\section{Exercices}

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% r:=1 [ [op(l, r..(r:=r+i)-1)] $ i in c] ;

\subsection{The degenerate Hecke Algebra}
\eng{ The Iwahori-Hecke algebra $H_n(q)$ is the ${\mathbb C}$-algebra
  generated by elements $T_i$ for $i<n$ with the relations:
\begin{alignat}{2}
  T_i^2&=(q-1)T_i+q    \qquad&&\text{for $1\leq i \leq n-1$,}\notag\\
  T_iT_j&=T_jT_i \qquad&&\text{for $|i-j|>1$,}\\
  T_iT_{i+1}T_i &=T_{i+1}T_iT_{i+1}T \qquad&&\text{for $1\leq i \leq n-2$,}
  \notag
\end{alignat}
The $0$-Hecke algebra is obtained by setting $q=0$ in these relations. Then,
the first relation becomes $T_i^2=-T_i$. Let $\pi_i := 1+T_i$. 
\begin{itemize}
\item Define a parameterized domain \Mex{perm(n)} for the permutations of $n$.
  Note that this may be automatic in the near future.
\item Define a parameterized domain \Mex{HeckePi(n)} for the Hecke algebra
  in the basis $(\pi_\sigma)_{\sigma \in {\mathfrak S}_n}$ with coefficient
  the rational (\Mex{Dom::rational}).
\item Define the elementary generator $\pi_i$. Check some braid
relations. 
\item Define a parameterized domain \Mex{HeckeT(n)} for the Hecke algebra
  in the basis $(T_\sigma)_{\sigma \in {\mathfrak S}_n}$ with coefficient
  the rational (\Mex{Dom::rational}).
\item Check for small $n$ that sum and alternating sum of the $T_\sigma$ are
two idempotents.
\item Define the change of basis between the two versions. This can be done by
  hands or using the Bruhat order (\Mex{permutations::smallerBruhat}). Make
  the conversion automatic.
\item We need to compute with different coefficients, in particular we often
need to compute with unknown coefficients taken out
\Mex{Dom::ExpressionField()}. Modify the two parameterized domains
\Mex{HeckePi(n)} and \Mex{HeckeT(n)} so that they accept a ring for second
parameter with \Mex{Dom::ExpressionField()} as default value.
\end{itemize}
As a application we want now to compute the radical of $Hn(0)$. We use a
characterization from Dickson.
\newtheorem{theorem}{Theorem}[section]
\begin{theorem}
  Let $A$ be a sub algebra of $M_n(F)$ for a field $F$ of zero characteristic.
  Then $x\in A$ is in the radical of $A$ if and only if the trace of each $y$
  in the two sided (resp. left, right) ideal generated by $x$ ($AxA$, resp.
  $Ax$, $xA$) is zero.
\end{theorem}
For finite dimensional algebra $A$, we realize $A$ as a matrix algebra by
considering any faithful representation, for example the regular one. 
\begin{itemize}
\item Let $g := \sum_{\sigma \in {\mathfrak S}_n} a_\sigma \Pi_{\sigma}$ be
  the generic element of $H_n(0)$. Compute $g$ and the generating family
  $(g\Pi_{\mu})_{\mu \in {\mathfrak S}_n}$ for the ideal.
\item Write a function \Mex{trace} that compute the trace of the endomorphism
$x\in Hn(0) \to x e$ where $e\in Hn(0)$.
\item Using Dickson criterion, compute the equation of the radical of $Hn(0)$.
\item What is the dimension of the radical, of the quotient ?
\item Write a function \Mex{radNormal} that normalize an element of $H_n(0)$
  modulo its radical.
\end{itemize}
Suppose that $A$ is an algebra with a basis $B$. In \mupad, $A$ is a
certain domain \Mex{Alg}. The basis $B$ of $A$ is indexed by the element of
the domain \Mex{Alg::basisIndices}. Like all combinatorial domain this domain may
have a list method which can be called by \Mex{Alg::basisIndices::list}.  
\begin{itemize}
\item Write a function \Mex{GetBasis} to compute a basis of an algebra or more
generally of a free module.
\item Write a function \Mex{Generic Element} to compute a generic element;
\item Write a function \Mex{Radical} to compute the radical of an algebra. 
\end{itemize}

}

