Add generic character table for SL(2,q), with q odd and non-square?

[fingolfin]

There may be good reasons this is not in here: e.g. it may not fit into the data format. Or it is not as useful due to the exceptions. Or whatever. But it might be worth figuring out...

That said, the table cane be found on page 58 (Table 5.4) in

Cédric Bonnafé, Representations of $\mathrm{SL}_2(\mathbb{F}_q)$, https://doi.org/10.1007/978-0-85729-157-8

[SoongNoonien]

Maybe I'm missing something but we have SL2.1, isn't that what you are looking for?

[fingolfin]

I think SL2.1 only covers the case $\mathrm{SL}_2(q^2)$ and $q$ odd, i.e. over fields of square size.

I adjusted the title to clarify that the missing case is $q$ odd and not a square.

[fingolfin]

That said, in SL2.0 it indeed says:

See SL2.1 for the generic character table of $SL_2(q)$, $q$ odd

[SoongNoonien]

In the info string of SL2.1 it says:

Information about the generic character table of $SL_2(q^2)$,
$q^2$ odd. The possible values for q are given by
$q^2 = p^m$ with m a non negative integer and $p$ a prime number.

I think it's $q^2$ to avoid $\sqrt{q}$ in the table.

[SoongNoonien]

I took a closer look and I'm pretty sure that I can match all character types and class types of SL2.1 with table 5.4 from the book. In SL2.1 the last conjugacy class of 5.4 is split into two distinct classes. Apparently one is for $\epsilon = 1$ and the other for \epsilon = -1, thus there are five class types in SL2.1. Similarly some character types are split, I suppose this is to minimize the number of class/character variables.

[fingolfin]

OK. We probably should rename q to q0 in SL2.1 then, and make sure someone reading the info text understands what is going on.

We might then also define an auxiliary variable q = q0^2 and use that to replace occurrences of q0^2, q0^4` etc. -- this way it is much easier to grep for places that really use this "square root of q"