Document (and improve) output of printinfochar and printinfoclass

[fingolfin]

From the manual

julia> g=genchartab("GL2");

julia> printinfochar(g)
1	Any["", [1, 0], Any["A_1", [2]]]
2	Any["", [1, 1], Any["A_1", [1, 1]]]
3	Any["", [2, 0], Any["A_0", [1]]]
4	Any["", [3, 0], Any["A_0", [1]]]

julia> printinfoclass(g)
1	Any["", [1, 0], Any["A_1", [1, 1]]]
2	Any["", [1, 1], Any["A_1", [2]]]
3	Any["", [2, 0], Any["A_0", [1]]]
4	Any["", [3, 0], Any["A_0", [1]]]

But what does that mean?

From the CHEVIE paper:

We get information about the class types and the corresponding centralizers with the command PrintInfoClass.

It then goes on to show the equivalent of this:

julia> printinfoclass(g)
1	Any["", [1, 0], Any["A_2", [1, 1, 1]]]
2	Any["", [1, 1], Any["A_2", [2, 1]]]
3	Any["", [1, 2], Any["A_2", [3]]]
4	Any["", [1, 3], Any["A_2", [3]]]
5	Any["", [1, 4], Any["A_2", [3]]]
6	Any["", [2, 0], Any["A_1", [1, 1]]]
7	Any["", [2, 1], Any["A_1", [2]]]
8	Any["", [3, 0], Any["A_0", [1]]]
9	Any["", [4, 0], Any["A_0", [1]]]
10	Any["", [5, 0], Any["A_0", [1]]]

and says:

The rows of this information table correspond to the columns of the generic character
table. Each row is a list with a certain number of entries (three in the above example), which may again be lists. For technical reasons, the first entry is empty. The third entry contains information about the labels of the class types (4.2 and 4.4).

For example in the row beginning with 7 the last entry means that the semisimple part of an element in the class type corresponding to the seventh column has Dynkin diagram of type $A_1$ and the unipotent part corresponds to the class with Jordan block of dimension 2 in this centralizer. The second entry of the information list is a list with two components. The first of these numbers the $\mathbb{G}(q)$-classes of centralizers of semisimple elements and the second the unipotent classes in these centralizers.

So it seems we can get rid of the first entry which is empty "for technical reasons" but those don't need to constrain us, right?

[SoongNoonien]

So it seems we can get rid of the first entry which is empty "for technical reasons" but those don't need to constrain us, right?

Yes, I would think so too. I don't see a "technical reason" in the current implementation.

[fingolfin]

Actually, both printinfochar and printinfoclass should also be replaced: instead we should have info methods for AbstractGenericCharacter resp. AbstractCharacterType (the later is yet to be introduced).

Of course that doesn't change the central point of this issue, which is about improving the documentation.