From: dheister@magnus.acs.ohio-state.edu (David J Heisterberg) Newsgroups: sci.chem Subject: Re: basic valence question Date: 11 Jan 1996 22:29:45 GMT In article <4d3tph$6ko@cloner2.ix.netcom.com>, <jaedba@ix.netcom.com> wrote: > Would you perhaps tell me what field of study this sort of thing in >concerned with? What classes might I take to learn more about this? You'll want (and indeed who wouldn't want!) to study group theory, especially representation theory. While basically a "pure math" book, one I really like (as I've read it 3 times and still haven't really understood all of it) is Simon Altmann's "Rotations, Quaternions, and Double Groups" from Oxford University Press. It's conversational almost, and goes out of its way not to initimidate, but yet it's very meaty. In chemistry there are several families of groups that are really important. There is R(4) and R(3)*, which apply to atomic problems; the point and space groups, which cover the symmetry of molecules and the solid state; the symmetric group S(N), which deals with the interchange of electrons; SU(2) (which has a very intimate relationship with R(3)), that describes electron spin; and the unitary group U(N), which comes into play when you look at the second quantized form of the hamilitonian (the H in the Schroedinger eqn). There's also a fascinating relationship between SU(2), S(N), and U(N) [the N's here need not be the same] that I understand has yet to be fully exploited. >obviously I had a very simplified view!) Join the club! *Actually there are some approaches to characterizing the atomic problem that use the properties of R(5) and higher dimensional versions of the rotation group. They lose me completely. Dave Heisterberg

From: Eric Lucas <lucas@superlink.net> Newsgroups: sci.chem,sci.math Subject: Re: Group Theory Date: Sun, 25 Aug 1996 13:16:44 -0400 Allen, Fascinating analysis of the relationships between the two disciplines! By the way, thank you. Herstein was the one that I was trying to think of when I posted, and it's been bugging me. *Excellent* book, and as far as I know, a standard for upper level undergrad math students. I think between Herstein and Cotton, he will slake his interests quite admirably. 'Course, if he also wanted to major in math (sounds like that might be a wise course, given his interests), the other ones are indispensible as well. Cheers, Eric Lucas Allen Adler wrote: > > > Can anyone recommend a good group theory book that doesn't proceed too > > fast? I am an organic chemist that has a secret love of math. Since > > I am a organic man, I probably won't see really complete coverage of > > group theory in any of my classes, but I am really interested in > > learning more than I find in the 50-60 pages in my PChem class that > > just blow over the subject. > > > > Ideally, I am looking for a book that doesn't blow over the math. > > Actually, I would love to find a book that gives at least as much > > attention to the mathematical concepts as the chemical applications (I > > want to learn about mathematical groups in general, not just point > > groups and how to determine bonding, vibration, etc...). FInally, the > > perfect book would be one designed to be read by a guy like me (it > > shouldn't require an acompaning lecture to get the full effect > > (preferably not a textbook, but ok if it is readable)). > > I've looked at books aimed at chemists and audited a few classes > for chemists. There are certain things that tend to isolate chemists > from mathematicians when it comes to learning group theory, so it > is important to address these things. Hopefully by doing so, it will > be easier for each clan to read the other's literature. > > (1) The groups that are most enphaasized for chemists are > subgroups of the group of rigid motions of 3 dimensional space. > This offers the pedagogical advantage that it is easier to > draw pictures and to motivate the concepts visually and > geometrically. It offers the simultaneous disadvantage that > often the concepts are not discussed in isolation from their > 3-dimensional context. > (2) The notation used by chemists to name the groups seems to be > notation going back to Schlaeffli in the 19th century. This > notation is no longer standard among mathematicians. This makes > it hard for chemists to compare notes with mathematicians. > (3) Although chemists learn the definition of a group (more or less, > depending on the book and the instructor), it is more accurate > to say that they learn about the isomorphism classes of pairs > (G,r) consisting of a group G and a 3-dimensional representation > of G. > (4) More generally, whereas math students are often taught about > groups but not about representations (and in some places, it is > possible to get into math grad school without knowing what a group > is), that which is taught to chemists is a mixture of group theory > and representation theory. So some of what the chemists are > learning is more advanced than that which is normally taught > to math majors. Furthermore, the distinctions between the group > theory and the representation theory is not emphasized. > (5) Since crystals are important in chemistry and crystals are > associated with lattices, the study of group theory in chemistry > is not separate from the study of 3-dimensional representations > of finite groups on lattices, something also beyond that which > is taught to math majors. > (6) There are literally hundreds of groups and representations > of importance in chemistry and there is an extensive notation > for them. > (7) In addition to the finite dimensional representations of these > finite groups, there is also their use in connection with > the Schroedinger equation, vibrational modes, etc. So a lot > of mathematics comes into the chemical perspective on groups. > (8) One stumbling block for chemists is the treatment of the > Dirac equation, since it is hard for them to visualize the > two sheeted cover of the orthogonal group. This is one place > where the emphasis on a purely visual treatment of groups > backfires. > (9) This and other examples also show that the study of groups in > chemistry is not the abstract algebraic theory of groups but > rather the study of Lie groups. It just happens that most of > the Lie groups that come up in chemistry are finite, but not > all. Most undergraduate math majors never hear about Lie groups. > (10) The study of Lie groups, especially SU(2) or O(3), would do a > lot to dispell the mystery which enshrouds the chemical pedagogy > when it discusses the different energy levels of electrons in > hydrogen and other substances. > > If I were a chemist wanting to learn group theory for its own sake, > I would first recognize that that which I have been calling group > theory comprises several subjects studied by mathematicians. > Accordingly, there are several things I would look at: > (1) Herstein's Topics in Algebra > (2) books on representation theory of finite groups. There are none > that I especially like, but you could look at: > (a) Dornhoff's book > (b) Curtis and Reiner (their first book) > (3) Schlaffli, if I could read the language. > (4) Chevalley's book on Lie groups > (5) A book on Lie algebras and their representations. This is closely > related to the corresponding study for Lie groups, at least the > compact ones. > (6) Cotton's book (which is pleasant reading, but not a primary source > for mathematics). > (7) A book which deals with the representation theory of SL(2,C), > including the infinite dimensional unitary representations. > I think Gelfand, Graev and Piatetskii-Shapiro does this > very concretely in vol.5 of their series on generalized > functions. > (8) Weyl's The Classical Groups. Hochschild said that one cannot > learn the subject for the first time from that book but that > once one already knows the subject, one never stops learning > from this book. I am inclined to agree. > (9) If you really want to study groups for their own sake without > worrying about chemistry, you might also look at Gorenstein's > book Finite Groups and his book Simple Groups: A Guide To Their > Classification. > (10) Another book that might be fun is Ehrenfeucht's The Cube Made > Interesting, which discusses group theory and comes with 3-d > glasses. > (11) The study of groups is greatly enriched by seeing their > applications in other parts of mathematics. One of the > very striking ones is Galois theory, which is treated in > Herstein's book and in Artin's book. Galois invented the > concept of a group as part of his study of the solvability > of equations by radicals. Another is the use of groups in > topology, such as the fundamental group. > (12) You might find the book of Singer and Thorpe useful for topology > and differentiable manifolds. > (13) Read Felix Klein's Lectures on the Icosahedron and Equations of > the Fifth Degree.

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