Cartan matrices

sage.combinat.root_system.cartan_matrix.cartan_matrix(t)

Returns the Cartan matrix corresponding to type t.

EXAMPLES:

sage: cartan_matrix(['A', 4])
[ 2 -1  0  0]
[-1  2 -1  0]
[ 0 -1  2 -1]
[ 0  0 -1  2]
sage: cartan_matrix(['B', 6])
[ 2 -1  0  0  0  0]
[-1  2 -1  0  0  0]
[ 0 -1  2 -1  0  0]
[ 0  0 -1  2 -1  0]
[ 0  0  0 -1  2 -1]
[ 0  0  0  0 -2  2]
sage: cartan_matrix(['C', 4])
[ 2 -1  0  0]
[-1  2 -1  0]
[ 0 -1  2 -2]
[ 0  0 -1  2]
sage: cartan_matrix(['D', 6])
[ 2 -1  0  0  0  0]
[-1  2 -1  0  0  0]
[ 0 -1  2 -1  0  0]
[ 0  0 -1  2 -1 -1]
[ 0  0  0 -1  2  0]
[ 0  0  0 -1  0  2]
sage: cartan_matrix(['E',6])
[ 2  0 -1  0  0  0]
[ 0  2  0 -1  0  0]
[-1  0  2 -1  0  0]
[ 0 -1 -1  2 -1  0]
[ 0  0  0 -1  2 -1]
[ 0  0  0  0 -1  2]
sage: cartan_matrix(['E',7])
[ 2  0 -1  0  0  0  0]
[ 0  2  0 -1  0  0  0]
[-1  0  2 -1  0  0  0]
[ 0 -1 -1  2 -1  0  0]
[ 0  0  0 -1  2 -1  0]
[ 0  0  0  0 -1  2 -1]
[ 0  0  0  0  0 -1  2]
sage: cartan_matrix(['E', 8])
[ 2  0 -1  0  0  0  0  0]
[ 0  2  0 -1  0  0  0  0]
[-1  0  2 -1  0  0  0  0]
[ 0 -1 -1  2 -1  0  0  0]
[ 0  0  0 -1  2 -1  0  0]
[ 0  0  0  0 -1  2 -1  0]
[ 0  0  0  0  0 -1  2 -1]
[ 0  0  0  0  0  0 -1  2]
sage: cartan_matrix(['F', 4])
[ 2 -1  0  0]
[-1  2 -1  0]
[ 0 -2  2 -1]
[ 0  0 -1  2]

This is different from MuPAD-Combinat, due to different node convention?

sage: cartan_matrix(['G', 2])
[ 2 -3]
[-1  2]
sage: cartan_matrix(['A',1,1])
[ 2 -2]
[-2  2]
sage: cartan_matrix(['A', 3, 1])
[ 2 -1  0 -1]
[-1  2 -1  0]
[ 0 -1  2 -1]
[-1  0 -1  2]
sage: cartan_matrix(['B', 3, 1])
[ 2  0 -1  0]
[ 0  2 -1  0]
[-1 -1  2 -1]
[ 0  0 -2  2]
sage: cartan_matrix(['C', 3, 1])
[ 2 -1  0  0]
[-2  2 -1  0]
[ 0 -1  2 -2]
[ 0  0 -1  2]
sage: cartan_matrix(['D', 4, 1])
[ 2  0 -1  0  0]
[ 0  2 -1  0  0]
[-1 -1  2 -1 -1]
[ 0  0 -1  2  0]
[ 0  0 -1  0  2]
sage: cartan_matrix(['E', 6, 1])
[ 2  0 -1  0  0  0  0]
[ 0  2  0 -1  0  0  0]
[-1  0  2  0 -1  0  0]
[ 0 -1  0  2 -1  0  0]
[ 0  0 -1 -1  2 -1  0]
[ 0  0  0  0 -1  2 -1]
[ 0  0  0  0  0 -1  2]
sage: cartan_matrix(['E', 7, 1])
[ 2 -1  0  0  0  0  0  0]
[-1  2  0 -1  0  0  0  0]
[ 0  0  2  0 -1  0  0  0]
[ 0 -1  0  2 -1  0  0  0]
[ 0  0 -1 -1  2 -1  0  0]
[ 0  0  0  0 -1  2 -1  0]
[ 0  0  0  0  0 -1  2 -1]
[ 0  0  0  0  0  0 -1  2]
sage: cartan_matrix(['E', 8, 1])
[ 2  0  0  0  0  0  0  0 -1]
[ 0  2  0 -1  0  0  0  0  0]
[ 0  0  2  0 -1  0  0  0  0]
[ 0 -1  0  2 -1  0  0  0  0]
[ 0  0 -1 -1  2 -1  0  0  0]
[ 0  0  0  0 -1  2 -1  0  0]
[ 0  0  0  0  0 -1  2 -1  0]
[ 0  0  0  0  0  0 -1  2 -1]
[-1  0  0  0  0  0  0 -1  2]
sage: cartan_matrix(['F', 4, 1])
[ 2 -1  0  0  0]
[-1  2 -1  0  0]
[ 0 -1  2 -1  0]
[ 0  0 -2  2 -1]
[ 0  0  0 -1  2]
sage: cartan_matrix(['G', 2, 1])
[ 2  0 -1]
[ 0  2 -3]
[-1 -1  2]