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 with as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)
The motivation behind this definition is a somewhat long story. When is the Weyl group of an algebraic group with Borel subgroup , the above relations describe the algebra of functions on which are bi-invariant with respect to the left and right actions of under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group , and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing and with and gives the Hecke operators in the theory of modular forms.
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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 also have served as a great introduction to this series before I started it:
There’s a widespread impression that number theory is about numbers, but I’d like to correct this, or at least supplement it. A large part of number theory – and by the far the coolest part, in my opinion – is about a strange sort of geometry. I don’t understand it very well, but that won’t prevent me from taking a crack at trying to explain it….
Before we talk about localization again, we need some examples of rings to localize. Recall that our proof of the description of also gives us a description of :
Theorem: consists of the ideals where is irreducible, and the maximal ideals where is prime and is irreducible in .
The upshot is that we can think of the set of primes of a ring of integers , where is a monic irreducible polynomial with integer coefficients, as an “algebraic curve” living in the “plane” , which is exactly what we’ll be doing today. (When isn’t monic, unfortunate things happen which we’ll discuss later.) We’ll then cover the case of actual algebraic curves next.
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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 . One way to see this is to look at their zeta functions.
The usual zeta function reflects the structure of prime factorization through its Euler product
where the product runs over all primes; this is essentially equivalent to unique factorization. Since we know that monic polynomials over also enjoy unique factorization, it’s natural to ask whether there’s a sum over monic polynomials that would give a similar Euler product.
In fact, there is such a product, and investigating it leads naturally to a seemingly unrelated subject: the combinatorics of words.
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