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Representation theory?

  1. Sep 20, 2007 #1
    I am going to ask some general questions about representation theory which may sound stupid as I don't know anything about it.

    How old or new is this theory? How did it come out?

    What is the general concensus of ths difficulty of the subject?

    Are there many open problems in this theory?

    Is it connected to modules in any way since they both deal with matrices in some way?

    It seems like a very general theory with deep connections to topology and analysis?

    How much connection has it got to physics and could it uncover some deep physical theories?

    I recall Matt Grime calling it 'abstract nonsense'. Why?
     
    Last edited: Sep 21, 2007
  2. jcsd
  3. Sep 20, 2007 #2
    I believe it's not that old -- but group theory isn't that old either. It's a pretty obvious way to study groups.

    The difficulty of a subject depends on yourself. You choose to go deep or not.

    There are always open problems.

    Yes. You can try to find representations (or some people call it realizations) over modules rather than linear spaces. The matrices are a red herring.

    All of maths is connected.

    In general, we often talk about representations of abstract structures -- not just groups over linear spaces. In particular, even in representations of infinite groups over linear spaces, other structures in the linear space may be important, e.g. Banach-ness, Hilbert-ness, etc.
     
  4. Sep 20, 2007 #3
    How old quantitatively?

    But some maths is more connected then others?

    'Representations of abstract structures'? Then this theory seems very general and 'connected'!
     
  5. Sep 20, 2007 #4

    Chris Hillman

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    Tiny sketch of history and scope of representation theory

    The theory of linear representations of finite groups was initiated c. 1896 by Georg Frobenius http://www-history.mcs.st-and.ac.uk/Biographies/Frobenius.html, who had earlier spent years working on the theory of permutation representations of finite groups. (In the first subject, given a finite group we study an isomorphic group which consists of linear transformations on some finite dimensional vector space; in the second, we study an isomorphic group which consists of permutations of some finite set. Or, sometimes, a quotient group.)

    If you know about the wonderful properties of the character table of a finite group, I urge you to take the time to learn about the analogous concept for permutation groups, the so-called table of Marks. This was introduced to math students by William Burnside, Theory of Groups of Finite Order, Cambridge University Press, 1897. In his forward, Burnside http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Burnside.html remarked that he had chosen not to cover the Greatest New Thing, namely Frobenius's theory of linear representations, because he didn't know of anything which that theory could do which couldn't be done with permutation representations. But he soon used the new theory to prove something important and became a passionate convert, as he explained in the forward to the second edition (1911) of his textbook, which did cover linear representations. So between 1898 and 1911 representation theory passed from being a promising innovation to an essential core topic.

    Incidently, legend has it that Burnside learned about the "table of Marks" from a brilliant amateur he met in the British Museum, who happened to know a German mathematician named Engel http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Engel.html who knew Frobenius and Lie http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Lie.html. But this man had a heavy accent, so Burnside misheard "Marx" and we wound up with the spectacularly misnamed "table of Marks", instead of "Frobenius table". Later, an Irish writer living in Paris, who had heard this story from a friend and was much amused by the collision of Scottish and German accents, tossed the phrase "three quarks for muster Mark" into his latest novel, which later inspired Gell-Mann... but I digress. My point was that while I am a huge fan of the table of Marks, it lacks the orthogonality properties which make the character table so useful for enumerating useful information about representations.

    (Burnside's textbook was responsible for another famous misnomer. He stated and proved a lemma which was well known to German and French mathematicians of the day, and which was due in part to Lagrange and in part to Frobenius. But some mathematicians mistakenly attributed it to Burnside, so for a long time this result was incorrectly known in the English language literature as the Burnside lemma. And it doesn't end there: the Burnside lemma, or Cauchy-Frobenius lemma as it is now known, is needed for Polya enumeration theory, which was indeed independently discovered by Polya http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Polya.html but which later turned out to have been earlier published by a forgotten American mathematician, Redfield.)

    Pretty soon mathematicians like Hermann Weyl http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Weyl.html jumped in and initiated the analogous theory of linear representations of Lie groups, especially the so-called classical Lie groups. Deep and beautiful connections soon came to light between the theory of representations of [itex]SL(2,C)[/itex] and [itex]S_n[/itex], and pretty soon there were highly developed and closely interrelated theories of invariants of, (linear) representations of, and harmonic analysis on these groups. Coxeter http://www-history.mcs.st-and.ac.uk/Biographies/Coxeter.html and Dynkin http://www-history.mcs.st-andrews.ac.uk/Biographies/Dynkin.html returned the favor to Lie theory by giving a beautiful proof of the earlier classification by Cartan http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cartan.html of the finite dimensional complex simple Lie algebras. (Hmm... "complex simple Lie algebra" sounds funny if you don't how mathematicians parse this, doesn't it?!)

    Many of the great mathematicians of the twentieth century made landmark contributions to the development of representation theory, including
    Issai Schur http://www-history.mcs.st-and.ac.uk/Biographies/Schur.html, Richard Brauer http://www-history.mcs.st-and.ac.uk/Biographies/Brauer.html, Harish-Chandra http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Harish-Chandra.html, Armand Borel http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Borel_Armand.html, George W. Mackey http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Mackey.html,
    and via harmonic analysis, one might even include Andre Weil http://www-history.mcs.st-andrews.ac.uk/Biographies/Weil.html.

    The multivolume book by Curtis and Reiner http://www-gap.dcs.st-and.ac.uk/~history/Printonly/Reiner.html contains much historical background, and a quick glance at this book (which covers basic representation theory up to the 1960s or so) should answer your question about whether this is a big subject. Or if not, well, my local research library has 152 books on representation theory, so this is a monster of a big subject! Whether you want to understand the "gauge symmetries" of some gauge theory in physics, or to study the distribution of polarized sky light (for which you need tensor spherical harmonics), you need to know about representations (particularly the basic "building blocks" of the theory, the irreducible representations, or irreps for short) of the appropriate group. And don't even get me started on ergodic theory!

    There are indeed deep connections with topology and analysis (and with much more besides, such as Kleinian geometry and Cartanian geometry--- the latter subject, the common generalization of Kleinian geometry and Riemannian geometry, is currently undergoing an impressive revival.). I already hinted at a connection with harmonic analysis (a vast generalization of the theory of Fourier transforms). As for a connection with topology, I might mention the theory of covering spaces, which is closely connected to homotopy groups in algebraic topology, and I might mention the concept of the universal covering group, a simply connected Lie group which plays an important role in much modern mathematics (among many other things, this notion provides some important examples of Lie groups which are not realizable as matrix groups). And there are indeed important connections with modules. Indeed, when R is the group ring of our group, R-modules--- oh never mind, I've said enough!

    For evidence that the theory continues to grow at a daunting rate, see http://www.arxiv.org/list/math.RT/recent

    I didn't see Matt Grimes's comment but I'll go out on a limb and guess that he was thinking of representations of categories, a subject which category theorists consider to be "concrete" but which many others might term "abstract", if not nonsense. Or he might have been referring to something like diagram chasing in homological algebra, a technique which certainly might have come up while discussing the cohomology of Lie groups.

    Classic references for one man's viewpoint:

    George W. Mackey, The Scope And History Of Commutative And Noncommutative Harmonic Analysis, American Mathematical Society, 1992.

    George W. Mackey, The theory of group representations, University of Chicago Press, 1955.

    George W. Mackey, Unitary group representations in physics, probability, and number theory, Benjamin/Cummings, 1978.

    C. C. Moore, editor. Group representations, ergodic theory, operator algebras, and mathematical physics: proceedings of a conference in honor of George W. Mackey.
    Springer, 1987.

    Also relevant:

    Charles W. Curtis, Pioneers of Representation Theory, Am. Math. Soc., 1999. (Same Curtis as in Curtis and Reiner.) See also this book review: www.ams.org/bull/2000-37-03/S0273-0979-00-00867-3/S0273-0979-00-00867-3.pdf by J. E. Humphreys, author of one of many textbooks on representation theory.

    (Edit: after having written the above, I discovered that this review is on-line and not surprisingly it is a much better summary of the history of representation theory than what I came up with above.)

    Er... did I answer the question? Or at least clarify why it might hard to concisely answer the question? :wink:
     
    Last edited: Sep 20, 2007
  6. Sep 21, 2007 #5
    Very nice post Chris. What field are you from? An academic?

    How much connection is representation theory in physics? Could it uncover some deep physical theory some day?

    I see now. http://en.wikipedia.org/wiki/Category_theory
    Stuff like that make me scared because for a person with lesser mathematical ability, the difference between abstract sense and nonsense may be blured at times.

    However representation theory seems like a good field to get into because of so many connections it offers.
     
    Last edited: Sep 21, 2007
  7. Sep 21, 2007 #6
    Representation theory is only a very small part of the physicist toolbox. No physics will come from pure maths -- otherwise we wouldn't call physics physics, but maths instead. When doing physics, always remember that maths is a tool that needs to be deftly wielded, but only a tool. Forgetting that what mathematically exists vs what physically exists is a quick way to get yourself into a nice quagmire of paradoxes.
     
  8. Sep 21, 2007 #7

    Chris Hillman

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    My thesis concerned a topic in symbolic dynamics, the most abstract branch of dynamical systems theory. (But this particular topic happens to highly visual, in fact I like to characterize it as a geometric visualization of phenomena in diophantine approximation, a classical topic in number theory.)

    Bah! Physics! Who cares about physics? Too many, that's who. grrr... I would like to see someone announce the career goal of using representation theory to advance the cause of... statistics. (No joke: see my example of a question in harmonic analysis above, and search for papers on the statistical analysis of--- oh dear, physics again--variations in the polarization of the CMB background.)

    That said, it would be hard to overestimate the importance of representation theory in physics, although this really only a reflection of the importance of representation theory (and allied concepts like symmetry and decompositions into irreducible pieces) in mathematics generally.

    When gauge theories were discovered, physicists like Gell-Mann rediscovered quite a bit of what the mathematicians had done some decades earlier, and I have the sense that there was indeed an expectation of something big in the air. Indeed, one might say there was something big in the air--- the Standard Model of particle physics. So one might say that representation theory has already played an essential role in the discovery of an important theory of fundamental physics. As the books by Mackey and the one by Curtis will show, the earliest development of representation theory was heavily influenced by the need to put quantum mechanics on a sound mathematical footing. Recall that in 1900, Hilbert space was an unknown concept; what we now call functional analysis, operator theory, ergodic theory all arose around the same time that mathematicians were developing representation theory into the rich subject found in modern textbooks. As I already said, the theory continues to advance--- oh, I forgot to say that there are indeed important open problems, in fact just recently there was a notable advance http://www.aimath.org/E8/ See Jeffrey Adams and David Vogan, Representation Theory of Lie Groups for short courses on advances since 1975, and see http://www.math.umd.edu/~jda/minicourse/ for some more exposition by Adams.
    http://www-math.mit.edu/~dav/ref.html

    Bah! Nothing to be scared of, if you only learn the elementary stuff. Think of category theory as an organizing principle which reduces keyspace when trying to crack some mathematical problem.

    Indeed. Like many highly developed subjects, it can seem dry (truth be told, some muttered that Jeffrey Adams oversold the much-hyped computation linked to above--- I have to confess my admiration for his mastery of the art of manipulating the popular press), but there is much beautiful stuff here.

    Er, what planet did you drop in from?

    That may be true on Planet 17XW41 orbiting Gliese 581, I dunno. (Someone please tell me how the Gliesans ever got their spacecraft into orbit, in that case.)

    Define "come from". I agree that physics is not a branch of math, although at a stretch one might call it a branch of applied mathematics.

    Alas, glibly referring to a "mathematical object" as "existing" is a good way to get into a conceptual quagmire (just Google for "is mathematics invented or discovered?"). This is even more true for "physical objects". Does an electron "exist"? Does the Earth "exist"? Does Canis lupus familiaris "exist"?

    I think what you are trying to say is that mathematics is, as I like to say, the art of reliable reasoning about simple phenomena, and that interpreting the results of mathematical analysis of some natural phenomenon can be fraught with conceptual (and even philosophical) difficulties which must never be underestimated.
     
    Last edited: Sep 21, 2007
  9. Sep 21, 2007 #8

    matt grime

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    I don't recall calling representation theory abstract nonsense. It is in fact what I do research in, so I'm sure I'd remember calling it that. I may well have said some results in Rep Theory were as a consequence of abstract nonsense, i.e. a purely technical argument usually from category theoretic considerations that has little underlying connection with the situation at hand.

    However, the term representation theory has a very wide scope, so you'd be mistake for thinking it is a priori about groups. That was how it started (Frobenius and Schur), but it is a huge subject now. A representation is typically a module for an algebra, rather than a ring.

    There are many important open questions. I'm particularly interested in Broue's conjecture, for example, which relates representations of G with representations of N_G(P), the normalizer of a Sylow subgroup, at least when P is abelian. It is also called the abelian defect group conjecture.
     
  10. Sep 21, 2007 #9
    I think we agree more than the cursory comments indicate. However, as a (mathematically able) physicist often I'm assaulted with the "but that's mathematically obvious". To me at least, the physics *ends* when I can write down the equations and say what the mean. The detailed issues of getting a number at the end is part of my mathematical toolbox. To be sure, it's a huge toolbox (and gets bigger all the time), and I often bemoan the lack of mathematical sophistication in physics texts. Nevertheless, I prefer to be clear (at least in my mind if I can't manage in written text) what's a deductive process (maths) and what's an absolute guess (physics).

    I would have to say that representation theory plays a small role in physics as a whole. Standard model not withstanding, even the whole of particle physics is only a small part of physics as a whole. It is certainly true that in our current models of particle physics and quantum mechanics representation theory plays a large role in the mathematical machinery (one might actually say the central role). But the application of those models, or even perhaps extensions of them one day, may not need the theory, but only the concrete products -- matrix expressions and their manipulations at the end. Many a physicist will do wonderful quantum mechanics always using the positional basis, and never really understand that it's only one of many possible representations.
     
  11. Sep 21, 2007 #10

    Chris Hillman

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    Representations: not just useful for quantum/fundamental physics

    So do I. That's why I tried to restate the point I thought you were trying to make in my own words.

    Did we lose a sentence here? I was expecting you to say "mathematically, my problems are solved when I have written down the general solution to my equations, which is the point where the physics resumes, as I try to interpret the solution physically".

    Number?

    Looks like I already tried to restate in my own words what I think you are trying to say :wink:

    I disagree.

    I agree, but this doesn't contradict what I said just above. Did you see my reference to harmonic analysis? C'mon, I bet you have recently taken a Fourier transform or two!

    Representation theory and closely allied subjects such as harmonic analysis have myriads of applications to PDEs, dynamics, and many other things, which are essential to physics independently of their role in theories of fundamental physics or in quantum physics.
     
  12. Sep 21, 2007 #11
    Aha! It would seem that we agree on things, but not on words. My idea of what "representation theory" is stops far short of the whole theory, plus all the trimmings, for better or worse (probably worse for me in the long run :sad:). I guess my point is that the words representation theory only consciously intrude into my mind when I'm explicitly finding representations of abstract structures, rather than just enjoying the fruits thereof. Let me be more specific. I'm currently embroiled in some quantum mechanics, which is arguably full of representation theoretically relevant things. However, what I do with it can be split into two things: using the fact that different representations exist and that they reflect an underlying structure, or finding new representations (of the operator algebra, usually). The former requires me to know almost nothing about the inner workings of the mathematical theory -- almost everything can be summed up by "here are some homomorphisms". The latter however, can get seriously hairy -- deep into using the big bag o' tricks. My point is that much of physics happens as the former -- we use the results all the time, sometimes almost subconsciously.

    That said -- you have a much better grasp of the maths (and probably the physics) than I do, perhaps my view is still not sophisticated enough to see deeply yet... would hardly be the first time... :shy:
     
  13. Sep 21, 2007 #12
    One thing is that representation theory could be too hyped up with all its applications and so many research papers? ALmost like topics in theoretical physics?
     
  14. Sep 22, 2007 #13

    matt grime

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    If you're going to start a flame war, keep on that track. Or could attempt to justify where that completely unfounded assertion comes from.
     
  15. Sep 22, 2007 #14

    Chris Hillman

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    I certainly never implied that I think that representation theory is over-hyped; quite the opposite! From this and from your reporting of a sentiment which you incorrectly ascribed to Matt Grime, I suspect you haven't yet begun to learn about representation theory. If you tell us something about your math background (and what applications you think you might be particularly interested in), I can probably recommend a first textbook.

    One thing to bear in mind is that representation theory is defined much more narrowly by some. It seems to fair to say that this term generally has the connotation of representations of groups, rings, or whatever in terms of groups, rings, or whatever of linear operators. I discussed only the representation theory of groups in my posts in this thread. I would say that the representation theory of finite groups is by far the easiest piece to learn first. It is a beautiful theory with fairly few prerequisites, which gives a fair first impression of some of the important themes in the more general theory.

    Ditto Matt about critiques of contemporary physics, which clearly do not belong in this subforum.
     
    Last edited: Sep 22, 2007
  16. Sep 22, 2007 #15
    My bad, the remark by Matt was on category theory. But is that connected to representation theory? If so how?

    https://www.physicsforums.com/showpost.php?p=1266975&postcount=16

    Why is category theory considered abstract nonsense? Is it because concrete examples don't exist in it? What does Grothendieck think of it?

    I didn't know Matt had views on physics. But then again I don't know lots of things.

    I have done an upper undergraduate level course in algebra. So will need the most basic book on representation theory.
     
    Last edited: Sep 22, 2007
  17. Sep 23, 2007 #16

    matt grime

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    Do you know what a category is? Did you read the post of mine that you linked to? Particlularly the phrase along the lines of 'the underpinning of most modern mathematics'. Categories are linked to absolutely everything.

    because it is very abstract....

    no more nor less than any other subject in mathematics.

    One of the two current schools of thought on algebraic geometry is that of the categorical, a school arguably founded by Grothendieck (with Serre, Verdier, et al - see the EGA)

    I do, but where in this thread have I brought them up?
     
  18. Sep 23, 2007 #17

    matt grime

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    I have no idea what this means. Is it aimed at me?
     
  19. Sep 23, 2007 #18
    I've always thought it slightly ironic that as a physicist who's very focused on physical interpretation of mathematics, I've had a bit of a side-fetish with category theory. I think I'm just a sucker for structuralism, to be honest.
     
  20. Sep 23, 2007 #19
    IF concrete examples exist in category theory then how can it be abstract nonsense?

    To me abstract nonsense is when there are no concrete examples because if there are then one can always find counter examples to proofs. So proofs in category theory are falsifable just like other areas of maths. So when someone dubs category theory nonsense, the category theorist can just challenge them to a counter example?
     
  21. Sep 23, 2007 #20

    matt grime

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    It's not nonsense because it makes 'no sense' or is wrong. Firstly, abstract category theory is not called 'abstract nonsense'. Read the post you linked to. I said some things were true from abstract nonsense _arguments_, which I have repeatedly explained is just a short hand for saying 'formal reasons'.

    E.g. property X holds because Mod(kG) is an abelian category with arbitrary direct sums. Thus we are completely ignoring the fact that we're dealing with modules for a group algebra, and deal with an argument that has nothing to do with groups, beyond establishing that Mod is indeed an abelian category (with arbitrary direct sums).


    This line of argument is very common, and very powerful. It means you don't have to verify it using particular examples. You don't have to verify that for any ring R and left R-module that X Hom_R(X,-) is left exact by writing anything down to do with modules or rings, or Hom sets. It follows from a so-called abstract nonsense argument because all functors with right adjoints are left exact (n.b. you might have to interchange the words left and right in some parts of that sentence, I can't be bothered to think about it).

    You want more? Every cohomological functor from a compactly generated triangulated category is representable. You probably don't want to prove that for a particular stable homotopy category, even if you could do it directly.

    How about, a functor is an equivalence of categories if it is fully faithful and essentially surjective? Thus negating the need to write down an actual inverse. Technically this is a weak equivalence, since it only defines things upto natural isomorphism, and assumes the axiom of choice.
     
    Last edited: Sep 23, 2007
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