So, I’ve never done any novel math. But personally doing highschool and university competition math problems beat the “seriously, check easy examples, check the next easiest examples after that, if it’s been more than a few minutes then I don’t care if you think you can solve it without doing that just go and do it for Christ’s sake, don’t you remember all those previous times you were kicking yourself for not even writing down a simple example when that would’ve immediately lead you to the answer” into me. The idea of going for a year and a half without checking simple examples is not something I think I would do. I don’t know much rep theory but GL_2 is indeed probably at the top of the list (after SL, SO, and SU). When I read Feynman’s anecdote about examples[1], I thought “duh, every mathematician worth their salt does this all the time” (no offense to you! it was honestly what I thought, though).
I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) — disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!” If it’s true, they get all excited, and I let them go on for a while. Then I point out my counterexample. “Oh. We forgot to tell you that it’s Class 2 Hausdorff homomorphic.” “Well, then,” I say, “It’s trivial! It’s trivial!” By that time I know which way it goes, even though I don’t know what Hausdorff homomorphic means.
So, I’ve never done any novel math. But personally doing highschool and university competition math problems beat the “seriously, check easy examples, check the next easiest examples after that, if it’s been more than a few minutes then I don’t care if you think you can solve it without doing that just go and do it for Christ’s sake, don’t you remember all those previous times you were kicking yourself for not even writing down a simple example when that would’ve immediately lead you to the answer” into me. The idea of going for a year and a half without checking simple examples is not something I think I would do. I don’t know much rep theory but GL_2 is indeed probably at the top of the list (after SL, SO, and SU). When I read Feynman’s anecdote about examples[1], I thought “duh, every mathematician worth their salt does this all the time” (no offense to you! it was honestly what I thought, though).