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Module: sage.combinat.root_system.cartan_matrix
Cartan matrices
Module-level Functions
| t) |
Returns the Cartan matrix corresponding to type t.
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', 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]
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