Importance of Group Representation theory

I was reading about the group representation, but couldn't really why is it important or interesting.

Can you someone explain about some of the important mathematical applications (not from physics, possibly from algebra or number theory) of group representation or why is it interesting at all?

Solution 1:

Although you specifically asked for applications not from physics, let me begin by mentioning that representation theory is of paramount importance in physics, and once you decide to look for such applications, you will find many! The same goes for chemistry.

Now for applications in pure mathematics. As Tobias mentioned in a comment, two famous applications are the Burnside $pq$-theorem and the structure theorem of Frobenius groups. Both of these are discussed in detail in chapter 6 of my representation theory notes. Isaacs's wonderful book on character theory contains a vast amount of applications of representation theory. For example the classification of finite simple groups is completely unthinkable without representation theory, both classical and modular, and Isaacs gives a glimpse of that. In fact, character theory was invented by Frobenius without representations in mind, and in the attempt to solve a purely group theoretic problem. It was pointed out later by Schur that what Frobenius had really done was representation theory. A sketch of this history is contained in the introduction to the aforementioned notes, but there are better sources.

Representation theory is extremely important in number theory. In particular, there are groups that we don't know how else to begin understanding, other than through their representations, most notably the absolute Galois group of $\mathbb{Q}$. It is very big, and it is not clear how to describe such big groups in a useful way (note that generators and relations are useless for most purposes, since there is not even an algorithm to tell whether a given presentation describes the trivial group). On the other hand, the Galois group by its very nature acts on lots of things, and it is very natural to try understanding it through these actions.

Finally, note that group representations are simply part of our world, so it would be foolish to try avoiding them. In particular, historically one could argue that group representations were born before groups were. This is not true literally, since the definition only appeared in the 20th century, but it is true morally: the first incarnation of groups that people considered was that of symmetries of geometric objects. And those are nothing other than group representations.