# Application of Schur orthogonality

Lemma. Let be a finite group of order , and write for the set of irreducible characters of . Then

1.

2.

Proof. First consider the case . This is just an example; it could also be something much better. Then the second statement is obvious, and the first is left as an exercise to the reader. The general case is similar.

Here is a trivial consequence:

Corollary. Let be a positive integer, and let . Then

Proof 1. Without loss of generality, has exact order . Set , let , and note that

Part 1 of the lemma gives the result.

Proof 2. Set as before, let be the homomorphism , and the homomorphism . Then part 1 of the lemma does not give the result, but part 2 does.

In fact, the corollary also implies the lemma, because both are true ().

# Perfectionism and the stages of completion

This week I finished another in a series of three preprints that in September 2017 I said were ‘almost done’. With a newer result added later, there are still two more papers left in a similar state.

However, the positive news is that I have now finished two things (oh, and there was this thing too), and I am becoming more honest with myself about what it means to finish something. Here are the stages of completion that I currently observe, and my (current/past) responses to them. I hope this reflection may be useful or somewhat amusing to someone.

Mathematical completion

By this I mean the internal conviction that there is probably a correct and complete argument, and you worked out the most important lemmas. This stage often happens in some part in my head, and not everything will have been written down yet (not even the important arguments).

Coincidentally, this is also the stage where my brain jumps to the next problem. Maybe not the best idea. I call this mathematical completion somewhat ironically, because it’s not: there are always more lemmas to prove in better ways, and things that you thought were obvious when they aren’t.

At this stage, I usually have enough down to feel confident giving talks about the matter, as my familiarity with the argument is pretty robust and immune to probing.

Nearly done

This is usually the point where the rough outline is laid down, a structure for the paper has been decided and set up, and the most important arguments are explained. It feels that there are still some small things to fix, but otherwise it shouldn’t take too long. This, like the previous stage, is based on a lie.

This was the stage that my three papers in 2017 were in, two of which have been posted now. The third one needs a more substantial rewrite for stylistic and formatting reasons, but it has long passed the Mathematical completion stage.

Fully written up

This means that there is a document from start to finish, complete with abstract, introduction, and correctly formatted references. Surely it shouldn’t take more than a week to finish, right? Wrong: proofreading. There’s often more missing than you think. Plus, I don’t like excessive updating, so I’d rather get it right the first time around.

There’s an aspect of optimisation too: every result should be written up in the easiest way in the correct generality (this is not always the biggest generality; more about this in a future post). Some results feel too unimportant to warrant a two-page proof, and I try to find a proof that does not take more space than the result deserves. This is hard to combine with the idea of completeness and self-containedness, and it certainly takes time to get it right. Local optimisation does not always align with global optimisation.

What is appropriate depends on the medium: in my dissertation I explain some well-known results in depth if no complete and modern argument is available in the literature. In papers, I try to give the briefest argument using (well-written and modern) accounts from the literature.

I’m done with this shit

Finally, once I (and maybe someone else too) spent enough time proofreading and fixing small gaps, confusions, notational inconsistencies, inaccurate references, and irrelevant distractions, I don’t want to think about it anymore and just dump it on arXiv. This allows other people to look at it, who will immediately find more mistakes or missed opportunities. (This is the best possible outcome; the alternative is that people don’t care at all.)

I keep a file with additional changes I want to make at some point in the future. I fix some of them and send it to a journal.

To be continued…

Since none of my papers have appeared in print yet, I cannot comment on the final stage of completion. Let’s hope it happens some day…

The backlog

In the weeks leading to completion, I am strongly inclined only to work on the paper, because it’s so nearly done (right?!). Also, other things don’t seem to matter (mumble something about microeconomics and optimising utility).

After I finish working obsessively on a paper for some weeks, I need some time to catch up on the rest of my life. Maybe as I’m getting more honest with myself, I can also change this habit (although it’s not entirely clear to me that this is desirable!)…

Closing remarks.

This post doesn’t really talk about my excessive¹ perfectionism, about which I may say a thing or two in the future.

¹Meaning it’s hard for me to finish anything at all, or sometimes even to start something.