Update, and the combinatorics of quintic equations

A brief update. I’ve been at Cambridge for the last week or so now, and lectures have finally started. I am, tentatively, taking the following Part II classes: Riemann Surfaces Topics in Analysis Probability and Measure Graph Theory Linear Analysis (Functional Analysis) Logic and Set Theory I will also attempt to sit in on Part [...]

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Test your intuition: consecutive tails

Something very unfortunate has happened: several things I have recently written that could have been blog entries are instead answers on math.SE! In the interest of exposition beyond the Q&A format I am going to “rescue” one of these answers. It is an answer to the following question, which I would like you to test [...]

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Walks on graphs and statistical mechanics

I finally learned the solution to a little puzzle that’s been bothering me for awhile. The setup of the puzzle is as follows. Let be a weighted undirected graph, e.g. to each edge is associated a non-negative real number , and let be the corresponding weighted adjacency matrix. If is stochastic, one can interpret the [...]

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Hecke algebras and the Kazhdan-Lusztig polynomials

The Hecke algebra attached to a Coxeter system is a deformation of the group algebra of defined as follows. Take the free -module with basis , and impose the multiplicative relations if , and otherwise. (For now, ignore the square root of .) Humphreys proves that these relations describe a unique associative algebra structure on [...]

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Chevalley-Bruhat order

Before we define Bruhat order, I’d like to say a few things by way of motivation. Warning: I know nothing about algebraic groups, so take everything I say with a grain of salt. A (maximal) flag in a vector space of dimension is a chain of subspaces such that . The flag variety of is, [...]

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The strong exchange condition

It’s nice that Weyl groups are Coxeter groups and all, but the definition of a Coxeter group as a group with a particular kind of representation doesn’t immediately tell us why this is the appropriate level of generalization (although the faithfulness of the geometric representation is a good sign). It turns out there is a [...]

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Coxeter groups

At SPUR this summer I’ll be working on the Kazhdan-Lusztig polynomials, although my mentor and I haven’t quite pinned down what problem I’m working on. I thought I’d take the chance to share some interesting mathematics and also to write up some background for my own benefit. I’ll mostly be following the second half of [...]

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Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on [...]

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The Jacobi-Trudi identities

Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let be a partition. A semistandard Young tableau of shape is a filling of the Young diagram of with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau is defined as where [...]

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The Lindstrom-Gessel-Viennot lemma

One of my favorite results in algebraic combinatorics is a surprisingly useful lemma which allows a combinatorial interpretation of the determinant of certain integer matrices. One of its more popular uses is to prove an equivalence between three other definitions of the Schur functions (none of which I have given yet), but I find its [...]

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