Posted in group theory, measure theory, number theory, probability, statistical mechanics, tagged compactness, partition functions, profinite groups, q-analogues, universal properties, zeta functions on November 9, 2010 |
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An interesting result that demonstrates, among other things, the ubiquity of in mathematics is that the probability that two random positive integers are relatively prime is . A more revealing way to write this number is , where
is the Riemann zeta function. A few weeks ago this result came up on math.SE in the following form: if you are standing at the origin in and there is an infinitely thin tree placed at every integer lattice point, then is the proportion of the lattice points that you can see. In this post I’d like to explain why this “should” be true. This will give me a chance to blog about some material from another math.SE answer of mine which I’ve been meaning to get to, and along the way we’ll reach several other interesting destinations.
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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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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 Humphreys.
A Coxeter system is a group together with a generating set and presentation of the form
where , and . (When there is no relation between , we write this as .) The group is a Coxeter group, and is usually understood to come with a preferred presentation, so we will often abuse terminology and use “group” and “system” interchangeably. is also referred to as the set of simple reflections in , and the rank. (We will only consider finitely-generated Coxeter groups.)
Historically, Coxeter groups arose as symmetry groups of regular polytopes and as Weyl groups associated to root systems, which in turn are associated to Lie groups, Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a non-technical overview of Coxeter groups and -analogues, I recommend John Baez’s week184 through week187. The slogan you should remember is that Weyl groups are “simple algebraic groups over .”
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