What Is Research?

June 29, 2006

The eighty page paper — begins

Filed under: Regular updates,Thinking and research,TIFR (VSRP) — vipulnaik @ 4:01 pm

The paper “Lie Group Representations on Polynomial Rings” (sorry, you can’t read the paper unless you have JSTOR access, and it’s illegal for me to put it up online) is eighty pages long. It was penned by Kostant in 1963, so it is about 43 years old. So how do I begin with it?

Every daunting task should be handled by breaking it into submodules, so I decided I’ll just concentrate on getting the gist of the kind of issues that the author is trying to address. I basically decided to begin from the beginning — Page 1. (Page 1 is publicly accessible, so you can read it yourself at the link). Talk of symbols watering in front of my eyes!

One of the things that has irritated me in the books I have read nad the lectures I have attended is an excessive use of symbols like letters to denote abstract concepts. Symbols are indeed indispensable because without them, abstract algebra would have been impossible. However, I think symbols should remain what they are: symbols, and not synonyms for the concepts they connote.

What sends me up is statements like: In our talk, R will always denote a commutative ring with identity. Or worse still, people assuming that the letter N, wherever it pops up, means a normal subgroup, and not even bothering to say so. I feel this has its own dangers as we lose out on the statement being made at the conceptual level.

And Kostant’s paper was full of such statements…

This brings me to the old issue of mathematicians being accused of deliberately trying to obscure their work in technicality. Richard Hamming, in his classical talk, says that giving an accessible talk is difficult because it forces the mathematician to step back and critically examine how his or her work fits into a larger perspective. Staying immersed in one’s own comfort zones, however murky they may be, seems so much easier! We mathematicians use jargon as a means of protection (largely imaginary) against a world where we feel we don’t belong.

May be that’s taking it a little too far, … and any way, I didn’t know why Kostant had chosen his conventions the way he did. It probably had more advantages than disadvantages and I needed to first understand what he was saying.

So a look at the first para.

Let G be a group of linear transformations on a finite dimensional real or complex vector spacve X. Assume X is completely reducible as a G module. Let S be the ring of all complex-valued polynomials on X, regarded as a G module in the obvious way and let J be the subring of S comprising the G invariant polynomials on X.

Dense, it seemed to me. Which brings me to another question. What do we have against dense material which makes it more difficult than light material which may be much longer? I think it is the fact that we are not used to stopping at the end of each sentence and evaluating what it means. We want to use the pipeline of our minds: read the next sentence as the back of the mind evaluates the previous one. This pipeline is effective only if the effort required to understand each sentence is minimal.

But I’d been sitting in front of this page for quite a long time, so I decided: “might as well seriously try making sense of it. I probably do know enough for that.”

Read again: sentence 1 “Let G be a …” absolutely clear. I know a lot about subgroups of the general linear group and I’ve studied the representation theory of finite groups. So nothing new in that.

sentence 2: “Assume X is…” A mundanity. Forget it, I know what it means, but I don’t have thecontext to understand its significance.

sentence 3: “Let S …” Now I did remember that when a group acts on a vector space, it does also act on the polynomial ring. In fact, I remembered reading somewhere about the invariant subring under the symmetric group being the subring generated by the elementary symmetric functions, which is a formulation of the Fundamental Theorem for elementary symmetric functions.

So I thought: “This is stuff I probably know and can make sense of. But how do I keep track of and remember the symbols? What is the author actually trying to do and compute? May be I can try working and computing the things myself. Reading the paper passively seems a very dull activity and may be if I work things out, I’ll better appreciate what Kostant has done.”

Which raises many interesting questions abotu how we go about reading stuff. What should we do and what do we end up doing?
(i) Stare at it: Is it completely effective or does it set the stage for back of the mind processing?
(ii) Skim through it: If we’re not following anything, is it still worth skimming through it?
(iii) Decipher it sentence by sentence: How much time does that take? Do we lose sight of the big picture in that process?
(iv) Get a general idea and then explore through other means: That’s what I’m usually very comfortable with. But is it always appropriate? Isn’t there a chance of my going too far astray?

Looknig forward to your comments…


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