HELP FOR:  root system data Structures,SYNOPSIS:   
  A root system data Structure is any one of the following: a name, a list
  of simple roots (i.e., a Base), a Cartan matrix, or a Coxeter matrix.,   
  The irreducible crystallographic root systems are named A1,A2,...,
  B2,B3,..., C3,C4,..., D4,D5,..., E6,E7,E8,F4,G2. The irreducible 
  non-crystallographic root systems are named I2[4],I2[5],...., H3,H4.
  A reducible root system name is obtained by forming a monomial out of
  the names of the irreducible root systems. For example, a root system
  consisting of the orthogonal direct sum of two copies of A2 and one
  copy of D4 would be named A2^2*D4.
  For convenience, B1,C1,C2,D2,D3,E3,E4,E5,I2[2],I2[3] are also allowed
  as names of root systems, even though they are isomorphic to others
  previously named.  Also, the integer 1 is allowed as the name of the
  empty root system.,  Bases:   
  A root system can also be specified by a list of simple roots; i.e.,
  a Base. The names e1,e2,e3,... are reserved as an orthonormal basis for
  the space containing the root system, and a root is a linear combination
  of these basis vectors. For example, the list [e2-e1,e3-e2] is a Base
  for the root system A2. In a crystallographic root system, the
  coefficients must be rational, but in a non-crystallographic root system,
  the coefficients can be rational or floating-point (or a mix of the two).
  Cartan matrices:
  A crystallographic root system can also be specified by a Cartan matrix.
  If [r_1,....,r_n] is a list of simple roots, then the corresponding 
  Cartan matrix is the n x n integer matrix whose (i,j)-entry is
  2*Iprod(r_i,r_j)/Iprod(r_j,r_j). Since Maple does not support 0 x 0
  matrices, the empty list [] is used as the Cartan matrix of the empty
  root system.,  Coxeter matrices:
  A root system can also be (imperfectly) specified by a Coxeter matrix.
  If s_1,...,s_n are the Reflections corresponding to the simple roots of
  a root system, then the associated Coxeter matrix is the n x n matrix
  whose (i,j) entry is the order of s_i * s_j. (The Coxeter "matrix" of
  the empty root system is the empty list [].) This matrix completely
  specifies the Coxeter group generated by s_1,...,s_n, but non-isomorphic
  root systems can have the same Coxeter matrix. In circumstances where
  this distinction matters, a root system (crystallographic, if possible)
  with this Coxeter matrix will be chosen by the function called.
EXAMPLE:   
  The root system A3 can be specified in any of the following four ways:
  A3                                    # by name
  [e2-e1,e3-e2,e1+e2]                   # a list of simple roots
  array([[2,-1,0],[-1,2,-1],[0,-1,2]])  # a Cartan matrix
  array([[1,3,2],[3,1,3],[2,3,1]])      # a Coxeter matrix
SEE ALSO:  Base, CartanMatrix, Coxeter_matrix, NameOf):
