Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such as we have for simple Lie algebras/groups.



Is the representation of finite simple groups fully understood?
A term that may fit in the scope of this problem is "generic character table", character tables of a whole family of groups of Lie type.


Example: Generic character table of $SL\_2(q)$, $q = 2^f$





| Representations | $I$ | $U$ | $S(a)$ | $T(b)$ |
| --- | --- | --- | --- | --- |
| Trivial | $1$ | $1$ | $1$ | $1$ |
| Steinberg | $q$ | $0$ | $1$ | $-1$ |
| Principal indexed by $k=1 \dots q/2 − 1$ | $q+1$ | $1$ | $\epsilon^{ak}$$+$$\epsilon^{−ak}$ | $0$ |
| Discrete indexed by $l=1 \dots q/2$ | $q-1$ | $-1$ | $0$ | $−\eta^{bl}$$−$$\eta^{−bl}$ |



where $ \epsilon = \exp (2πi/(q − 1))$, $η = exp (2πi/(q + 1))$.


The generic character tables of the groups of Lie type $D\_5(q)$ and $E\_6(q)$ are still unknown, let alone $E\_7(q)$ and $E\_8(q)$.


And even if we worked out the whole character table, there is still a gap from the character table to representations. To see this, you can try to derive the representations from the character table above.


