Going to the mat with representation, characters and group theory

Willock’s book (see https://luysii.wordpress.com/category/willock-molecular-symmetry/) convinced me of the importance of the above in understanding vibrational spectroscopy.  I put it aside because the results were presented, rather than derived.  From the very beginning (the 60’s for me) we were told that group theory was important for quantum mechanics, but somehow even a 1 semester graduate course back then didn’t get into it.  Ditto for the QM course I audited a few years ago.

I’ve been nibbling about the edges of this stuff for half a century, and it’s time to go for it.  Chemical posts will be few and far between as I do this, unless I run into something really interesting (see https://luysii.wordpress.com/2012/02/18/a-new-way-to-study-reaction-mechanisms/).  Not to worry –plenty of interesting molecular biology, and even neuroscience will appear in the meantime, including a post about article showing that just about everything we thought about hallucinogens is wrong.

So, here’s my long and checkered history with groups etc.  Back in the day we were told to look at “The Theory of Groups and Quantum Mechanics” by Hermann Weyl.  I dutifully bought the Dover paperback, and still have it (never throw out a math book, always throw out medical books if more than 5 years old).  What do you think the price was — $1.95 — about two hours work at minimum wage then.  I never read it.

The next brush with the subject was a purchase of Wigner’s book “Group Theory and its Application to the Quantum Mechanics of Atomic Spectra”  — also never read but kept.  A later book (see Sternberg later in the post)  noted that the group theoretical approach to relativity by Wigner produced the physical characteristics of mass and spin as parameters in the description of irreducible representations.  The price of this one was $6.80.

Then as a neurology resident I picked up “Group Theory” by Morton Hammermesh (Copyright 1962).  It was my first attempt to actually study the subject.  I was quickly turned off by the exposition.  As groups got larger (and more complicated) more (apparently) ad hoc apparatus was brought in to explain them — first cosets, then  subgroups, then normal subgroups, then conjugate classes.

That was about it, until retirement 11 years ago.  I picked up a big juicy (and cheap) Dover paperback “Modern Algebra” by Seth Warner — a nice easy introduction to the subject.

Having gone through over half of Warner, I had the temerity to ask to audit an Abstract Algebra course at one of the local colleges.  I forget the text, but I didn’t like it (neither did the instructor).  We did some group theory, but never got into representations.

A year or so later, I audited a graduate math abstract algebra course given by the same man.  I had to quit about 3/4 through it because of an illness in the family.  We never got into representation theory.

Then, about 3 years ago, while at a summer music camp, I got through about 100 pages of “Representations and Characters of Groups” by James and Liebeck.  The last chapter in the book (starting on p. 366) was on an application to molecular vibration.  The book was hard to use because they seemed to use mathematical terms differently than I’d learned — module for example.   I was used to.  100 pages was as far as I got.

Then I had the pleasure of going through Cox’s book on Galois theory, seeing where a lot of group theory originated  (along with a lot of abstract algebra) — but there was nothing about representations there either.

Then after giving up on Willock, a reader suggested  “Elements of Molecular Symmetry” by Ohrn.  This went well until p. 28, when his nomenclature for the convolution product threw me.

So I bought yet another book on the subject which had just come out “Representation Theory of Finite Groups” by Steinberg.  No problems going through the first 50 pages which explains what representations, characters and irreducibles are.  Tomorrow I tackle p. 52 where he defines the convolution product.  Hopefully I’ll be able to understand it an go back to Ohrn — which is essentially chemically oriented.

The math of it all is a beautiful thing, but the immediate reason for learning it is to understand chemistry better.  I might mention that I own yet another book “Group Theory and Physics” by Sternberg, which appears quite advanced.  I’ve looked into it from time to time and quickly given up.

Anyway, it’s do or die time with representation theory.  Wish me luck

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