What are...characters?

[@aurelayrault663]

Did representation theory grow out of Dirichlet's work on arithmetic progression (L-functions) ?

[@VisualMath]

Excellent question and observation, that is exactly right! The comments are probably not the right place to do the history justice. So maybe this way: There is a beautiful book about the history of the subject that can be found here: bookstore.ams.org/view?ProductCode=HMATH/15 Freely available and based on that book, Geordie gave a nice talk about the history of the subject that can be found here: www.maths.usyd.edu.au/u/geordie/BMS.pdf vimeo.com/148780725 You will find the connection to Dirichlet’s L-functions about 50% in, but the rest is also great!

[@aurelayrault663]

Thanks for the reference, My understanding is that the natural progression is : Gauss sums (Cyclotomy/Quadratic reciprocity) ==> L - Functions ==> Representations Which is why the guiding concept is really Characters in my view, representations came after.

[@VisualMath]

Yes, that is correct. There are many nontrivial steps involved from left to right, so its a bit of a miracle that things worked out the way they did. However, one might argue that representations are quite a natural concept and would have been discovered eventually anyway.