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\chapter{SYMMETRY: Symmetric matrices}
\label{SYMMETRY}
\typeout{{SYMMETRY: Operations on symmetric matrices}}

{\footnotesize
\begin{center}
Karin Gatermann\\
Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\
Takustra\"se 7 \\
D--14195 Berlin--Dahlem, Germany \\[0.05in]
e--mail: gatermann@zib.de
\end{center}
}
\ttindex{SYMMETRY}

The SYMMETRY package provides procedures
that compute symmetry-adapted bases and block diagonal forms
of matrices which have the symmetry of a group.

\section{Operators for linear representations}

The data structure for a linear representation, a {\em
representation}, is a list consisting of the group identifier and
equations which assign matrices to the generators of the group.

{\bf Example:}
\begin{verbatim}
   rr:=mat((0,1,0,0),
           (0,0,1,0),
           (0,0,0,1),
           (1,0,0,0));

   sp:=mat((0,1,0,0),
           (1,0,0,0),
           (0,0,0,1),
           (0,0,1,0));

   representation:={D4,rD4=rr,sD4=sp};
\end{verbatim}

For orthogonal (unitarian) representations the following operators
are available.

{\tt canonicaldecomposition(representation);}\ttindex{canonicaldecomposition}

returns an equation giving the canonical decomposition of the linear
representation.

{\tt character(representation);}\ttindex{character}

computes the character of the linear representation. The result is a list
of the group identifier and of lists consisting of a 
list of group elements in one equivalence class and a real or complex number.

{\tt symmetrybasis(representation,nr);}\ttindex{symmetrybasis}

computes the basis of the isotypic component corresponding to the irreducible
representation of type nr. If the nr-th irreducible representation is
multidimensional, the basis is symmetry adapted. The output is a matrix.
 
{\tt symmetrybasispart(representation,nr);}\ttindex{symmetrybasispart}

is similar as {\tt symmetrybasis}, but for multidimensional 
irreducible representations only the first part of the 
symmetry adapted basis is computed.

{\tt allsymmetrybases(representation);}\ttindex{allsymmetrybases}

is similar as {\tt symmetrybasis} and {\tt symmetrybasispart}, 
but the bases of all
isotypic components are computed and thus a 
complete coordinate transformation is returned.

{\tt diagonalize(matrix,representation);}\ttindex{diagonalize}

returns the block diagonal form of matrix which has the symmetry 
of the given linear representation. Otherwise an error message occurs.


\section{Display Operators}

Access is provided to the information for a group, and for adding
knowledge for other groups.  This is explained in detail in the
Symmetry on-line documentation.