How to use a character table

Question

This is quite an open-ended question. I have learned representation theory of finite groups and have an understanding of how to obtain the character table of a finite group. I see how the character table is useful from a representation-theoretical point of view as it can easily help me understand how an arbitrary representation can decompose as a product of irreducibles. At some point I even recall having learned how to extract information about the centre of the group using the character table. However, it still seems very mysterious to me how people used character tables to understand large groups like the Monster. And what seems even more intriguing is how the character tables were obtained before the groups were constructed!

Every book on representation theory that I have read seems to just concern itself with building the theory of representations of groups or algebras, but I have yet to find one that shows the full potential of these techniques.

Here are a couple of concrete questions I am interested in having an answer to:

(1) What would be a good reference to learn what the character table can tell me about the group, the limits of what it can tell me, and the techniques that people use to read interesting information from the table. The best would be if such a reference would include a good deal of exercises so that I can test my knowledge of the techniques.

(2) How can one build a character table for a hypothetical group without having a construction for it?

Answer 1

I recommend I.M. Isaacs's Character Theory of Finite Groups to see finite groups from the point of view of their characters.

I couldn't think of a really short example of finding a character table of an unknown group, but I relate a story that can be very briefly summarized as: "Here is an argument in a textbook showing how to uniquely classify two simple groups by the centralizer of an involution. People did that a lot with known simple groups, until Janko found a previously unknown simple group." I only tell the textbook part.

Often we don't need to compute the entire character table to determine how to construct the group. One class of result I found interesting were the "recognition theorems" which often have a large character theory step. However, in order to have an answer that is shorter, I chose an earlier recognition theorem, theorem 7.10 from Isaacs's book:

Suppose a finite perfect group $G=G'$ contains an element $\tau$ of order 2 whose centralizer $C_G(\tau)$ is a dihedral group of order 8. Then $G$ itself must be the 3x3 general linear group $\operatorname{GL}(3,2)$ of order 168, or the degree 6 alternating group $A_6$ of order 360.

Sketch of proof: Let $M \subset D=C_G(\tau)$ be cyclic of order 4 and centralizing $\tau$. Then $M$ is a "trivial intersection" set (similar to Frobenius complements if you've read about those) in that $M \cap M^x = \begin{cases} M & \text{if } x \in N_G(M)=C_G(\tau) \\ 1 & \text{otherwise}\end{cases}$.

Now class functions on $D=C_G(\tau)$ which vanish on $M$ have a very nice isometry property: If $\theta,\phi$ are class functions on $C_G(\tau)$ which vanish on $M$ and $\theta(1)=0$, then the induced character $\theta^G(x) = \theta(x)$ doesn't change for $x\in D$, and moreover neither does the inner product $[\theta^G,\phi^G] = [\theta,\phi]$.

This lets us deduce part of the character table of $G$ from that of $C_G(\tau)$

(I consider that statement to more or less be my answer to your question, but I think it helps to see a little more of how this actually works.)

In particular, take one of the two Galois conjugate faithful linear characters of $M$, call it $\lambda$, then $\theta=(1_M-\lambda)^D$ is a class function on $D$, $\theta(1)=0$, and $\theta$ vanishes on $D \setminus M$ by the formula for induced character. Since $[\theta,\theta]=3$ by an explicit calculation in $D$ (a very small group with an easy character table), we also have $[\theta^G,\theta^G] = 3$ which is now occurring in an unknown group $G$. This means we can write $\theta^G = 1_G + \chi - \psi$ for irreducible characters $\chi,\psi$ of $G$. Since $\theta^G(x) = \theta(x)$ for $x \in D$, we can nearly compute the values of $\chi$ and $\psi$ on $D$!

In $D$, we have $\theta(1)= 0$ and $\theta(\tau)=4$ (again easy calculations in a tiny group), so in $G$ we have $0 = \theta^G(1) = 1 + \chi(1) - \psi(1)$ and $4 = 1 + \chi(\tau) - \psi(\tau)$.

(Isaacs now uses a class function that counts solutions of equations, a technique from Frobenius, to show that:)

$|G|=\dfrac{2^8}{\frac{1^2}{1} + \frac{\chi(\tau)^2}{\chi(1)} - \frac{\psi(\tau)^2}{\psi(1)}}$

A simpler argument shows $|D| \geq 1 + \chi(\tau)^2 + \psi(\tau)^2$

Now a little number crunching finds all possible values of $\chi(\tau)$ and $\psi(\tau)$ which gives all possible orders of $G$: $|G|=168$ or $|G|=360$. Since perfect groups of such small order are quite rare, we have it must be one of the two well-known simple groups.

So this doesn't exactly answer your question: instead of using a character table to find an unkown group, we use it to show a potentially unknown group is actually on old friend. This sort of thing happened several times until Janko accidentally broke it when trying $C_G(\tau) = C_2 \times A_5$ and we found our first new sporadic simple group in a hundred years. The part of the argument I've highlighted here is echoed on page 153 of Janko's 1966 Journal of Algebra paper -- I believe I learned these steps from Suzuki's CA papers, but I can't seem to find my notes at the moment.