Tutorial

Occurrence Matrices

We mentioned before that a Cartan subgroup may appear in different real forms of an inner class. More precisely, for a given inner class there is a fixed set of Cartan subgroups which may or may not appear in the different real forms associated to this inner class. If we ask for the number of Cartan classes of real form \(G\), we get the same number regardless of which real form in the inner class we are using. This can be misleading unless we understand what the software is doing. This is because atlas is implicitly assuming that you are asking for the number of Cartan subgroups in the inner class of these real forms. For example:

atlas> set G=SO(5,4)
Identifier G: RealForm
atlas> nr_of_Cartan_classes (G)
Value: 9
atlas> G:=SO(9,0)
Value: compact connected real group with Lie algebra 'so(9)'
atlas> nr_of_Cartan_classes (G)
Value: 9
atlas>

So, to find out which Cartan subgroups appear in each individual real form there is an occurrence matrix:

atlas> occurrence_matrix (G)
Value:
| 1, 0, 0, 0, 0, 0, 0, 0, 0 |
| 1, 0, 1, 0, 0, 0, 0, 0, 0 |
| 1, 1, 1, 1, 0, 0, 0, 0, 0 |
| 1, 1, 1, 1, 0, 1, 0, 1, 0 |
| 1, 1, 1, 1, 1, 1, 1, 1, 1 |

Which again, is the same matrix for all the real forms in the inner class.

This matrix has 9 columns for all the Cartan classes and 5 rows for all the real forms of the group of type B4. Recall that the real forms for this inner class can be listed as follows:

atlas> void: for H in real_forms (G) do prints(H) od
compact connected real group with Lie algebra 'so(9)'
disconnected real group with Lie algebra 'so(8,1)'
disconnected real group with Lie algebra 'so(7,2)'
disconnected real group with Lie algebra 'so(6,3)'
disconnected split real group with Lie algebra 'so(5,4)'
atlas>

Remember that we type void: to avoid getting the empty values [(),(),(),(),()]

So, the occurrence matrix says that all 9 Cartan subgroups appear in the split form \(SO(5,4)\), that only the compact Cartan subgroup appears in the compact real form \(SO(9,0)\), that \(SO(6,3)\) has only 6 Cartan subgroups, etc. Also note that the Compact Cartan subgroup appears in all real forms but the split Cartan subgroup only appears in the split real form.