ufs_dynkintype - calculates the Dynkin type of a quadratic form

Calling sequence:

ufs_dynkintype(M)

Parameters:

M - a symmetric matrix definig a non negative unit form q

Synopsis:

ufs_dynkintype calculates the Dynkin type of q, as described below.

The argument must be a symmetric matrix definig a unit form, otherwise an error occurs.

It is a classic result, that the equivalence classes of positive unit forms are characterized by the Dynkin diagrams

( ), ( ) and ( ),

namely a Dynkin diagram defines a unit form in the following way: Suppose has 1,..., as vertices then is a form in the variables ,..., and where is the number of edges between and in . Now each positive unit form is equivalent to a unit form where is one of those Dynkin diagrams, called the Dynkin type of . It has been shown in [1], that the equivalence classes of non-negative unit forms are characterized by two data: the corank and a Dynkin diagram as above, namely each unit form in the variables ,..., which has corank is equivalent to some , where is one of those Dynkin diagrams with vertices. Again, is called the Dynkin type of .

Example:

> A:=matrix([[2,1,1,0,0],[1,2,1,1,0],[1,1,2,1,1],[0,1,1,2,1],[0,0,1,1,2]]);
B:=linalg[diag](A,A):

> ufs_dynkintype(A);
ufs_dynkintype(B);

>