous variables produces the input vector $x$ of size 75.

The neural network $f(\cdot)$ has a sequential form and is defined as follows:
\begin{itemize}
\item an input is processed by $h(\cdot)$ which is as follows:
	\begin{itemize}
	\item $Linear(75, 64)$,
	\item $ReLU$,
	\item $Linear(64, 128)$,
	\item $ReLU$,
	\end{itemize}
\item afterwards an output of $h(\cdot)$ is fed to a simple function $g(\cdot)$ which is just a $Linear(128, 1)$.
\end{itemize}
The \emph{drawered} neural network with incorporated $s(\cdot)$ is as follows:
\begin{itemize}
\item as in the original $f(\cdot)$, the same $h(\cdot)$ processes an input,
\item an output of $h(\cdot)$ is duplicated and processed independently by $g(\cdot)$ which is the same as in the original $f(\cdot)$ and $s(\cdot)$ which is as follows:
	\begin{itemize}
	\item $Linear(128, 1024)$,
	\item $ReLU$,
	\item $Dropout(0.5)$,
	\item $Linear(1024, 19)$,
	\item $Sigmoid$.
	\end{itemize}
\end{itemize}
\emph{The torch notation were used, here:
\begin{itemize}
\item $Linear(a, b)$ is a linear transformation -- vector of size $a$ into vector of size $b$,
\item $ReLU$ is the rectifier function applied pointwise,
\item $Sigmoid$ ia the sigmoid function applied pointwise,
\item $Dropout$ is a dropout layer \cite{srivastava2014dropout}.
\end{itemize}
}

The \emph{drawered} neural network has roughly $150k$ more parameters. It is a huge advantage, but these additional parameters are used only to calculate new target and additional calculations may be skipped during an evaluation. We believe that patterns found to answer the additional target, which is related to the original one, were transferred to the core part $h(\cdot)$.

We used dropout only in $s(\cdot)$ since incorporating dropout to $h(\cdot)$ causes instability in learning. While work on regression tasks we noticed that it may be a general issue and it should be investigated, but it is not in the scope of this paper.

Fifty learning procedures for both the original and the extended neural network were performed. They were stopped after fifty iterations without any progress on validation set (and at least one hundred iterations in total). The iteration of the model which performed the best on validation set was chosen and evaluated on the test set. The loss function used was a classic square error loss.

The minimal error on the test set achieved by the \emph{drawered} neural network is $4.481$, which is $7.5\%$ better than the best original neural network. The difference between the average of Top5 scores is also around $7.5\%$ in favor of \emph{drawering}. While analyzing the average of all fifty models per method the difference seems to be blurred. It is caused be the fact that a few learning procedures overfited too much and achieved unsatisfying results. But even in this case the average for \emph{drawered} neural networks is about $3.8\%$ better. All these scores with standard deviations are showed in the Table \ref{rossmannScores}.

\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{Rossmann Store Sales Scores}
\label{rossmannScores}
\centering
\begin{tabular}{c||c|c|c|c|c}
Model & Min & Top5 mean & Top5 std & All mean & All std\\
\hline
Original & $4.847$ & $4.930$ & $0.113$ & $5.437$ & $0.259$\\
Extended & $4.481$ & $4.558$ & $0.095$ & $5.232$ & $0.331$\\
\end{tabular}
\end{table}

\textit{We have to note that extending $h(\cdot)$ by additional $150k$ parameters may result in ever better performance, but it would drastically slow an evaluation. However, we noticed that simple extensions of the original neural netwok $f(\cdot)$ tend to overfit and did not achieve better results.}

The train errors may be also investigated. In this case the original neural network performs better which supports our thesis that \emph{drawering} works as a regularization. Detailed results are presented in the Table \ref{rossmannScoresTrain}.

\begin{table}[!h]
\renewcommand{\arraystretch}{1.3}
\caption{Rossmann Store Sales Scores on Training Set}
\label{rossmannScoresTrain}
\centering
\begin{tabular}{c||c|c|c|c|c}
Model & Min & Top5 mean & Top5 std & All mean & All std\\
\hline
Original & $3.484$ & $3.571$ & $0.059$ & $3.494$ & $0.009$\\
Extended & $3.555$ & $3.655$ & $0.049$ & $3.561$ & $0.012$\\
\end{tabular}
\end{table}

\subsection{Conversion value task}
This dataset provides detailed users descriptions which are consisted of 6 categorical features and more than 400 continuous ones. After encoding the original neural network $f(\cdot)$ takes an input vector of size 700. The core part $h(\cdot)$ is the neural network with 3 layers that outputs a vector of size 200. The function $g(\cdot)$ and the extension $s(\cdot)$ are simple, $Linear(200,1)$ and $Linear(200, 21)$, respectively.

In case of the conversion value task we do not provide a detailed model description since the dataset is private and this experiment can not be reproduced. However, we decided to incorporate this comparison to the paper because two versions of \emph{drawers} were tested on this dataset (\emph{disjoint} and \emph{nested}). We also want to point out that we invented \emph{drawering} method while working on this dataset and afterwards decided to check the method out on public data. We were unable to achieve superior results without \emph{drawering}. Therefore, we believe that work done on this dataset (despite its privacy) should be presented.

To obtain more reliable results ten learning procedures were performed for each setup:
\begin{itemize}
\item \textit{Original} -- the original neural network $f(\cdot)$,
\item \textit{Disjoint} -- \emph{drawered} neural network for \emph{disjoint drawers},
\item \textit{Nested} -- \emph{drawered} neural network for \emph{nested drawers}.
\end{itemize}

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{cv}
\caption{Sample evolutions of scores on validation set during learning procedures.}
\label{cvcurves}
\end{figure}

In the Figure \ref{cvcurves} six learning curves are showed. For each of three setups the best and the worst ones were chosen. It means that the minimum of the other eight are between representants showed. The first 50 iterations were skipped to make the figure more lucid. Each learning procedure was finished after 30 iterations without any progress on the validation set.

It may be easily inferred that all twenty \emph{drawered} neural networks performed significantly better than neural networks trained without the extension. The difference between \textit{Disjoint} and  \textit{Nested} versions is also noticeable and $\textit{Disjoint}$ \emph{drawers} tends to perform slightly better.

In the Rossmann Stores Sales case we experienced the opposite, hence the version of \emph{drawers} may be understood as a hyperparameter. We suppose that it may be related with the size of a given dataset.

\section{Analysis of $s(h(x))$ values}

Values of $s(h(x))$ may be analyzed. For a pair \mbox{$(x,y) \in D$} the $i$-th value of the vector $s(h(x))$ is the probability that target $y$ belongs to the \emph{drawer} $f_i$. In this section we assume that \emph{drawers} are nested, hence values of $s(h(x))$ should be descending. Notice that we do not force this property by the architecture of \emph{drawered} neural network, so it is a side effect of the nested structure of \emph{drawers}.

\begin{figure}[!t]
\centering
\includegraphics[width=\columnwidth]{rss}
\caption{Sample values of $s(h(x))$ for a randomly chosen model solving Rossmann Store Sales problem (nested \emph{drawers}).}
\label{rss}
\end{figure}

In the Figure \ref{rss} a few sample distributions are showed. Each according label is ground truth ($i$ such that $e_i$ contains target value). The values of $s(h(x))$ are clearly monotonous as expected. It seems that $s(h(x))$ performs well -- values are close to one in the beginning and to zero in the end. A switch is in the right place, close to the ground truth label and misses by maximum one \emph{drawer}.

\section{Conclusion}
The method presented, \emph{drawering}, extends a given regression neural network which makes training more effective. The modification affects the learning procedure only, hence once \emph{drawered} model is trained, the extension may be easily omitted during evaluation without any change in prediction. It means that the modified model may be evaluated as fast as the original one but tends to perform better.
\newpage
We believe that this improvement is possible because \emph{drawered} neural network has bigger expressive power, is provided with better behaved gradients, can model arbitrary distribution and is regularized. It turned out that the knowledge gained by the modified neural network is contained in the parameters shared with the given neural network.

Since the only cost is an increase in learning time, we believe that in cases when better performance is more important than training time, \emph{drawering} should be incorporated into a given regression neural network.

\bibliographystyle{IEEEtran}


\section{Introduction}

At the meeting of the American Mathematical Society in Hayward, California, in April 1977, Olga Taussky-Todd \cite{TausskyTodd} asked whether one could characterize the values of the group determinant when the entries are all integers.
For a prime $p,$ a complete description was obtained for $\mathbb Z_{p}$ and $\mathbb Z_{2p}$, the cyclic groups of order $p$ and $2p$,  in \cite{Newman1} and \cite{Laquer}, and  for $D_{2p}$ and $D_{4p}$ the dihedral groups of order $2p$ and $4p$ in \cite{dihedral}. The values for $Q_{4n}$, the dicyclic group of order $4n$ were explored in \cite{dicyclic}
with a near complete description for $Q_{4p}$.  In general though this quickly becomes a hard problem,
with only partial results known even for $\mathbb Z_{p^2}$ once $p\geq 7$ (see \cite{Newman2} and \cite{Mike}).

The remaining groups of order less than 15 were tackled in  \cite{smallgps} and $\mathbb Z_{15}$ in \cite{bishnu1}.
The integer group determinants have been determined for  all  five  abelian groups of order 16 ($\mathbb Z_2 \times \mathbb Z_8$, $\mathbb Z_{16}$, $\mathbb Z_2^4$, $\mathbb Z_4^2$, $\mathbb Z_2^2 \times\mathbb Z_4$  in  \cite{Yamaguchi1,Yamaguchi2,Yamaguchi3,Yamaguchi4,Yamaguchi5}), and for three of the  non-abelian groups
($D_{16}$, $\mathbb Z_2\times D_8$, $\mathbb Z_2 \times Q_8$ in \cite{dihedral,ZnxH}).

Here we determine the the group determinants for $Q_{16}$,  the dicyclic or generalized quaternion group of order 16.
$$ Q_{16}=\langle X,Y \; | \; X^8=1,\; Y^2=X^4,\; XY=YX^{-1}\rangle. $$
 This leaves five  unresolved non-abelian groups of order 16

\begin{theorem} The even integer group determinants for $Q_{16}$ are exactly the multiples of $2^{10}$.

The odd integer group determinants are all the integers $n\equiv 1$ mod 8 plus  those $n\equiv 5$ mod 8 of the form
$n=mp^2$ where $m\equiv 5$ mod 8 and $p\equiv 7$ mod $8$ is prime.

\end{theorem}

We shall think here of the  group determinant as being defined on elements of the group ring $\mathbb Z [G]$
$$ \mathcal{D}_G\left( \sum_{g\in G} a_g g \right)=\det\left( a_{gh^{-1}}\right) .$$
\begin{comment}
We observe  the multiplicative property
\begin{equation} \label{mult}  \mathcal{D}_G(xy)= \mathcal{D}_G(x)\mathcal{D}_G(y), \end{equation}
using that
$$ x=\sum_{g \in G} a_g g,\;\;\;  y=\sum_{g \in G} b_g g \; \Rightarrow  \; xy=\sum_{g\in G} \left(\sum_{hk=g}a_hb_k\right) g. $$
\end{comment}
Frobenius \cite{Frob} observed that the group determinant can be factored using the  groups representations (see for example \cite{Conrad} or \cite{book})
and an explicit  expression for a dicyclic group determinant was given in \cite{smallgps}. For $Q_{16}$, arranging the
16 coefficients into two polynomials of degree 7
$$ f(x)=\sum_{j=0}^7 a_j x^j,\;\; g(x)=\sum_{j=0}^7 b_jx^j, $$
and writing the primitive 8th root of unity $\omega:=e^{2\pi i/8}=\frac{\sqrt{2}}{2}(1+i)$, this becomes
\begin{equation} \label{form}\mathcal{D}_G\left( \sum_{j=0}^7 a_j X^j + \sum_{j=0}^7 b_j YX^j\right)  =ABC^2D^2 \end{equation}
with integers $A,B,C,D$ from
\begin{align*}
A=& f(1)^2- g(1)^2\\
B=& f(-1)^2-g(-1)^2\\
C=& |f(i)|^2-|g(i)|^2 \\
D=& \left(|f(\omega)|^2+|g(\omega)|^2\right)\left(|f(\omega^3)|^2+|g(\omega^3)|^2\right).
\end{align*}
From \cite[Lemma 5.2]{dicyclic} we know that the even values must be multiples of $2^{10}$.  The odd values must be
1 mod 4 (plainly $f(1)$ and $g(1)$ must be of opposite parity and $A\equiv B\equiv \pm 1$  mod 4 with $(CD)^2\equiv 1$ mod 4).

\section{Achieving the values $n\not \equiv 5$ mod 8}

We can achieve all the multiples of $2^{10}$.

Writing $h(x):=(x+1)(x^2+1)(x^4+1),$ we achieve the $2^{10}(-3+4m)$ from
$$
f(x) = (1-m)h(x),\quad
g(x)=1+x^2+x^3+x^4-mh(x), $$
the $2^{10}(-1+4m)$  from
$$ f(x)= 1+x+x^4+x^5-mh(x),\;\;\;\;
g(x)= 1+x-x^3-x^7-mh(x), $$
the $2^{11}(-1+2m)$ from
$$ f(x)= 1+x+x^2+x^3+x^4+x^5-mh(x),\;\;\quad
g(x)=1+x^4-mh(x), $$
and the $2^{12}m$ from
$$ f(x)= 1+x+x^4+x^5-x^6-x^7-mh(x),\;\;
g(x)= 1+x-x^3+x^4+x^5-x^7+mh(x). $$

We can achieve all the $n\equiv  1$ mod 8; the $1+16m$ from
$$ f(1)=1+mh(x),\;\; g(x)=mh(x), $$
and the $-7+16m$ from
$$f(x)= 1-x+x^2+x^3+x^7- mh(x),\;\;
g(x)= 1+x^3+x^4+x^7-mh(x). $$

\section{ The form of the $n\equiv 5$ mod 8}

This leaves the $n\equiv 5$ mod 8. Since $(CD)^2\equiv 1$ mod 8 we must have $AB\equiv 5$ mod 8. Switching $f$ and $g$ as necessary we assume that $f(1),f(-1)$ are odd and $g(1),g(-1)$ even. Replacing $x$ by $-x$  if needed we can assume that $g(1)^2\equiv 4$ mod 8 and $g(-1)^2\equiv 0$ mod 8.

We write
$$ F(x)=f(x)f(x^{-1})= \sum_{j=0}^7 c_j (x+x^{-1})^j, \quad  G(x)=g(x)g(x^{-1})= \sum_{j=0}^7 d_j (x+x^{-1})^j, $$
with the $c_j,d_j$ in $\mathbb Z$.

From $F(1),F(-1)\equiv 1$ mod 8 we have
$$ c_0+2c_1+4c_2 \equiv 1 \text{ mod }8, \quad  c_0-2c_1+4c_2 \equiv 1 \text{ mod }8,  $$
and $c_0$ is odd and $c_1$ even.
From $G(1)\equiv 4$, $G(-1)\equiv 0$ mod 8 we have
$$  d_0+2d_1+4d_2 \equiv 4 \text{ mod 8}, \quad d_0-2d_1+4d_2 \equiv 0 \text{ mod } 8, $$
and $d_0$ is even and $d_1$ is odd.
Since $\omega+\omega^{-1}=\sqrt{2}$ we get
\begin{align*} F(\omega)  & = (c_0+2c_2+4c_4+\ldots ) + \sqrt{2}(c_1+2c_3+4c_5+\cdots),\\
G(\omega)  & = (d_0+2d_2+4d_4+\ldots ) + \sqrt{2}(d_1+2d_3+4d_5+\cdots),
\end{align*}
and
$$|f(\omega)|^2+|g(\omega)|^2=  F(\omega)+G(\omega) = X+ \sqrt{2} Y>0, \;\; \quad X, Y \text{odd, }  $$
with $ |f(\omega^3)|^2+|g(\omega^3)|^2=F(\omega^3)+G(\omega^3) = X- \sqrt{2} Y>0$. Hence  the positive integer $D=X^2-2Y^2\equiv -1$ mod 8.
Notice that primes 3 and 5 mod 8 do not split in $\mathbb Z[\sqrt{2}]$ so only their squares can occur in $D$. Hence
$D$ must contain at least one prime $p\equiv 7$ mod 8, giving the claimed form of the values 5 mod 8.

\section{Achieving the specified values 5 mod 8}
Suppose that $p\equiv 7$ mod 8 and $m\equiv 5$ mod 8. We need to achieve $mp^2$.

Since $p\equiv 7$ mod 8 we know that $\left(\frac{2}{p}\right)=1$ and $p$ splits in $\mathbb Z[\sqrt{2}].$ Since $\mathbb Z[\sqrt{2}]$  is
a UFD, a generator for the prime factor gives a  solution to
$$ X^2-2Y^2=p, \;\; X,Y\in \mathbb N. $$
Plainly $X,Y$ must both be odd and $X+\sqrt{2}Y$ and $X-\sqrt{2}Y$ both positive.
Since $(X+\sqrt{2}Y)(3+2\sqrt{2})=(3X+4Y)+\sqrt{2}(2X+3Y)$ there will be $X,Y$ with $X\equiv 1$ mod 4 and with
$X\equiv -1$ mod 4.

Cohn \cite{Cohn} showed that $a+b\sqrt{2}$ in $\mathbb Z[\sqrt{2}]$ is a sum of four squares in $\mathbb Z[\sqrt{2}]$ if and only if $2\mid b$. Hence we can write
$$ 2(X+\sqrt{2}Y)= \sum_{j=1}^4 (\alpha_j + \beta_j\sqrt{2})^2, \;\;\alpha_j,\beta_j\in \mathbb Z. $$
That is,
$$ 2X=\sum_{j=1}^4 \alpha_j^2+ 2\sum_{j=0}^4 \beta_j^2,\;\;\quad Y=\sum_{j=1}^4\alpha_j\beta_j.$$
Since $Y$ is odd we must have at least one pair, $\alpha_1$, $\beta_1$ say,  both odd. Since $2X$ is even we must have two
or four of the $\alpha_i$ odd. Suppose that $\alpha_1$, $\alpha_2$ are odd and $\alpha_3,\alpha_4$ have the same parity.
We get
\begin{align*} X+\sqrt{2}Y & = \left(  \frac{\alpha_1+\alpha_2}{2} + \frac{\sqrt{2}}{2}(\beta_1+\beta_2)\right)^2+ \left(  \frac{\alpha_1-\alpha_2}{2} + \frac{\sqrt{2}}{2}(\beta_1-\beta_2)\right)^2 \\
  & \quad +  \left(  \frac{\alpha_3+\alpha_4}{2} + \frac{\sqrt{2}}{2}(\beta_3+\beta_4)\right)^2+ \left(  \frac{\alpha_3-\alpha_4}{2} + \frac{\sqrt{2}}{2}(\beta_3-\beta_4)\right)^2.
\end{align*}
Writing
$$ f(\omega)=a_0+a_1\omega+a_2\omega^2+a_3\omega^3=a_0+ \frac{\sqrt{2}}{2}(1+i)a_1+a_2i+ \frac{\sqrt{2}}{2}(-1+i)a_3,$$
we have
$$ \abs{f(\omega)}^2 =\left(a_0+ \frac{\sqrt{2}}{2}(a_1-a_3)\right)^2  +  \left(a_2+ \frac{\sqrt{2}}{2}(a_1+a_3)\right)^2  $$
and can make
$$ |f(\omega)|^2+|g(\omega)|^2 = X + \sqrt{2}Y $$
with the selection of integer coefficients for $f(x)=\sum_{j=0}^3a_jx^j$ and $g(x)=\sum_{j=0}^3 b_jx^j$
\begin{align*}  a_0=&\frac{1}{2}(\alpha_1-\alpha_2),\quad  a_1 =\beta_1,\quad   a_2=\frac{1}{2}(\alpha_1+\alpha_2), \quad a_3=\beta_2, \\
    b_0=& \frac{1}{2}(\alpha_3-\alpha_4),\quad  b_1 =\beta_3,\quad   b_2=\frac{1}{2}(\alpha_3+\alpha_4), \quad b_3=\beta_4.
\end{align*}
These $f(x)$, $g(x)$ will then give $D=p$ in \eqref{form}.
We can also determine the parity of the coefficients.

\vskip0.1in
\noindent
{\bf Case 1}: the $\alpha_i$ are all odd.

Notice that $a_0$ and $a_2$ have opposite parity, as do $b_0$ and $b_2$. Since $Y$ is odd we must have one or three of the
$\beta_i$ odd.

 If $\beta_1$ is odd and $\beta_2,\beta_3,\beta_4$ all even, then $2X\equiv 6$ mod 8 and $X\equiv -1$ mod 4.
Then $a_0,a_1,a_2,a_3$ are either odd, odd, even, even or even, odd, odd, even and $f(x)=u(x)+2k(x)$
with $u(x)=1+x$ or $x(1+x)$. Likewise $b_0,b_1,b_2,b_3$  are odd, even, even, even or even, even, odd, even
and $g(x)=v(x)+2s(x)$ with $v(x)=1$ or $x^2$. Hence if we take
\begin{equation} \label{shift}  f(x)=u(x)+(1-x^4)k(x)-mh(x),\quad g(x)=v(x)+(1-x^4)s(x)-mh(x), \end{equation}
we get $A=3-16m$, $B=-1$, $C=1$, $D=p$ and we achieve $(16m-3)p^2$ in \eqref{form}.

If three $\beta_i$ are odd then $2X\equiv 10$ mod 8 and $X\equiv 1$ mod 4. We assume $\beta_1,\beta_2,\beta_3$ are
odd and $\beta_4$ even. Hence  $a_0,a_1,a_2,a_3$ are either odd, odd, even, odd or even, odd, odd, odd and
$f(x)=u(x)+2k(x)$ with $u(x)=1+x+x^3$ or $x(1+x+x^2)$ and $b_0,b_1,b_2,b_3$  are odd, odd, even, even or even, odd, odd, even
and $g(x)=v(x)+2s(x)$ with $v(x)=1+x$ or $x(1+x)$. In this case \eqref{shift} gives
$A=(5-16m)$, $B=1$, $C=-1$, $D=p$ achieving $(5-16m)p^2$.

\vskip0.1in
\noindent
{\bf Case 2}: $\alpha_1$, $\alpha_2$ are odd, $\alpha_3$, $\alpha_4$ are even.

In this case $a_0$, $a_2$ will have opposite parity and $b_0$, $b_2$ the same parity.
Since $Y$ is odd we must have $\beta_1$ odd, $\beta_2$ even. Since $2X\equiv 2$ mod 4 we must have one more odd $\beta_i$, say $\beta_3$ odd and $\beta_4$ even.
If $\alpha_3\equiv \alpha_4$ mod 4 then $2X\equiv 6$ mod 8 and $X\equiv -1$ mod 4. Hence
 $a_0,a_1,a_2,a_3$ are either odd, odd, even, even or even, odd, odd, even, that is $u(x)=1+x$ or $x(1+x)$  and  $b_0,b_1,b_2,b_3$  are even, odd, even, even and $v(x)=x^2$ and again \eqref{shift} gives $(16m-3)p^2$.

If $\alpha_3\not\equiv \alpha_4$ mod 4 then $2X\equiv 10$ mod 8 and $X\equiv 1$ mod 4. In this case
 $a_0,a_1,a_2,a_3$ are either odd, odd, even, even or even, odd, odd, even, that is $u(x)=1+x$ or $x(1+x)$  and  $b_0,b_1,b_2,b_3$  are odd, odd, odd, even and $v(x)=1+x+x^2$ and again \eqref{shift} gives $(5-16m)p^2$.

Hence, in either case, starting with an $X\equiv 1$ mod 4 gives the $mp^2$ with $m\equiv 5$ mod 16 and
an  $X\equiv -1$ mod 4 the $mp^2$ with $m\equiv -3$ mod 16.

\section*{Acknowledgement}
\noindent
We thank Craig Spencer for directing us to Cohn's four squares theorem in $\mathbb Z[\sqrt{2}]$.


\section{Introduction}

Seven institutes\footnote{CEA Saclay, Paris, France; CAB-INTA, Madrid, Spain; MPIA, Heidelberg, Germany; University College London, U.K.; University of Leicester, U.K.; SRON, Utrecht, NL; Universitat Wien, Austria}
in Europe have combined their expertise in the field of exoplanetary research to develop the European Horizon-2020
ExoplANETS-A\footnote{{\it Exoplanet Atmosphere New Emission Transmission Spectra Analysis};
https://cordis.europa.eu/project/rcn/212911\_en.html ;
The ExoplANETS-A project has received funding from the EU's Horizon-2020 programme; Grant Agreement no.~776403.}
project under the coordination of CEA Saclay. In the framework of the project, novel data calibration and spectral extraction tools, as well as novel retrieval tools, based on 3D models of exoplanet atmospheres, will be developed to exploit archival data from space- and ground-based observatories, and produce a homogeneous and reliable characterization of the atmospheres of transiting exoplanets. Additionally, to model successfully the exoplanet atmosphere, it is necessary to have a sound knowledge of the host star. To this end, we will collect a coherent and uniform database of the relevant properties of host stars from online archives (e.g. XMM-Newton, Gaia) and publications. These exoplanet and host-star catalogues will be accompanied by computer models to assess the importance of star--planet interactions, for example the `space weather' effects of the star on its planetary system. The knowledge gained from this project will be published through peer-reviewed scientific journals and modelling tools will be publicly released.

The project has six work packages (WPs); the focus in this paper is on the WP `Host-star properties: the active environment of exoplanets'. Fig.\ref{fig_dataflow}(a) illustrates the flow of data through the host-stars WP, and interfaces to the overall project.

We outline the activities concerning the host stars, and present early results from the host-star investigations, including the basic stellar observational and physical properties, and indications of future observations needed to maximize the coverage of the target list of $\sim 100$ stars. We also discuss some of the modelling aspects.

\begin{figure}[h]
\begin{center}
 \includegraphics[width=50mm]{ExoplanetsA-HostStarDataFlow-Fig-v01.eps}
 \includegraphics[width=80mm]{SpType_Histogram_v03.eps}
  \caption{ {\it (a, left).} Schematic of the flow of data through the host-stars workpackage, and interfaces to the overall project. {\it (b, right).} The target sample, in terms of the distribution of host stars with spectral type, showing the breakdown by archival-observation types.}
   \label{fig_dataflow}
\end{center}
\end{figure}

\section{The host-stars activities}

In addition to compiling the observational data, we will
where necessary, e.g.\ in the EUV, interpolate, extrapolate and scale spectral and other information, to cover gaps in the observational data, in order to provide the full XUV spectral range for modelling the host-star influence on the exoplanet atmosphere (e.g.\ Nemec \etal\ 2019).
This work will be accompanied by computer models to assess the importance of star--planet interactions, for example the `space weather' effects of the star on its planetary system; We will also model the possible evolutionary scenarios for the stellar activity over the star's lifetime, in order to gain insight into the past environment of the exoplanet (e.g.\ Johnstone \etal\ 2015).

\section{The sample}

The sample of exoplanets and host stars considered by the project comprises all transiting-exoplanet systems observed by HST or Spitzer. Currently, this corresponds to 135 exoplanets, of which 85 have HST data; the associated number of stars is 113, with 76 having HST data (Fig.\ref{fig_dataflow}(b)).
For the host stars, the primary online archival databases to be used are HST (for UV spectra), XMM-Newton (for X-ray), Gaia (for astrometric properties) and SIMBAD (for spectral types etc). The searches also include: GALEX (UV photometry); Chandra, ROSAT and other X-ray catalogues, together with results retrieved from the published literature. At X-ray wavelengths, most of the detections are from the 3XMM (DR8) serendipitous source catalogue (Rosen \etal\ 2016).

In summary, to date (November 2018), we have the following statistics:
\begin{itemize}
	\item X-ray (3XMM-DR8 cat.): 31 stars observed, with 17 detections in the public archive, and the outcomes awaited for the others;
	\item UV photometry (GALEX, GR6 cat.): 70 stars detected (51 in the HST sample)
	\item UV spectra (HST COS and/or STIS): 26 stars observed
	\item X-ray observation \& UV photometry: 23 stars
	\item X-ray observation \& UV spectra: 20 stars
\end{itemize}
All the stars with XMM observations also have HST visible/IR spectra.

Fig.\ref{fig_examples} illustrates a few of the stellar parameters from our initial assessments to date.

\begin{figure}[h]
\begin{center}
 \includegraphics[width=50mm]{Dist_AppMagGaia_v03.eps}
 \includegraphics[width=40mm]{AppMagGaia_Xflux_v01.eps}
 \includegraphics[width=40mm]{Dist_Lx_v01.eps}
  \caption{Examples of host-star properties, plotted from the catalogue content. }
   \label{fig_examples}
\end{center}
\end{figure}

In order to fill in gaps in the stellar parameter space (e.g.\ to have adequate UV and X-ray measurements across spectral types) we are planning observing proposals, principally to HST and XMM-Newton.

\section{Determination of the physical parameters of the host stars}

A coherent and uniform determination of the stellar properties is essential to avoid any bias in the final results; use of Virtual Observatory (VO) tools facilitates this goal.
For example, we have used VOSA (http://svo2.cab.inta-csic.es/theory/vosa/) to build observational Spectral Energy Distributions (SED) and  compare them with theoretical SEDs (Fig.\ref{fig_sed_rot}) to obtain physical parameters such as effective temperature,
stellar radius and luminosity.
Effective temperatures range from 3000 to 6500K.
We have also used VOSA to search for infrared excess in our sources, with negative outcome.

Medium/high resolution spectra for 48 of our sources
observed with FEROS, HARPS, FORS 1-2, X-SHOOTER, FLAMES and UVES were gathered from the ESO archive.
Visible-light photometry from Kepler/K2 and TESS will also be used to help characterise the stars in terms of rotational and aperiodic variability.
One third of our sources have
Kepler light-curves while TESS will probably provide data for the remainder.

\begin{figure}[h]
\begin{center}
 \includegraphics[width=65mm]{sed_example_v01.eps}
 \includegraphics[width=65mm]{RotCurves_v02.eps}
  \caption{{\it (a, left).} SED fitting using VOSA. Blue spectrum represents the theoretical model (Allard \etal\ 2012) that best fits while red dots represent the observed photometry. {\it (b, right).} Stellar rotation rate (days) as a function of log(age, years) for 3 cases: wind + Alf\'en wings torques (light blue curve), wind + tides (red) and wind + tides + Alfv\'en wings torques (dark blue) (Ahuir \etal\ 2019 in prep; see Benbakoura \etal\ 2018 for details of the ESPEM code used for the study). }
   \label{fig_sed_rot}
\end{center}
\end{figure}

The ExoplANETS-A database of stellar properties and spectral models will be made publicly available, via a web-based interface, by the end of 2020.

\section{Scientific interpretation of the data: star--planet interactions}
Most exoplanets live around active stars, whose magnetism and intense activity can have
a direct impact on habitability conditions and more generally on the exosphere of the planet.
Intense wind and storms can lead to atmospheric loss. Further, a large fraction of exoplanets live
close to their host star (within 10--20 solar radii), to a point where in many cases they orbit within the star's Alfv\'en surface.
This has direct consequence on star--planet interaction and planet migration, as magnetic torques through
Alfv\'en wings directly connecting  the planet to its host star can occur. We have developed both
ab-initio 3D MHD simulation of such close-in systems (Strugarek \etal\ 2015, Strugarek \etal\ 2017) as well as a simpler 1D secular evolution model
of star--planet systems (Benbakoura \etal\ 2018).
In Fig.\ref{fig_sed_rot}(b) we show the evolution of a star--planet system subject to intense
magnetic torques through direct Alfv\'en wings connection (on top of tides and wind effect) and compare it to the case with magnetic torques and stellar wind only (light blue curve)
or tides + wind (red curve).
We note that adding all the effects leads to a quick demise of the planet, that impacts the star's rotation rate.
Such Alfv\'en wings can also lead to hot spots on the star's surface, impacting directly their light-curve and hence the transit curves.
We intend to develop 3D and secular models for the most important systems identified in the ExoplANETS-A project,
as well as obtaining spectropolarimetric magnetic maps of the host star in order to model the stellar wind and assess as accurately as possible the
space environment around these exoplanetary systems.


\section{Introduction}

The Fock space representation of the
quantum affine algebra $U_q(\widehat{sl}_n)=U_q(A^{(1)}_{n-1})$
was constructed by Hayashi \cite{H}.
A combinatorial version of this construction was then used by
Misra and Miwa \cite{MM} to describe  Kashiwara's crystal basis of
the basic representation $V(\Lambda_0)$.
This made it possible to compute the global crystal basis of
$V(\Lambda_0)$ \cite{LLT}. Then, it was conjectured
that the degree $m$ part of the transition matrices
giving the coefficients of the global basis
on the natural basis of the Fock space
were $q$-analogues of the decomposition matrices of the  type $A$
Hecke algebras $H_m$ at an $n$th root of unity \cite{LLT}.
According to a conjecture
of James \cite{J}, these should coincide, for $n$ prime and
large enough,
with the decomposition matrices of symmetric groups ${\rm S}_m$ over a field
of characteristic $n$.
The conjecture of \cite{LLT} has been proved by Ariki \cite{Ar},
and by Grojnowski \cite{Gr} using the results of \cite{G}.

There is another approach to the calculation of decomposition
matrices of type $A$ Hecke algebras, relying upon Soergel's
results on tilting modules for quantum groups at roots of
unity \cite{Soe1,Soe2}.
This approach also leads to $q$-analogues of decomposition
numbers expressed in terms of Kazhdan-Lusztig polynomials.
It seems that these $q$-analogues are the same as those
of \cite{LLT} but there is no proof of this coincidence.
In fact, the relationship between the two approaches is still unclear.

The results of \cite{LLT,Ar,Gr} have been applied recently
by Foda {\it et al.} \cite{FLOTW} to determine which simple
$H_m$-modules remain simple after restriction to $H_{m-1}$
and to show that this problem is equivalent to the decomposition
of a tensor product of level 1 $A_{n-1}^{(1)}$-modules.
This provided an explanation for an intriguing correspondence
previously observed in \cite{FOW} between a class of RSOS models
and modular representations of symmetric groups.

Another description of the $U_q(A^{(1)}_{n-1})$-Fock space,
as a deformation of the infinite wedge realization of
the fermionic Fock space, was obtained by Stern \cite{St}.
In \cite{KMS}, the $q$-bosons needed for the decomposition
of the Fock space into irreducible $U_q(A^{(1)}_{n-1})$-modules
were introduced. This construction was used in \cite{LLTrib}
to give a combinatorial formula for the highest weight
vectors, and in \cite{LT} to define a  canonical basis
of the whole Fock space which was conjectured to
yield the decomposition matrices
of $q$-Schur algebras at roots of unity.
Moreover, strong support in favor of this conjecture was
obtained by establishing its compatibility with a version
of the Steinberg tensor product theorem proved by James
in this context \cite{J,LT}.

Recently, the theory of perfect crystals \cite{KMN1,KMN2} allowed
Kashiwara {\it et al.} \cite{KMPY} to define a general
notion of $q$-Fock space, extending the results of \cite{KMS}
to several series of affine algebras.
Their results apply in particular to the twisted affine algebra
of type $A^{(2)}_{2n}$, which is the case considered in this note.

It has been noticed by Nakajima and Yamada \cite{NY} that the combinatorics
of the basic representation
$V(\Lambda_n)$ of $A^{(2)}_{2n}$ was similar to the
one encountered in the $(2n+1)$-modular representation theory of the spin
symmetric groups ${\rm \widehat{S}}_m$ by Morris \cite{Mo1} as early as 1965.
This can be explained to a certain extent by observing that
the $(r,\bar{r})$-inducing operators of Morris and Yaseen \cite{MY}
coincide with the Chevalley lowering operators of the
Fock space representation of $A^{(2)}_{2n}$. This provides
a further example of the phenomenon observed in \cite{LLT}
in the case of symmetric groups and $A_{n-1}^{(1)}$-algebras.

In this note, we give the analogues for $U_q(A^{(2)}_{2n})$ of the
results of \cite{LLT}.
Using the level~1 $q$-Fock spaces of \cite{KMPY},
we describe an algorithm for computing
the canonical basis of the basic representation $V(\Lambda_n)$, which
allows us to prove that this basis is in the ${\bf Z}[q]$-lattice
spanned by the natural basis of the $q$-Fock space, and that
the transition matrices have an upper triangle of zeros
(Theorem 4.1).

We conjecture that the specialization $q=1$ gives, up to splitting of rows and
columns for pairs of associate characters, and for sufficiently
large primes $p=2n+1$, the decomposition matrices of spin symmetric groups.
However, the reduction $q=1$ is more tricky than in the $A_{n-1}^{(1)}$ case.
Indeed, the $q$-Fock space of $A^{(2)}_{2n}$ is strictly larger than
the classical one, and one has to factor out the null space
of a certain quadratic form \cite{KMPY} to recover the usual
description.

The missing ingredient in the spin case when we compare it to
\cite{LLT} is that, since the spin symmetric groups are
not Coxeter groups, there is no standard way of associating
to them a Hecke algebra, and this is an important obstruction
for proving our conjecture.
What we can actually prove is that all self-associate
projective characters of ${\rm \widehat{S}}_m$ are linear combinations
of characters obtained from smaller groups by a sequence
of $(r,\overline r)$-inductions (Theorem 6.1).
This proof is constructive in the sense that the intermediate
basis $\{A(\mu)\}$ of our algorithm for the canonical basis,
suitably specialized at $q=1$, is a basis for the space spanned
by such characters.

This should have implications on the way of labelling the
irreducible modular spin representations of ${\rm \widehat{S}}_m$.
Up to now, a coherent labelling scheme has been found
only for $p=3$ \cite{BMO} and $p=5$ \cite{ABO}.
The case $p\ge 7$ led to formidable difficulties.
To overcome this problem, we propose to use the labels
of the crystal graph of $V(\Lambda_n)$, which may contain
partitions with repeated parts not arising in the
representation theory of ${\rm \widehat{S}}_m$, and corresponding to  ghost vectors
of the $q$-Fock space at $q=1$.

\section{The Fock space representation of $U_q(A^{(2)}_{2n})$}

The Fock space representation of the affine Lie algebra
$A^{(2)}_{2n}$ can be constructed by means of its
embedding in $b_\infty=\widehat{go}_\infty$, the completed infinite
rank affine Lie algebra of type $B$  \cite{DJKM1,DJKM2}.

The (bosonic) Fock space of type $B$ is
the polynomial algebra ${\cal F} = {\bf C}[p_{2j+1}, j\ge 0 ]$ in an infinite
number of  generators $p_{2j+1}$ of odd degree $2j+1$. If one identifies
$p_k$ with the power sum symmetric function $p_k=\sum_i x_i^k$
in some infinite set of variables, the natural basis of weight
vectors for $b_\infty$ is given by Schur's $P$-functions $P_\lambda$
(where $\lambda$ runs over the set ${\rm DP}$ of partitions
into distinct parts) \cite{DJKM1,You,JY}.

The Chevalley generators $e^\infty_i$, $f^\infty_i$ ($i\ge 0$)
of $b_\infty$ act on $P_\lambda$ by
\begin{equation}\label{FP}
e^\infty_i P_\lambda = P_\mu \ , \qquad f^\infty_i P_\lambda = P_\nu
\end{equation}
where $\mu$ (resp. $\nu$) is obtained from $\lambda$ by replacing its part $i+1$
by $i$ (resp. its part $i$ by $i+1$), the result being $0$
if $i+1$ (resp. $i$) is not a part of $\lambda$.
Also, it is
understood that $P_\mu=0$ as soon as $\mu$ has a multiple part.
For example, $f^\infty_0 P_{32}=P_{321}$,
$f^\infty_3 P_{32}=P_{42}$,
$e^\infty_1 P_{32}= P_{31}$ and $e^\infty_2 P_{32}=P_{22}=0$.

Let $h=2n+1$.
The Chevalley generators $e_i$, $f_i$ of $A^{(2)}_{2n}$
will be realized as
\begin{equation}\label{FNP}
f_i=\sum_{j\equiv n\pm i} f^\infty_j  \qquad
(i=0,\ldots ,n) \,,
\end{equation}
\begin{equation}
e_i=\sum_{j\equiv n\pm i } e^\infty_j \qquad
(i=0,\ldots, n-1)\,,\qquad
e_n=e^\infty_0 +
2 \sum_{\scriptstyle j>0 \atop \scriptstyle j \equiv 0,-1} e^\infty_j \,,
\end{equation}
where all congruences are taken modulo $h$.
Let $A_{2n}^{(2)}{}'$ be the derived algebra of $A_{2n}^{(2)}$
(obtained by omitting the degree operator $d$).
The action of $A_{2n}^{(2)}{}'$ on ${\cal F}$ is centralized
by the Heisenberg algebra generated by the operators
$\displaystyle{\partial\over\partial p_{hs}}$ and $p_{hs}$ for odd $s\ge 1$.
This implies that the Fock space decomposes under $A_{2n}^{(2)}$ as
\begin{equation}\label{DEC1}
{\cal F} = \bigoplus_{k\ge 0} V(\Lambda_n-k\delta)^{\oplus p^* (k)}
\end{equation}
where $p^*(k)$ is the number of partitions of $k$ into odd parts.
In particular, the subrepresentation generated by the vacuum vector
$|0\rangle=P_0 = 1$ is the basic
representation $V(\Lambda_n)$ of $A_{2n}^{(2)}$, and its principally
specialized character is \cite{KKLW}
\begin{equation}\label{CHAR}
{\rm ch}_t\,V(\Lambda_n) =
\sum_{m\ge 0}\dim V(\Lambda_n)_m\,t^m =
\prod_{\scriptstyle i \ {\rm odd} \atop \scriptstyle i\not\equiv 0{\ \rm mod\ } h}
{1\over 1-t^i}\,.
\end{equation}

The $q$-deformation of this situation has been discovered
by Kashiwara {\it et al.}  \cite{KMPY}.
Contrary to the case of
$A^{(1)}_{n-1}$, the $q$-Fock space is strictly larger than
the classical one.
We recall here briefly their construction, referring to
\cite{KMPY} for details and notation.

Let ${\rm DP}_h(m)$ be the set
of partitions $\lambda=(1^{m_1}2^{m_2}\ldots r^{m_r})$
of $m$ for which $m_i\le 1$ when $i\not\equiv 0 {\ \rm mod\ } h$.
For example, ${\rm DP}_3(7)=\{(7),(61),(52),(43),(421),(331)\}$.
Set ${\rm DP}_h=\bigcup_m {\rm DP}_h(m)$.
Then, the $q$-Fock space of type $A_{2n}^{(2)}$ is
\begin{equation}
{\cal F}_q = \bigoplus_{\lambda\in {\rm DP}_h} {\bf Q}(q)\, |\lambda\>
\end{equation}
where for $\lambda=(\lambda_1,\ldots,\lambda_r)$,
$|\lambda\>$ denotes the infinite $q$-wedge product
\[
|\lambda\> = u_\lambda = u_{\lambda_1}\wedge_q u_{\lambda_2}\wedge_q\cdots\wedge_q
u_{\lambda_r}\wedge_q u_0 \wedge_q u_0 \wedge_q \cdots
\]
of basis vectors $u_i$ of the representation $V_{\rm aff}$.
The quantum affine algebra $U_q(A_{2n}^{(2)})$ acts on
$V_{\rm aff}=\bigoplus_{i\in{\bf Z}}{\bf Q}(q) u_i$
by
\begin{eqnarray}
f_i u_j = \cases{ u_{j+1} & if $j\equiv n\pm i{\ \rm mod\ } h$ \\
                      0   & otherwise \\}
\qquad (i=0,\ldots,n-1)
\\
\label{ACTF}
f_n u_j = \cases{ u_{j+1} & if $j\equiv -1 {\ \rm mod\ } h$ \\
       (q+q^{-1}) u_{j+1} & if $j\equiv 0 {\ \rm mod\ } h$ \\
                        0 & otherwise \\}
\\
e_i u_j = \cases{ u_{j-1} & if $j\equiv n+1\pm i{\ \rm mod\ } h$ \\
                     0    & otherwise \\}
\qquad (i=0,\ldots,n-1)
\\
e_n u_j = \cases{ u_{j-1} & if $j\equiv 1 {\ \rm mod\ } h$ \\
       (q+q^{-1}) u_{j-1} & if $j\equiv 0 {\ \rm mod\ } h$ \\
                        0 & otherwise \\}
\\
t_0 u_j = \cases{ q^4 u_j & if $j\equiv n{\ \rm mod\ } h$ \\
              q^{-4}  u_j & if $j\equiv n+1 {\ \rm mod\ } h$ \\
                      u_j & otherwise \\}
\\
t_i u_j = \cases{ q^2 u_j & if $j\equiv n\pm i{\ \rm mod\ } h$ \\
              q^{-2}  u_j & if $j\equiv n+1\pm i {\ \rm mod\ } h$ \\
                      u_j & otherwise  \\ }
\qquad (i=1,\ldots,n-1)
\\
t_n u_j = \cases{ q^2 u_j & if $j\equiv -1{\ \rm mod\ } h$ \\
              q^{-2}  u_j & if $j\equiv 1 {\ \rm mod\ } h$ \\
                      u_j & otherwise  \\}
\end{eqnarray}
The only commutation rules we will need to describe the
action of $e_i$ and $f_i$ on ${\cal F}_q$ are:
\begin{eqnarray}
u_j \wedge_q u_j &=& 0 \ {\rm if}\ j\not\equiv 0 {\ \rm mod\ } h \\
u_j \wedge_q u_{j+1} &=& -q^2 u_{j+1}\wedge_q u_j \ {\rm if} \label{STR2}
 j\equiv 0,-1 {\ \rm mod\ } h \ .
\end{eqnarray}
The action on the vacuum vector
$|0\> = u_0\wedge_qu_0\wedge_q\cdots $
is given by
\begin{equation}
e_i|0\> = 0, \qquad
f_i|0\> = \delta_{i n}|1\>, \qquad
t_i|0\> = q^{\delta_{i n}}|0\>,
\end{equation}
and on a $q$-wedge
$|\lambda\>=u_{\lambda_1}\wedge_q\cdots\wedge_q u_{\lambda_r}
\wedge_q |0\>$,
\begin{eqnarray}
f_i |\lambda\>
=&
f_iu_{\lambda_1}\wedge_q t_iu_{\lambda_2}\wedge_q\cdots t_iu_{\lambda_r}
\wedge_q t_i|0\> \nonumber \\
& +
u_{\lambda_1}\wedge_q f_iu_{\lambda_2}\wedge_q\cdots t_iu_{\lambda_r}
\wedge_q t_i|0\>  \nonumber \\
& + \cdots +
u_{\lambda_1}\wedge_q u_{\lambda_2}\wedge_q\cdots u_{\lambda_r}
\wedge_q f_i|0\>
\end{eqnarray}
\begin{eqnarray}
e_i |\lambda\>
=&
t_i^{-1} u_{\lambda_1}\wedge_q t_i^{-1}u_{\lambda_2}\wedge_q\cdots t_i^{-1}u_{\lambda_r}
\wedge_q e_i|0\> \nonumber \\
& +
t_i^{-1}u_{\lambda_1}\wedge_q t_i^{-1}u_{\lambda_2}\wedge_q\cdots e_iu_{\lambda_r}
\wedge_q |0\>  \nonumber \\
& + \cdots +
e_i u_{\lambda_1}\wedge_q u_{\lambda_2}\wedge_q\cdots u_{\lambda_r}
\wedge_q |0\>
\end{eqnarray}
\begin{equation}
t_i |\lambda\> =
t_i u_{\lambda_1}\wedge_q t_i u_{\lambda_2}\wedge_q\cdots\wedge_q
t_i u_{\lambda_r}\wedge_q t_i|0\> \ .
\end{equation}
For example, with $n=2$, one has
\[
f_2 |542\>= (q^4+q^2)|642\>+q|552\>+|5421\>,
\]
and
\[
f_2 |552\> = (q^2+1)(|652\>+|562\>)+|5521\>
= (1-q^4)|652\>+|5521\>,
\]
the last equality resulting from (\ref{STR2}).

It is proved in \cite{KMPY} that ${\cal F}_q$ is an integrable
highest weight $U_q(A_{2n}^{(2)})$-module whose decomposition
into irreducible components, obtained by means of $q$-bosons, is
\begin{equation}\label{DEC2}
{\cal F}_q = \bigoplus_{k\ge 0} V(\Lambda_n-k\delta)^{\oplus p(k)}
\end{equation}
where $p(k)$ is now the number of all partitions of $k$
(compare (\ref{DEC1})).
Thus, the submodule $U_q(A_{2n}^{(2)}) \,|0\>$ is a realization
of the basic representation $V(\Lambda_n)$.

\section{The crystal graph of the $q$-Fock space}

The first step in computing the global basis
of $V(\Lambda_n) \subset {\cal F}_q$ is to determine
the crystal basis of ${\cal F}_q$ whose description
follows from \cite{KMPY,KMN1,KMN2}.
Let $A$ denote the subring of ${\bf Q}(q)$ consisting of
rational functions without pole at $q=0$.
The crystal lattice of ${\cal F}_q$ is
$L = \bigoplus_{\lambda\in {\rm DP}_h} A\,|\lambda\>$,
and the crystal basis of the ${\bf Q}$-vector space $L/qL$ is
$B=\{|\lambda\> {\ \rm mod\ } qL, \lambda \in {\rm DP}_h\}$.
We shall write $\lambda$ instead of $|\lambda\> {\ \rm mod\ } qL$.

The Kashiwara operators $\tilde{f}_i$ act on $B$ in
a simple way recorded on the crystal graph $\Gamma({\cal F}_q)$.
To describe this graph, one starts with the crystal graph
$\Gamma(V_{\rm aff})$ of $V_{\rm aff}$. This is the graph with vertices
$j\in {\bf Z}$, whose arrows labelled by $i\in \{0,1,\ldots ,n\}$
are given, for $i \not = n$, by
\[
j \stackrel{i}{\longrightarrow} j+1 \quad \Longleftrightarrow \quad
j \equiv n \pm i {\ \rm mod\ } h \,,
\]
and for $i=n$ by
\[
j \stackrel{n}{\longrightarrow} j+1 \quad \Longleftrightarrow \quad
j \equiv -1,0 {\ \rm mod\ } h \,.
\]
Thus for $n=2$ this graph is
\[
\cdots \stackrel{1}{\longrightarrow} -1
\stackrel{2}{\longrightarrow} 0
\stackrel{2}{\longrightarrow} 1
\stackrel{1}{\longrightarrow} 2
\stackrel{0}{\longrightarrow} 3
\stackrel{1}{\longrightarrow} 4
\stackrel{2}{\longrightarrow} 5
\stackrel{2}{\longrightarrow} 6
\stackrel{1}{\longrightarrow} 7
\stackrel{0}{\longrightarrow}
\cdots
\]
The graph $\Gamma({\cal F}_q)$ is obtained inductively
from $\Gamma(V_{\rm aff})$ using the following rules.
Let $\lambda = (\lambda_1,\ldots ,\lambda_r)\in B$,
and write $\lambda = (\lambda_1,\lambda^*)$
where $\lambda^* = (\lambda_2,\ldots ,\lambda_r)$.
Then one has $\tilde{f}_i (0) = \delta_{in} (1)$,
$\varphi_i(0)= \delta_{in}$, and
\[
\tilde{f}_i\lambda = \left\{
\matrix{(\tilde{f}_i \lambda_1, \lambda^*)
\ {\rm if} \ \varepsilon_i(\lambda_1) \ge \varphi_i(\lambda^*), \cr
(\lambda_1,\tilde{f}_i\lambda^*)
\ {\rm if} \ \varepsilon_i(\lambda_1) < \varphi_i(\lambda^*). } \right.
\]
Here, $\varepsilon_i(\lambda_1)$ means the distance in $\Gamma(V_{\rm aff})$
from $\lambda_1$ to the origin of its
$i$-string, and $\varphi_i(\lambda^*)$ means the distance in
$\Gamma({\cal F}_q)$ from $\lambda^*$ to the end of its $i$-string.

Thus for $n=1$ one computes successively the following $1$-strings
of $\Gamma({\cal F}_q)$
\[
(0) \stackrel{1}{\longrightarrow} (1)
\]
\[
(2)=(2,0)\stackrel{1}{\longrightarrow}(2,1)
\stackrel{1}{\longrightarrow} (3,1)
\stackrel{1}{\longrightarrow} (4,1)
\]
\[
(3,2)=(3,2,0)\stackrel{1}{\longrightarrow}(3,2,1)
\stackrel{1}{\longrightarrow} (3,3,1)
\stackrel{1}{\longrightarrow} (4,3,1)
\]
from which one deduces that $\tilde{f_1}(3,3,1) = (4,3,1)$
and $\varphi_1(3,3,1)=1$.

The first layers of the crystal $\Gamma({\cal F}_q)$ for $n=1$
are shown in Fig.~\ref{FIG1}.
\begin{figure}[t]
\begin{center}
\leavevmode
\epsfxsize = 15cm
\epsffile{crystalFock.eps}
\end{center}
\caption{\label{FIG1} The graph $\Gamma({\cal F}_q)$ for $A_2^{(2)}$ up to degree $7$}
\end{figure}
One can observe that the decomposition of $\Gamma({\cal F}_q)$ into connected
components reflects the decomposition (\ref{DEC2})
of ${\cal F}_q$ into simple modules.
More precisely, the connected components of
$\Gamma({\cal F}_q)$ are all isomorphic as colored graphs
to the component $\Gamma(\Lambda_n)$ containing the
empty partition.
Their highest vertices are the partitions $\nu$ whose
parts are all divisible by $h$.
This follows from the fact, easily deduced from the
rules we have just explained, that if
$\nu = h\mu = (h\mu_1,\ldots ,h\mu_r)$ is such a partition,
then the map
\begin{equation}\label{MAP}
\lambda \mapsto \lambda + \nu = (\lambda_1+h\mu_1,\lambda_2+h\mu_2,\ldots \ )
\end{equation}
is a bijection from $\Gamma(\Lambda_n)$ onto the connected component
of $\Gamma({\cal F}_q)$ containing $\nu$, and this bijection commutes with
the operators $\tilde{e}_i$ and $\tilde{f}_i$.
This implies that the vertices of $\Gamma(\Lambda_n)$
are the partitions $\lambda=(\lambda_1,\ldots ,\lambda_r,0)\in {\rm DP}_h$ such that
for $i=1,2,\ldots ,r$, one has $\lambda_i- \lambda_{i+1} \le h$ and
$\lambda_i- \lambda_{i+1} < h$ if $\lambda_i \equiv 0 {\ \rm mod\ } h$.
We shall call a partition that satisfies these conditions
$h$-regular.
The set of $h$-regular partitions of $m$ will be denoted by ${\rm DPR}_h(m)$,
and we shall write ${\rm DPR}_h=\bigcup_m {\rm DPR}_h(m)$.

For example,
\[
{\rm DPR}_3(10) = \{ (3331), (4321), (532), (541) \} \,.
\]

\section{The canonical basis of $V(\Lambda_n)$}\label{SECT4}

In this section, we describe an algorithm for computing
the canonical basis (global lower crystal basis) of the
basic representation $V(\Lambda_n)=U_q(A_{2n}^{(2)}) |0\>$
in terms of the natural basis $|\lambda\>$ of the
$q$-Fock space. To characterize the canonical basis, we
need the following notations
\begin{equation}
q_i =
\cases{q & if $i=n$ \\
      q^2& if $1\le i<n$ \\
      q^4& if $i=0$ \\ }
\qquad
t_i =
\cases{q^{h_n}& if $i=n$\\
      q^{2h_i}& if $1\le i<n$ \\
      q^{4h_0}& if $i=0$ \\}
\end{equation}
and
\begin{equation}
[k]_i = {q_i^k-q_i^{-k}\over q_i-q_i^{-1}}\ ,
\qquad
[k]_i! = [k]_i [k-1]_i \cdots [1]_i \ .
\end{equation}
The $q$-divided powers of the Chevalley generators are defined by
\begin{equation}
e_i^{(k)} = {e_i^k\over [k]_i!}\ ,\qquad
f_i^{(k)} = {f_i^k\over [k]_i!}\ .
\end{equation}
The canonical basis is defined in terms of an involution
$v\mapsto\overline{v}$ of $V(\Lambda_n)$.
Let $x\mapsto \overline{x}$ be the ring automorphism of
$U_q(A_{2n}^{(2)})$ such that $\overline{q}=q^{-1}$,
$\overline{q^h}=q^{-h}$ for $h$ in the Cartan subalgebra
of $A_{2n}^{(2)}$, and $\overline{e_i}=e_i$,
$\overline{f_i}=f_i$. Then, for $v=x|0\>\in V(\Lambda_n)$,
define $\overline{v}=\overline{x}|0\>$.

We denote by $U_{\bf Q}^-$ the sub-${\bf Q}[q,q^{-1}]$-algebra
of $U_q(A_{2n}^{(2)})$ generated by the $f_i^{(k)}$
and set $V_{\bf Q}(\Lambda_n)=U_{\bf Q}^-|0\>$.
Then, as shown by Kashiwara \cite{K}, there exists a unique
${\bf Q}[q,q^{-1}]$-basis $\{G(\mu), \mu\in {\rm DPR}_h\}$
of $V_{\bf Q}(\Lambda_n)$, such that
\begin{quote}
(G1) $G(\mu) \equiv |\mu\> {\ \rm mod\ } qL$

(G2) $\overline{G(\mu)}= G(\mu)$.
\end{quote}

To compute $G(\mu)$, we follow the same strategy as in
\cite{LLT}. We first introduce an auxiliary basis
$A(\mu)$ satisfying (G2), from which we  manage to construct
combinations satisfying also (G1). More precisely,
let ${\cal F}_q^m$ be the subspace of ${\cal F}_q$ spanned
by $|\lambda\>$ for $\lambda \in {\rm DP}_h(m)$ and set
$V(\Lambda_n)_m={\cal F}_q^m\cap V(\Lambda_n)$. Denote
by $\unlhd$ the natural order on partitions.
Then, the auxiliary
basis will satisfy
\begin{quote}
(A0) $\{A(\mu),\mu\in {\rm DPR}_h(m)\}$ is a ${\bf Q}[q,q^{-1}]$-basis
of $V_{\bf Q}(\Lambda_n)_m$,

(A1) $A(\mu)=\sum_\lambda a_{\lambda\mu}(q)|\lambda\>$,
where $a_{\lambda\mu}(q)=0$ unless $\lambda\unrhd\mu$,
$a_{\mu\mu}(q)=1$ and $a_{\lambda\mu}(q)\in{\bf Z}[q,q^{-1}]$,

(A2) $\overline{A(\mu)}=A(\mu)$.
\end{quote}
The basis $A(\mu)$ is obtained
by applying monomials in the $f_i^{(k)}$ to the highest weight vector,
that is, $A(\mu)$ is of the form
\begin{equation}\label{defA}
A(\mu) = f_{r_s}^{(k_s)}f_{r_{s-1}}^{(k_{s-1})}\cdots f_{r_1}^{(k_1)}|0\>
\end{equation}
so that (A2) is satisfied.

The two sequences $(r_1,\ldots,r_s)$ and $(k_1,\ldots,k_s)$ are, as
in \cite{LLT}, obtained by peeling off the $A_{2n}^{(2)}$-ladders
of the partition $\mu$, which are defined as follows. We first fill
the cells of the Young diagram $Y$ of $\mu$ with integers
(called residues), constant in
each column of $Y$. If $j\equiv n\pm i {\ \rm mod\ } h$
($0\le i\le n$), the numbers filling
the $j$-th column of $Y$ will be equal to $i$. A ladder of $\mu$
is then a sequence of cells with the same residue, located
in consecutive rows at horizontal distance $h$, except when the residue
is $n$, in which case two consecutive $n$-cells  in a row belong also
to the same ladder. For example, with $n=3$ and $\mu=(11,7,7,4)$,
one finds $22$ ladders (indicated by subscripts), the longest
one being the 7th, containing three 3-cells:
\[
\young{
3_{19} & 2_{20} & 1_{21} & 0_{22} \cr
3_{13} & 2_{14} & 1_{15} & 0_{16} & 1_{17} & 2_{18} & 3_{19} \cr
3_{7}&2_8&1_9&0_{10}&1_{11}&2_{12}&3_{13}\cr
3_1&2_2&1_3&0_4&1_5&2_6&3_7&3_7&2_8&1_9&0_{10}\cr}
\]
Note that this definition of ladders agrees with that of \cite{BMO}
for $n=1$, but differs from that of \cite{ABO} for $n=2$.

Then, in (\ref{defA}), $s$ is the number of ladders,
$r_i$ the residue of the $i$th ladder, and $k_i$ the number of its
cells. Thus, proceeding with our example,
\[
\fl
A(11,7,7,4)=
f_0f_1f_2f_3^{(2)}f_2f_1f_0f_1f_2f_3^{(2)}f_2f_1f_0^{(2)}
f_1^{(2)}f_2^{(2)}f_3^{(3)} f_2f_1f_0f_1f_2f_3 |0\> \ .
\]
The proof of (A0) and (A1) can be readily adapted from
\cite{LLT}. In particular, (A1) follows from the fact that
a partition $\lambda$ belongs to ${\rm DPR}_h$ if and only if all cells of a given
ladder intersecting $\lambda$ occupy the highest
possible positions on this ladder.

Another choice of an intermediate basis,
more efficient for practical computations,
would be to use inductively the vectors $G(\nu)$ already computed
and to set $A(\mu)=f_{r_s}^{(k_s)} G(\nu)$, where
$\nu$ is the partition obtained from $\mu$ by removing
its outer ladder.

Define now the coefficients $b_{\nu\mu}(q)$ by
\begin{equation}
G(\mu) =\sum_\nu b_{\nu\mu}(q) A(\nu) \ .
\end{equation}
Still following \cite{LLT}, one can check that $b_{\nu\mu}(q)=0$
unless $\nu\ge \mu$, where $\ge$ denote the   lexicographic
ordering on partitions, and that $b_{\mu\mu}(q)=1$. Therefore,
one can apply the triangular process of \cite{LLT} as follows.

Let $\mu^{(1)} < \mu^{(2)} <\ldots < \mu^{(t)}$ be the set
${\rm DPR}_h(m)$ sorted in  lexicographic order, so that
$A(\mu^{(t)})=G(\mu^{(t)})$.  Suppose that the expansion
on the basis $|\lambda\>$ of $G(\mu^{(i+1)}),\ldots, G(\mu^{(t)})$
has already been calculated. Then,
\begin{equation}
G(\mu^{(i)})=
A(\mu^{(i)})-\gamma_{i+1} (q) G(\mu^{(i+1)}) - \cdots - \gamma_t(q)G(\mu^{(t)}) \ ,
\end{equation}
where the coefficients are  determined by the conditions
\[
\gamma_s(q^{-1})=\gamma_s(q), \qquad G(\mu^{(i)}) \equiv |\mu^{(i)}\> {\ \rm mod\ } qL.
\]

Thus, for $n=1$, the first partition for which $A(\mu)\not = G(\mu)$
is $\mu=(3321)$ and
\begin{eqnarray}
\fl A(3321) =
|3321\> + q|333\> + (q^2-q^6)|432\> + (1+2q^2)|531\> + (q^2+q^4)|54\>
\nonumber \\
 \lo+ (2q^2+q^4)|621\> +2q^3 |63\> + (q^4+q^6)|72\> +q^4|81\> +q^5|9\>
\end{eqnarray}
Indeed, $A(3321)\equiv |3321\>+|531\> {\ \rm mod\ } qL$.
On the other hand, $ A(531)=|531\>+q^2|54\>+q^2|621\>+q^3|63\>+q^6|72\> $
is equal to $G(531)$,
and one finds by subtracting this from $A(3321)$ that
\begin{eqnarray}
\fl G(3321) =  |3321\> + q|333\> + (q^2-q^6)|432\> + 2q^2|531\> + q^4|54\> \\
\lo+ (q^2+q^4)|621\> +q^3 |63\> + q^4|72\> +q^4|81\> +q^5|9\> \ .
\end{eqnarray}
Since $A(432)=|432\>+q^4|531\>+q^2|72\>+q^6|81\>$ satisfies
(G1) and (G2), it has to be equal to
$G(432)$, which completes the determination of the canonical
basis for $m=9$. For $m=10$, the results are displayed as the
columns of Table \ref{TAB1}.

\begin{table}
\caption{\label{TAB1}
The canonical basis for $n=1$ and $m=10$.
}
\begin{indented}
\item[]\begin{tabular}{@{}llllllll}
\br
&$(3 3 3 1)$&$(4 3 2 1)$&$(5 3 2)$&$(5 4 1)$\\
\mr
$(3 3 3 1)$&1&0&0&0\\
$(4 3 2 1)$&$q-q^{5}$&1&0&0\\
$(4 3 3)$&$q^{2}$&$q$&0&0\\
$(5 3 2)$ &0&0&1&0\\
$(5 4 1)$&$q+q^{3}$&$q^{2}+q^{4}$&0&1\\
$(6 3 1)$&$2\ q^{2}$&$q^{3}$&0&$q$ \\
$(6 4)$&$q^{4}$&0&0&$q^{3}$\\
$(7 2 1)$&$q^{3}+q^{5}$&$q^{2}$&0&$q^{4}$\\
$(7 3)$&$q^{4 }$&$q^{3}$&0&$q^{5}$\\
$(8 2)$ & 0 & 0 & $q^2$ & 0 \\
$(9 1)$&$q^{4}$&$q^{5}$&0&0\\
$(10)$&$q^{6}$&0&0&0\\
\br
\end{tabular}
\end{indented}
\end{table}

In the Fock space representation of $A_{n-1}^{(1)}$,
the weight of a basis vector $|\lambda\>$ is determined by
the $n$-core of the partition $\lambda$ (and its degree) \cite{ANY,LLT}.
There is a similar result of Nakajima and Yamada \cite{NY}
for $A_{2n}^{(2)}$, in terms of the notion of $\overline{h}$-core
of a strict partition
introduced by Morris \cite{Mo1} in the context of the modular
representation theory of spin symmetric groups.

One way to see this is to use a theorem of \cite{MY1}
according to which $\lambda, \mu \in {\rm DP}(m)$
have the same $\overline{h}$-core if and only if
they have, for each $i$, the same number $n_i$ of nodes of residue $i$.
On the other hand, it follows from the
implementation of the Chevalley generators
that $|\lambda\>$ has $A_{2n}^{(2)}$-weight
$\Lambda_n - \sum_{0\le i \le n} n_i \alpha_i$,
and the statement follows.

The definition of $\overline{h}$-cores can be extended to ${\rm DP}_h$
by deciding that if $\lambda$ has repeated parts, its $\overline{h}$-core
is equal to that of the partition obtained by removing those repeated parts.
Then it is clear that if $|\lambda\>$ and $|\mu\>$ have the same
$U_q(A_{2n}^{(2)})$-weight, the two partitions $\lambda$ and $\mu$
have the same $\overline{h}$-core.
It follows, since $G(\mu)$ is obviously a weight vector, that its
expansion on the basis $|\lambda\>$ involves only partitions
$\lambda$ with the same $\overline{h}$-core as $\mu$.

Summarizing the discussion, we have:

\begin{theorem}\label{TH}
For $\mu \in {\rm DPR}_h(m)$, define $d_{\lambda\mu}(q)$ by
$\displaystyle G(\mu)=\sum_{\lambda \in {\rm DP}_h(m)} d_{\lambda\mu}(q)|\lambda\>$.
Then,

{\rm (i)} $d_{\lambda\mu}(q)\in {\bf Z}[q]$,

{\rm (ii)} $d_{\lambda\mu}(q)=0$ unless $\lambda\unrhd\mu$,
and $d_{\mu\mu}(q) = 1$,

{\rm (iii)} $d_{\lambda\mu}(q)=0$ unless $\lambda$ and $\mu$
have the same $\overline{h}$-core.
\end{theorem}

\section{The reduction $q=1$}
As observed by Kashiwara {\it et al.} \cite{KMPY}, to recover the
classical Fock space representation ${\cal F}$ of $A_{2n}^{(2)}$, one has
to introduce the inner product on ${\cal F}_q$ for
which the vectors $|\lambda\>$ are orthogonal and the adjoint
operators of the Chevalley generators are
\begin{equation} \label{ADJOINT}
f_i^{\dag} = q_i e_i t_i, \qquad
e_i^{\dag} = q_i f_i t_i^{-1}, \qquad
t_i^{\dag} = t_i.
\end{equation}
It can be checked that, for $\lambda \in {\rm DP}_h$,
\begin{equation} \label{NORM}
\<\lambda|\lambda\> = \prod_{k>0}\prod_{i=1}^{m_{kh}} (1-(-q^2)^i),
\end{equation}
where $m_{kh}$ is the multiplicity of the part $kh$ in $\lambda$.

Let ${\cal F}_1$ denote the $A_{2n}^{(2)}$-module obtained by specializing
$q$ to 1 as in \cite{KMPY}. This space is strictly larger than the classical Fock space
${\cal F}$, since the dimension of its $m$th homogeneous component
(in the principal gradation) is $|{\rm DP}_h(m)|$ whereas that of ${\cal F}$
is only $|{\rm DP}(m)|$.
Let ${\cal N} = {\cal F}_1^\perp$ denote the nullspace. It follows
from (\ref{ADJOINT}) that ${\cal N}$ is a $A_{2n}^{(2)}$-module, and
from (\ref{NORM}) that ${\cal N}$ is the subspace of ${\cal F}_1$
spanned by the wedge products $|\lambda\>$ labelled by $\lambda \in {\rm DP}_h - {\rm DP}$.
Therefore ${\cal F}_1/{\cal N}$ is a $A_{2n}^{(2)}$-module that can
be identified with ${\cal F}$.

In this identification one has, for
$\lambda=(\lambda_1,\ldots,\lambda_r) \in {\rm DP}$,
\begin{equation}
P_\lambda = 2^{\sum_{i=1}^r\lfloor (\lambda_i-1)/h \rfloor} |\lambda \>.
\end{equation}
The power of $2$ comes from the fact that if $\lambda_i = kh$ for $k>0$,
and $\nu$ denotes the partition obtained from $\lambda$ by replacing $\lambda_i$
by $\nu_i = \lambda_i+1$, then it follows from (\ref{FP}), (\ref{FNP})
that $f_n P_\lambda$ contains $P_\nu$ with
coefficient 1, while
$f_n |\lambda\>$ contains $|\nu\>$ with coefficient 2 by (\ref{ACTF}).
For later use we set
\begin{equation}\label{AHN}
a_h(\lambda) = \sum_{i=1}^r\left\lfloor {\lambda_i-1\over h} \right\rfloor \,.
\end{equation}

\section{Modular representations of ${\rm \widehat{S}}_m$}

We refer the reader to \cite{B} for an up-to-date review of the
representation theory of the spin symmetric groups
and their combinatorics.

Let ${\rm \widehat{S}}_m$ be the spin symmetric group as defined by Schur \cite{S},
that is,
the group of order $2\,m!$ with generators $z,s_1,\ldots,s_{m-1}$
and relations $z^2=1$, $zs_i=s_iz$, $s_i^2 = z$, $(1\le i\le m-1)$,
$s_is_j=zs_js_i$ ($|i-j|\ge 2$) and $(s_is_{i+1})^3=z$
($1\le i\le m-2)$.

On an irreducible representation of ${\rm \widehat{S}}_m$, the central element $z$ has to act
by $+1$ or by $-1$. The representations for which $z=1$ are
actually linear representations of the symmetric group ${\rm S}_m$,
and those with $z=-1$, called spin representations
correspond to two-valued representations of ${\rm S}_m$. The irreducible spin
representations over a field of characteristic $0$
are labelled, up to association, by strict partitions
$\lambda\in {\rm DP}(m)$. More precisely, let ${\rm DP}_+(m)$ (resp. ${\rm DP}_-(m)$)
be the set of strict partitions of $m$ having an even (resp. odd)
number of even parts. Then, to each $\lambda\in{\rm DP}_+(m)$ corresponds
a self-associate irreducible spin character $\pr{\lambda}$, and to each
$\lambda\in{\rm DP}_-(m)$ a pair of associate irreducible spin characters denoted
by $\pr{\lambda}$ and $\pr{\lambda}'$.

According to Schur \cite{S}, the values $\pr{\lambda}(\rho)$
of the spin character $\pr{\lambda}$ on conjugacy classes of cycle-type
$\rho=(1^{m_1},3^{m_3},\ldots )$ are given by the expansion
of the symmetric function $P_\lambda$ on the basis of power sums,
namely
\begin{equation}
P_\lambda = \sum_\rho 2^{\lceil (\ell(\rho)-\ell(\lambda))/2\rceil}
            \pr{\lambda}(\rho) {p_\rho\over z_\rho}
\end{equation}
where $z_\rho=\prod_j j^{m_j} m_j!$ and $\ell(\lambda)$ stands for the length
of $\lambda$, that is the number of parts of $\lambda$.

For $\lambda\in{\rm DP}(m)$, one introduces the self-associate spin character
\begin{equation}
\prh{\lambda} = \cases{\pr{\lambda} & if  $\lambda\in{\rm DP}_+(m)$,\\
                     \pr{\lambda}+\pr{\lambda}'& if  $\lambda\in{\rm DP}_-(m)$.\\}
\end{equation}
The branching theorem for spin characters of Morris \cite{Mo1} implies that
if $\prh{\lambda}$ gets identified with a weight vector of ${\cal F}$
by setting
\begin{equation}\label{IDENT}
P_\lambda = 2^{\lfloor (m - \ell(\lambda))/2\rfloor} \, \prh{\lambda} ,
\end{equation}
then the $b_\infty$-operator $f = \sum_{i\ge 0} f^{\infty}_i$ implements
the induction of self-associate spin characters
from ${\rm \widehat{S}}_m$ to ${\rm \widehat{S}}_{m+1}$.
Similarly, $e= e^{\infty}_0 + 2\sum_{i>0} e^{\infty}_i$ implements
the restriction from ${\rm \widehat{S}}_m$ to ${\rm \widehat{S}}_{m-1}$.
Thus, the Fock space representation of $b_\infty$ may be viewed
as the sum
${\cal F} = \bigoplus_m {\cal C}(m)$
of additive groups generated by self-associate spin characters
of ${\rm \widehat{S}}_m$ in characteristic 0.
In this setting, the Chevalley generators of $b_\infty$ act as
refined induction and restriction operators.

Now, similarly to the case $A_{n-1}^{(1)}$, the reduction from
$b_\infty$ to $A_{2n}^{(2)}$ parallels the reduction modulo $p=h=2n+1$
of representations of ${\rm \widehat{S}}_m$ (from now on we assume that $h$
is an odd prime).

More precisely, using (\ref{FP}) (\ref{FNP}) (\ref{IDENT}),
one sees immediately that the Chevalley generators $f_i$ of
$A_{2n}^{(2)}$ act on $\prh{\lambda}$ as
the $(r,\overline{r})$-induction operators of Morris and
Yaseen $(r=n+1-i)$ \cite{MY}.
Hence the vectors of degree $m$ of
$V(\Lambda_n) = U(A_{2n}^{(2)})^-\,|0\>$ can be identified
with linear combinations of self-associate spin characters
obtained by a sequence of $(r,\overline{r})$-inductions.
It is known from modular representation theory that the maximal
number of linearly independent self-associate projective spin characters
of ${\rm \widehat{S}}_m$ in characteristic $p$ is equal to the number of partitions
of $m$ into odd summands prime to $p$.
Therefore the following result follows at once from (\ref{CHAR}).
\begin{theorem}
The  self-associate projective spin characters
of ${\rm \widehat{S}}_m$ in characteristic $p$ are linear combinations
of characters obtained by a sequence of $(r,\overline{r})$-inductions.
\end{theorem}
This was proved by Bessenrodt {\it et al.} for $p=3$ \cite{BMO}
and Andrews {\it et al.} for $p=5$ \cite{ABO}, but the question
remained open for $p\ge 7$ \cite{B}.

Moreover, the construction of
Section~\ref{SECT4} gives an explicit basis for the space spanned
by such characters.
Denote by $\underline{A}(\mu)$ the column vector obtained
from $A(\mu)$ by reduction $q=1$ and expansion on the basis
$\prh{\lambda}$.
Then, $\underline{A}(\mu)$ is a projective character by
(\ref{defA}) and
$\{\underline{A}(\mu)\ | \ \mu \in {\rm DPR}_p(m) \}$ is a
basis of the ${\bf Q}$-vector space of self-associate projective spin characters
of ${\rm \widehat{S}}_m$ in characteristic $p$.

These observations and the results of \cite{LLT,Ar,Gr,LT} lead us to
formulate a conjecture relating the global basis of $V(\Lambda_n)$
and the decomposition matrices for spin characters of the groups ${\rm \widehat{S}}_m$.

Let $\mu\in {\rm DPR}_p(m)$ and let $\underline{G}(\mu)$ stand for the image of the
global basis $G(\mu)$ in ${\cal F}={\cal F}_1/{\cal N}$, that is,
\begin{equation}
\underline{G}(\mu) = \sum_{\lambda \in {\rm DP}(m)}
2^{b(\lambda) - a_p(\lambda)}
d_{\lambda\mu}(1) \prh{\lambda} \,,
\end{equation}
where $a_p(\lambda)$ is given by (\ref{AHN}) and
\begin{equation}
b(\lambda)= \left\lfloor {m - \ell(\lambda)\over 2}\right\rfloor\,.
\end{equation}
Then denote by $\underline{\underline{G}}(\mu)$ the vector obtained
by factoring out the largest power of $2$ dividing the coefficients
of $\underline{G}(\mu)$ on the basis $\prh{\lambda}$.
For simplicity of notation, we shall identify $\underline{\underline{G}}(\mu)$
with the column vector of its coordinates on $\prh{\lambda}$.

Finally, let us call reduced decomposition matrix of ${\rm \widehat{S}}_m$ in characteristic
$p$ the matrix obtained from the usual decomposition matrix for spin characters
by adding up pairs of associate columns and expanding the column vectors
so obtained on the basis $\prh{\lambda}$.
This is a matrix with $|{\rm DP}(m)|$ rows and $|{\rm DPR}_p(m)|$ columns.
The definition is illustrated in Table~\ref{TAB2} and Table~\ref{TAB3}.
(Table~\ref{TAB2} is taken from \cite{MY}, except for the column labels
which are ours and will be explained in the next section.)
\begin{table}
\caption{\label{TAB2}
The decomposition matrix of ${\rm \widehat{S}}_{10}$ in characteristic 3.}
\begin{indented}
\item[]\begin{tabular}{@{}llllllll}
\br
             &(3331)&(3331)'&(4321)&(4321)'&(532)&(541)&(541)'\\
\mr
$\pr{4321}$    &0&0&1&1&0&0&0\\
$\pr{532}$     &0&0&0&0&1&0&0\\
$\pr{532}'$    &0&0&0&0&1&0&0\\
$\pr{541}$     &1&1&1&1&0&0&1\\
$\pr{541}'$    &1&1&1&1&0&1&0\\
$\pr{631}$     &2&2&1&1&0&1&1\\
$\pr{631}'$    &2&2&1&1&0&1&1\\
$\pr{64}$      &1&1&0&0&0&1&1\\
$\pr{721}$     &1&1&0&1&0&0&1\\
$\pr{721}'$    &1&1&1&0&0&1&0\\
$\pr{73}$      &1&1&1&1&0&1&1\\
$\pr{82}$      &0&0&0&0&1&0&0\\
$\pr{91}$      &1&1&1&1&0&0&0\\
$\pr{10}$      &0&1&0&0&0&0&0\\
$\pr{10}'$     &1&0&0&0&0&0&0\\
\br
\end{tabular}
\end{indented}
\end{table}

\begin{table}
\caption{\label{TAB3}
The reduced decomposition matrix of ${\rm \widehat{S}}_{10}$ in characteristic 3.}
\begin{indented}
\item[]\begin{tabular}{@{}lllll}
\br
             &(3331)&(4321)&(532)&(541)\\
\mr
$\prh{4 3 2 1}$& 0 &2&0&0\\
$\prh{5 3 2}$ &0&0&1&0\\
$\prh{5 4 1}$&2&2&0&1\\
$\prh{6 3 1}$&4&2&0&2 \\
$\prh{6 4}$&2 &0&0&2\\
$\prh{7 2 1}$&2  &1 &0&1\\
$\prh{7 3}$&2 & 2&0& 2 \\
$\prh{8 2}$& 0&0&1&0\\
$\prh{9 1}$&2&2&0&0\\
$\prh{10}$&1 &0&0&0 \\
\br
\end{tabular}
\end{indented}
\end{table}

\begin{conjecture}
(i) The set of column vectors of the reduced decomposition matrix of ${\rm \widehat{S}}_m$
in odd characteristic $p$ such that $p^2 > m$
coincides with $\{\underline{\underline{G}}(\mu) \ | \ \mu \in {\rm DPR}_p(m)\}$.

(ii) For $p^2\le m$,
the reduced decomposition matrix of ${\rm \widehat{S}}_m$
is obtained by postmultiplying the matrix whose columns are
$\underline{\underline{G}}(\mu)$ by a unitriangular matrix with
nonnegative entries.
\end{conjecture}

Our conjecture has been checked on  the numerical tables
computed by Morris and Yaseen ($p=3$) \cite{MY} and
Yaseen ($p=5,7,11$) \cite{Ya}.
Thus, for $p=3$, $m=11$, the columns of the reduced decomposition matrix
are
\[
\underline{\underline{G}}(3332),\
\underline{\underline{G}}(4331)+\underline{\underline{G}}(641),\
\underline{\underline{G}}(5321),\
\underline{\underline{G}}(542),\
\underline{\underline{G}}(641).
\]

\section{Labels for irreducible modular spin characters
and partition identities}

The labels for irreducible modular representations of symmetric
groups form a subset of the ordinary labels
\cite{JK}. It is therefore natural to look for a labelling scheme
for irreducible modular spin representations of
${\rm \widehat{S}}_m$  using a subset of ${\rm DP}(m)$. This was accomplished for
$p=3$ by Bessenrodt {\it et al.} \cite{BMO}, who found
that the Schur regular partitions of $m$ form a convenient
system of labels. These are the partitions
$\lambda=(\lambda_1,\ldots,\lambda_r)$  such that
$\lambda_i-\lambda_{i+1}\ge 3$ for $i=1,\ldots,r-1$, and
$\lambda_i -\lambda_{i+1}>3$ whenever $\lambda_i\equiv 0{\ \rm mod\ } 3$.

In \cite{BMO}, it was also conjectured that for $p=5$, the  labels
should be  the partitions $\lambda=(\lambda_1,\ldots,\lambda_r)$
satisfying the following conditions: (1) $\lambda_i>\lambda_{i+1}$
for $i\le r-1$,
(2) $\lambda_i -\lambda_{i+2} \ge 5$ for $i\le r-2$,
(3) $\lambda_i -\lambda_{i+2} > 5$ if $\lambda_i\equiv 0{\ \rm mod\ } 5$
or if $\lambda_i+\lambda_{i+1}\equiv 0{\ \rm mod\ } 5$ for $i\le r-2$,
and (4) there are no subsequences of the following types
(for some $j\ge 0$): $(5j+3,5j+2)$, $(5j+6,5j+4,5j)$,
$(5j+5,5j+1,5j-1)$, $(5j+6,5j+5,5j,5j-1)$.
This conjecture turned out to be equivalent to a
$q$-series identity conjectured long ago by Andrews
in the context of extensions of the Rogers-Ramanujan identities,
and  was eventually proved by Andrews {\it et al.} \cite{ABO}.
The authors of \cite{ABO} observed however that such a labelling
scheme could not be extended to $p=7,11,13$ (see also \cite{B}).

In terms of canonical bases, the obstruction can be understood as
follows.
Assuming our conjecture and using the results of \cite{BMO,ABO},
one can see that for $p=3,5$, the labels of \cite{BMO} and \cite{ABO}
are exactly the partitions indexing the lowest nonzero entries
in the columns of the matrices $D_m(q) = [d_{\lambda\mu}(q)]_{\lambda,\mu\vdash m}$.
For example, in Table \ref{TAB1}, these are
$(10),(91),(82)$ and $(73)$, which are indeed the Schur regular partitions
of $10$. The problem is that for $p\ge 7$, it can happen that
two columns have the same partition indexing the lowest nonzero
entry. For example, with $p=7$ ($n=3$) and $m=21$, the two canonical basis
vectors

$
G(75432)
=
\ket{7 5 4 3 2}+q^{2}\ket{7 6 4 3 1}+q\ket{7 7 5 2}+q^{3}\ket{7 7 6 1}
+q^{2}\ket{8 6 4 3}+\left (q^{2}+q^{4}\right )\ket{8 6 5 2}+
q^{3}\ket{8 7 6}+q^{4}\ket{9 5 4 3}+\left (q^{4}+q^{6}\right )\ket{9 6 5 1}+
q^{5}\ket{9 7 5}
$

\noindent and

$
G(654321)
=
\ket{6 5 4 3 2 1}+q\ket{7 5 4 3 2}+ q\ket{7 6 4 3 1}+ q\ket{7 6 5 2 1}+
q^{2}\ket{7 7 4 3}+q^{2}\ket{7 7 5 2}+q^{2}\ket{7 7 6 1}+q^{3}\ket{7 7 7}+
\left (q^{3}+q^{5} \right )\ket{8 6 4 3}+
\left (q^{3}+q^{5}\right )\ket{8 6 5 2}+\left (q^{4}-q^{8}\right )\ket{8 7 6}+
\left (q^{3}+q^{5}\right )\ket{9 6 5 1}+\left (q^{4}+q^{6}\right )\ket{9 7 5}
$

\noindent have the same bottom partition $(975)$
(compare \cite{B}, end of Section 3).

On the other hand the partitions
indexing the highest nonzero entries in the columns of $D_m(q)$ are
the labels of the crystal graph (by Theorem \ref{TH}(ii)),  so that
they are necessarily distinct. Therefore, we propose to use the set
\begin{eqnarray*}
{\rm DPR}_p(m) = &\{ \lambda = (\lambda_1,\ldots ,\lambda_r)\vdash m \ | \
0<\lambda_i-\lambda_{i+1}\le p \ {\rm if} \ \lambda_i \not \equiv 0 {\ \rm mod\ } p, \\
& 0 \le\lambda_i-\lambda_{i+1} < p \ {\rm if} \ \lambda_i \equiv 0 {\ \rm mod\ } p,
(1\le i \le r)\}
\end{eqnarray*}
for labelling the irreducible spin representations of ${\rm \widehat{S}}_m$
in characteristic $p$.
Indeed its definition is equally simple for all $p$.
Moreover, because of Theorem~\ref{TH}(iii), this labelling would be compatible with
the $p$-block structure, which can be read on the
$\overline{p}$-cores.
Also, it is adapted to the calculation of the vectors
$\underline{A}(\mu)$ which give an approximation to the
reduced decomposition matrix.

Finally, we note that since ${\rm DPR}_p$ provides the right number
of labels we have the following partition identity
\begin{equation}\label{PARTID}
\sum_{m\ge 0} | {\rm DPR}_p(m) |  t^m
=
\prod_{\scriptstyle i \ {\rm odd} \atop \scriptstyle i\not\equiv 0{\ \rm mod\ } p}
{1\over 1-t^i}
\end{equation}
which for $p=3,5$ is a counterpart to the
Schur and Andrews-Bessenrodt-Olsson identities.

This happens to be a particular case of a theorem of Andrews
and Olsson \cite{AO}.
Namely, one gets (\ref{PARTID}) by taking
$A=\{1,2,3,\ldots ,p-1\}$ and $N=p$ in Theorem~2 of~\cite{AO}.
A combinatorial proof of a refinement of the Andrews-Olsson
partition identity has been given by Bessenrodt \cite{B1}.

One can also get a direct proof of (\ref{PARTID})
without using representation theory by
simply considering the bijections (\ref{MAP}).

\section{Discussion}

We have used the level 1 $q$-deformed Fock spaces of Kashiwara {\it et al.}
to compute the canonical basis of the basic representation
of $U_q(A_{2n}^{(2)})$, and we have formulated a conjectural
relation with the decomposition matrices of the spin symmetric groups
in odd characteristic $p=2n+1$.

As in the case of $A_{n-1}^{(1)}$,
it is reasonable to expect that in general,
that is when $2n+1$ is not required to be a prime, the canonical basis
is related to a certain family of Hecke
algebras at $(2n+1)$th roots of unity. A good candidate might be the
Hecke-Clifford superalgebra introduced by Olshanski \cite{Ol}.

The case of $2n$th roots of unity should then be related
to the Fock space representation of the affine Lie algebras
of type $D_{n+1}^{(2)}$.
In particular we believe that the fact used by Benson \cite{Be}
and Bessenrodt-Olsson \cite{BO} that the 2-modular irreducible
characters of ${\rm \widehat{S}}_m$ can be identified with the 2-modular irreducible
characters of ${\rm S}_m$ corresponds in the realm of affine Lie algebras
to the isomorphism $D_2^{(2)} \simeq A_1^{(1)}$.

\section*{Acknowledgements}

We thank T. Miwa and A.O. Morris for stimulating discussions,
and G.E. Andrews for bringing references \cite{AO,B1} to
our attention.

\section*{References}