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A ‘Useless’ Perspective That Transformed Mathematics
- memexy 6y agoThis is a really good article on group theory and their representations but the article title is unfortunate. It could have just been "Representation theory and how it transformed mathematics".
- klyrs 6y agoPerhaps The Unreasonable Effectiveness of Representation Theory
- memexy 6y agoAlso a good title.
- DC-3 6y agoDon't we have enough of these memetic titles? It's hard to go a day without 'X for fun and profit' or 'Y considered harmful'.
- memexy 6y agoHe's referencing "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" by the physicist Eugene Wigner (https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...). > In the paper, Wigner observed that the mathematical structure of a physical theory often points the way to further advances in that theory and even to empirical predictions. I think it's a good analogy since it applies exactly how Wigner meant it except in this case it is a branch of mathematics being applied to mathematics instead of physics. Representation theory has been "unreasonably useful and effective" which is what the original article was about.
- klyrs 6y agoHi, I'm a woman. We still exist in tech spaces. Please use "they" when you don't know somebody's gender, or "she" for me now that you know. Thanks
- hanche 6y agoIn Frederik Pohl’s The years of the city, gendered pronouns are completely gone. Instead of he or she, there is “e”. His/her is “er”, and him/her is “um”. Too bad it didn’t catch on.
- memexy 6y agoNoted. But I don't pay much attention to user names so if we run into each other again I'm likely to make the same mistake.
- klyrs 6y agoI don't care if you remember me. Remember this. > Please use "they" when you don't know somebody's gender
- deleted 6y ago[deleted]
- hrktb 6y agoWe collectively accepted that titles are supposed to be memetic though. For instance readability is second to shortness. Accuracy is lower priority to impact. Our whole history with titles points to short and punchy catch copies. Purely descriptive titles only work when the reader will read the paper anyway (e.g. peer reviewing), which is far from the setting we have on HN.
- woopwoop 6y agoI like the title, although maybe it could have included the phrase "representation theory". I did not know that Burnside expected so little of representation theory; I think it's an interesting and important part of the article.
- memexy 6y agoEveryone is biased in one way or another. I'm sure Burnside had good reasons at the time.
- biddlesby 6y agoEver since my undergraduate I've wanted to understand what representation theory is and the rough outline of how Andrew Wile's proof was constructed. This article gave me both in a very understandable and engaging way. Thank you to the author!!!
- quickthrower2 6y agoI still want to know what was the proof Fermat wrote was that couldn't fit in a margin.
- dllthomas 6y agoIt seems there's a fair chance it was a flawed proof, in which case I doubt we'll ever be particularly confident although we could possibly surface some candidates.
- DC-3 6y agoSomething I always wondered is if we know of the existence of a proof that is both simple and also non-obviously flawed and as such could have been Fermat's solution?
- pathsjs 6y agoYes, Lamé solution, which assumes unique factorization in the rings of integers of cyclotomic fields, and can in fact be salvaged to prove the case of regular primes
- quickthrower2 6y agoThanks while I don't fully understand what this means I think I get the idea. I assume it's a proof that has the "n" over infinite many cases (primes) but not all of them? Googling it was hard because there are other Lamé things. I'll just imagine he wrote this proof. Not the full proof, but not lame!
- ur-whale 6y agoVery good article in that it taught me something I didn't know existed, and explained it very well. However, I wish the article gave an example on how to actually construct a mapping between e.g. a small finite group and the actual matrices in the representation, to - sort of - get a feel for why that actually works. I suspect there must be some sort of canonical method/algorithm to get from the "multiplication table" of a finite group to each matrix in a representation, but I haven't been able to find a reference. Would anyone have pointers? Also, jumping from the group to the character table (which seem to imply that there is indeed an algorithm to compute all possible representations) without having been told how the mapping is constructed feels like a rather big mental jump (and what makes the trace of the matrices important, btw - rather than, say, the determinant?).
- eigenket 6y agoGiven the multiplication table of a (finite) group there are a couple of "obvious" representations you can pick. Firstly theres always the trivial representation that just sends everything to the identity. More interestingly you can use the group elements as labels for an orthonormal basis for a vector space (e.g. your orthonormal set is {v_g for all g in G}). Then have the representation act by the usual group operation on the labels R_h[v_g] = v_{hg}. Then once you have an operation defined on a basis extending it linearly to the rest of the vector space is easy. This is (left) regular representation. https://en.wikipedia.org/wiki/Regular_representation https://en.wikipedia.org/wiki/Regular_representation Edit: Intuition for why the trace is important comes from looking at permutation matrices - the trace is the sum of the diagonal elements of the matrix and the number of 1s on the diagonal is exactly the number of points that are invariant under the permutation - e.g. the permutation [[1,0,0,0], [0,1,0,0], [0,0,0,1], [0,0,1,0]] has trace 2, and the permutation leaves the first two elements unchanged and flips the last two.
- JadeNB 6y agoTwo remarks on this nice example: (1) The word 'orthonormal' doesn't have to be there; representations are a priori just linear actions; if they carry an invariant inner product, that is a bonus ('unitary representation'). For compact (including finite) groups, the invariant inner product is no extra information, because we could pick any (not necessarily invariant) inner product and average it with respect to Haar measure to get an invariant one. This product is unique up to a scalar for an irreducible representation, but not in general (and the left-regular representation is never irreducible for a non-trivial finite group). (2) For people who are more used to cycle notation than the matrix representation of permutations, your permutation is (3 4). In roster notation, it's 1 2 3 4 1 2 4 3
- chmaynard 6y agoPeter Woit posted some comments: https://www.math.columbia.edu/~woit/wordpress/?p=11776 https://www.math.columbia.edu/~woit/wordpress/?p=11776
- emmelaich 6y agoGeordie Williamson has a fascinating history. He's the youngest living Fellow of the Royal Society. https://en.wikipedia.org/wiki/Geordie_Williamson https://en.wikipedia.org/wiki/Geordie_Williamson
- nickray 6y agoAnybody have an actual quote of Burnside's doubt? Article etc.
- CrazyStat 6y agoI believe the Burnside quote comes from the preface of his 1897 book: > It may then be asked why, in a book which professes to leave all applications on one side, a considerable space is devoted to substitution groups; while other particular modes of representation, such as groups of linear transformations, are not even referred to. My answer to this question is that while, in the present state of our knowledge, many results in the pure theory are arrived at most readily by dealing with properties of substitution groups, it would be difficult to find a result that could be most directly obtained by the consideration of groups of linear transformations.
- sva_ 6y ago>“Mathematicians basically know everything there is to know about matrices. It’s one of the few subjects of math that’s thoroughly well understood,” said Jared Weinstein of Boston University. I'm genuinely curious how such a statement can be made. I've recently been wondering about if it were possible to prove that one's theorems are exhaustive about a 'space' of possible theorems in an axiomatic system (or some subset thereof, since I'd assume such a space might be infinite (perhaps)). How can we know that there aren't some really surprising properties of matrices that we've previously been unaware of? As far as I can see, we can merely make a statement about that it fits well together with other (limited) findings we've made so far?
- bollu 6y agoIt's more that in linear algebra, we have "the nicest theorems we can hope for", so there's no "nasty surprises" lurking around. This is qualitatively different from pretty much all other algebraic structures one studies, where there is some amount of nastiness which we attempt to control.
- Koshkin 6y agoIndeed, “linear” and “nice” are almost synonymous in mathematics and, especially, in its applications. (For example, the entire differential calculus is based on the idea of approximating arbitrary “nice” functions with linear maps.)
- eru 6y agoThere's also lots of clever tricks to make things more linear. There's things like linear approximations. Or, my favourite, picking a clever semi-ring in which your problem is linear. See eg the analysis of Google's BBR TCP variant. Eg https://labs.xjtudlc.com/labs/wldmt1/papers/Traffic%20management%20congestion%20control,%20router%20mechanisms%20and%20pricing/BH00.pdf https://labs.xjtudlc.com/labs/wldmt1/papers/Traffic%20manage... Or 'Linear regression over the max-plus semiring:algorithms and applications' (https://arxiv.org/pdf/1712.03499.pdf https://arxiv.org/pdf/1712.03499.pdf)
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- poletopole 6y agoSometimes I go deep wiki some nights in mathematics. Representation theory and Set Theory are the only sane ways I know of on how to approach higher mathematics. But it’s their practical applications in software that amazes me. Take Young Tabs (https://en.wikipedia.org/wiki/Young_tableau https://en.wikipedia.org/wiki/Young_tableau) and Hook Lengths for example. I’ve been playing with the concept that you could use Young Tabs and Hook Lengths to represent groups of FSMs in metric space if you wanted to know mathematically if one FSM could be topologically sorted into a congruent FSM.
- ssivark 6y agoFascinating! Any references elaborating on this? (By FSMs, I suppose you mean Finite State Machines?)
- IngoBlechschmid 6y agoRepresentation theory, the subject of the submitted article, is also at the core of our understanding of the particle zoo in physics. A tutorial aimed at mathematicians is here: John Baez, John Huerta. The Algebra of Grand Unified Theories. https://arxiv.org/abs/0904.1556 https://arxiv.org/abs/0904.1556
- benrbray 6y agoGreat link, thanks! I alway enjoy reading summaries of other fields written by/for mathematicians, since the assumption of mathematical background allows a lot of fluff to be cut out (compared to, e.g. reading a textbook about physics intended for physics students). For background reading, the notes by Teleman 2005, "Representation Theory" [1] are a good intro to the topic. Learning about representation theory helped me understand the power of thinking about mathematical objects in terms of their action on other objects. Just as representation theory studies group actions on vector spaces, the theory of modules is best described as the study of ring actions on commutative groups. Many things happen to be rings (e.g. endomorphism rings of functions, where addition=pointwise addition and multiplication=composition), and modules allow us to apply (almost all of) vector space theory to better understand ring-like objects. [1] https://math.berkeley.edu/~teleman/math/RepThry.pdf https://math.berkeley.edu/~teleman/math/RepThry.pdf [2] Another favorite from John Baez: https://groups.google.com/d/msg/sci.physics.research/aiMUJrOjE8A/jGy2N3IaajwJ https://groups.google.com/d/msg/sci.physics.research/aiMUJrO...
- lawrenceyan 6y agoAre there any textbooks that provide a good introduction into Representation Theory, assuming a decent level of undergraduate mathematics as a foundation?
- cosmic_ape 6y agoNot a standard source for algebraists, but Percy Diaconis's book, mentioned in other comments, is an excellent introduction. Also contains applications in probability.
- benrbray 6y agoI recently did a deep dive into the topic, so here's a list of things I came across that helped me: [1] Teleman 2005: https://math.berkeley.edu/~teleman/math/RepThry.pdf https://math.berkeley.edu/~teleman/math/RepThry.pdf [2] Khonvanov List of Repr Theory Resources http://www.math.columbia.edu/~khovanov/resources/ http://www.math.columbia.edu/~khovanov/resources/ [3] Huang 2010, "Fourier-Theorietic Probabalistic Inference over Permutations" [4] Woit, Topics in Representation Theory Course, http://www.math.columbia.edu/~woit/repthy.html http://www.math.columbia.edu/~woit/repthy.html [5] Gallier 2013, "Spherical Harmonics and Linear Representations of Lie Groups" https://www.cis.upenn.edu/~cis610/sharmonics.pdf https://www.cis.upenn.edu/~cis610/sharmonics.pdf
- lawrenceyan 6y agoThank you!
- krackers 6y agoSee also the newer post: https://news.ycombinator.com/item?id=23549897 https://news.ycombinator.com/item?id=23549897 which has another presentation of this material.