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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The arithmetic plane

If you haven’t seen them already, you might want to read John Baez’s week205 and Lieven le Bruyn’s series of posts on the subject of spectra. I especially recommend that you take a look at the picture of to which Lieven le Bruyn links before reading this post. John Baez’s introduction to week205 would probably [...]

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The cyclotomic identity and Lyndon words

In number theory there is a certain philosophy that is a good toy model for the integers . The two rings share an important property: they are basically the canonical examples of Euclidean domains, hence PIDs, hence UFDs. However, many number-theoretic questions involving prime factorization over are much easier than their corresponding questions over . [...]

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Newton’s sums, necklace congruences, and zeta functions

The goal of this post is to give a purely combinatorial proof of Newton’s sums which would have interrupted the flow of the previous post. Recall that, in the notation of the previous post, Newton’s sums (also known as the first Newton-Girard identity) state that . One way to motivate a combinatorial proof is to [...]

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Young diagrams, q-analogues, and one of my favorite proofs

I’ve decided to start blogging a little more about the algebraic combinatorics I’ve learned over the past year. In particular, I’d like to present one of my favorite proofs from Stanley’s Enumerative Combinatorics I. The theory of Young tableaux is a great example of the richness of modern mathematics: although they can be defined in [...]

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The magic of the Frobenius map II

Once upon a time I discussed some interesting uses of the Frobenius map to solve some Putnam-style problems. Unfortunately, I wrote that post before becoming really interested in combinatorics, so I neglected to develop that particular side of the story, which I’d like to do now. The beginning of this story is the folklore combinatorial [...]

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