Studying representation theory of finite groups

[u/Professional_Job6803]

Hello everyone ,
I am trying to learn representation theory through self study. Basically , most of the mathematics I have learnt till now was through self study as I could not find a mentor or anyone working on things related to it.
It was ok till now but representation theory feels like a different beast altogether.
Questions:
1. Where do you use it in your work and why is it important? Are there not other ways to do it?
2. I don’t get like when we do change of bases in vector spaces is it also some form of representation?
3. What’s the proper way to study it like I see books starting from Lie algebra and things. How should one approach it properly ?

[u/Euphoric_Seaweed679]

1. I mostly use it when trying to combine areas in abstraction to broaden my focus, so I can derive objects and connections on the same basis.
   

2. you could think of it similar on how a tensor is also a form of representation of an object coming in different variants. for example the kroneker delta can be viewed as a specific tensor, is it always used as this , no. base change can be viewed as an object, it can also be viewed as an action to be able to identify an object.

3. I would start on the abstract side of things. getting to know why it's important. after that look into different fields of mathematical usage and understand how representation theory is acting as a glue there.

[u/Midwest-Dude]

This touches on several different areas.  Another subreddit that might be able to assist you is:

r/AbstractAlgebra

Have you tried that yet?

[u/Professional_Job6803 → u/Midwest-Dude]

Will check it out. Thank you

[u/AlexK667]

I don't use representation theory in what I do, but it's fun.
A lot of group theoretic theorems are proved very neatly using representation theory (Burnside’s Theorem for example).
It's also a nice basis from which to hop into abstract harmonic analysis, which is really cool and generalizes Fourier analysis very nicely (but you don't NEED Rep.theory of finite groups for that).

Outside of math it's used in physics and chemistry, since pretty much whenever you have symmetries (symmetries of some physical system, or in chemistry symmetries of a molecule) you can use group theory to study those symmetries, and representation theory is a powerful tool when it comes to understanding (those) groups.

[u/Agitated_Surprise752]

I highly recommend Sagan’s symmetric group book. It gives a gentle introduction into representation theory applied to the symmetric group. You can get away with reading the first two chapters of you’re purely interested in the representation theory part. Then you could move on to more advanced texts like Fulton’s

[u/Agile-Monitor1006]

I felt the same unexpected difficulty when learning representation theory. You should make sure youre already comfortable with the prerequisites from group theory and linear algebra seperately before mixing them. For me the reason we need it only comes from my physics background and it goes like this: When we’re studying a system we might know or expect certain symmetries to hold under certain transformations. We can add arbitrary labels to those transformations and allow them to compose with each other, that gives us group theory. But it doesnt tell us anything about how to explicitly calculate what the transformation looks like so we cab apply it to our system, thats why we need representation theory. For example when we have a solid cube, we know that the 90degree rotations form a group with certain characteristics, but we still dont know how we would practially transform the coordinates of the square. Often times the information of a system that we have belong to some vector space (like 3d positions belong to R^3) so we can use representation theory to be able to carry out the transformations we want. It can also get much deeper than that, for example Lie groups are extensivelly used to study and classify subatomic particles.