Ever since my undergraduate I've wanted to understand what representation theory is and the rough outline of how Andrew Wile's proof was constructed.<p>This article gave me both in a very understandable and engaging way. Thank you to the author!!!
Representation 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:<p>John Baez, John Huerta. The Algebra of Grand Unified Theories. <a href="https://arxiv.org/abs/0904.1556" rel="nofollow">https://arxiv.org/abs/0904.1556</a>
Very good article in that it taught me something I didn't know existed, and explained it very well.<p>However, I wish the article gave an example on how to actually construct a mapping between e.g. a small finite group and the <i>actual</i> matrices in the representation, to - sort of - get a feel for why that actually works.<p>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.<p>Would anyone have pointers?<p>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 <i>how</i> 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?).
Peter Woit posted some comments:<p><a href="https://www.math.columbia.edu/~woit/wordpress/?p=11776" rel="nofollow">https://www.math.columbia.edu/~woit/wordpress/?p=11776</a>
This 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".
Geordie Williamson has a fascinating history. He's the youngest living Fellow of the Royal Society.<p><a href="https://en.wikipedia.org/wiki/Geordie_Williamson" rel="nofollow">https://en.wikipedia.org/wiki/Geordie_Williamson</a>
Sometimes 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.<p>Take Young Tabs (<a href="https://en.wikipedia.org/wiki/Young_tableau" rel="nofollow">https://en.wikipedia.org/wiki/Young_tableau</a>) 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.
>“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.<p>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)).<p>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?
Are there any textbooks that provide a good introduction into Representation Theory, assuming a decent level of undergraduate mathematics as a foundation?
See also the newer post: <a href="https://news.ycombinator.com/item?id=23549897" rel="nofollow">https://news.ycombinator.com/item?id=23549897</a> which has another presentation of this material.