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Next year I have choose a few optative courses from a big list, but syllabi are not available yet, so I have to make a choice based on names only.

I am thinking about taking one titled "Groups and representations". I have really enjoyed my Abstract Algebra courses so far, specially the part that covered Group theory. However, we never got into Representation theory, and I ignore what mathematical interest it might have.

I was wondering if someone could provide me with a text explaining what is representation theory about and what is the mathematical interest or motivation behind it. I have found some books on the topic, but they all just start giving definitions and theorems and never give a good introduction.

Shaun
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Frank William Hammond
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2 Answers2

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Have you tried googling to find motivatations for the subject? Some are here, here, and here. Historical background for the subject is here. The question that first led Frobenius to develop representation theory of finite groups has always struck me as a bad motivation for the subject. Keep in mind that one person's inspiring motivation is someone else's dull problem or excessive technicality.

Example. When I first saw representation theory, I did not understand why it was worthwhile. The accessible applications that I heard about to pure group theory (proving the $p^aq^b$ theorem and showing nonexistence of simple groups of some peculiar size) felt uninspiring. What sold me on the utility of representation theory of finite groups was learning about Artin's creation of $L$-functions of representations of Galois groups, and how they lead to a decomposition of zeta-functions of Galois extensions of $\mathbf Q$ that generalizes the simpler decomposition of zeta-functions of quadratic fields and cyclotomic fields into products of Dirichlet $L$-functions. Reread the last sentence of my first paragraph, however.

KCd
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  • Yes, indeed, there are many "orthodox" reasons to pay attention to "representation theory" (whatever that is), but/and, also, it is also explanatory and helpful, which is not always mentioned. That is, it does not merely make up and solve new problems, but, rather, helps us resolve prior issues. :) – paul garrett May 09 '23 at 19:15
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It is hard not to begin with a definition. To me, a (linear!) representation of a group $G$ on a vector space $V$ is a homomorphism $\varphi : G \longrightarrow GL(V).$ You can define, i.e. phrase it differently via group actions but the homomorphism picture already illustrates the answer to your question: $$\varphi (g\cdot h)\stackrel{(*)}{=}\varphi (g)\cdot\varphi (h).$$ We transport the usually very abstract group multiplication among its elements into a multiplication of matrices, of linear functions. We usually can handle matrices far better than group multiplications. We have with linear algebra a mighty tool to investigate the right-hand-side of $(*)$ while the left-hand-side could be as abstract as e.g. $G=\langle a,b\,|\,a^2=aba=1 \rangle.$ If the representation is furthermore faithful, i.e. $\ker \varphi =\{1\},$ then we even have matrices that represent the group elements one-to-one. That's why we investigate group representations: we know how to deal with matrices. The homomorphism tells us what we can switch to the left what we have learned on the right (of (*)).

Marius S.L.
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  • Your presentation isn't so abstract. Since $$b=a^2ba^2=a(aba)a=a^2=1,$$ the group it defines is isomorphic to $\Bbb Z_2$. – Shaun May 09 '23 at 20:13
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    Yes, I know. I have written it without thought. After I had this thought and considered making it more complicated, I had a second thought: It actually emphasizes my statement. Yes, I have defined $\mathbb{Z}_2.$ But imagine how more obvious it would have been if I had written $G=\langle \begin{pmatrix}-1&0\0&-1\end{pmatrix} \rangle$! – Marius S.L. May 09 '23 at 20:17