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Archive for July, 2010

SPUR is finally over! Instead of continuing my series of blog posts, I thought I’d just link to my paper, Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials, and the first few blog posts should more or less provide enough background to read it.

My project ended up changing direction. The formula I was working with for the quadratic coefficient was so unwieldy that I ended up spending the whole time trying to simplify it, and instead of saying anything about non-negativity I ended up saying something about combinatorial invariance. The combinatorial invariance conjecture, which goes back to Lusztig and, independently, Dyer, says that the Kazhdan-Lusztig polynomial P_{u,v}(q) depends only on the poset structure of [u, v]. In the special case that u = e this was proven in 2006 by Brenti, Caselli, and Marietti. However, the conjecture is still open in general.

In particular, explicit nonrecursive formulas in which each term only depends on poset-theoretic data are not known in general. They are known in the case that the length \ell(u, v) of the interval [u, v] is less than or equal to 4, and there is also such a formula for the coefficient of q of P_{e,v}(q) where e is the identity. The main result of the paper is a formula for the coefficient of q^2 of P_{u,v}(q) in which all but three of the terms depend only on poset data, which is a simplification of a general formula due to Brenti for P_{u,v}(q) in terms of lattice paths. It reduces to

  • a formula for the coefficient of q^2 of P_{e,v}(q) in which all but one of the terms depends only on poset data,
  • a formula for the coefficient of q^2 of P_{u,v}(q) where \ell(u, v) = 5 in which all but one of the terms (but a different term) depends only on poset data (not in the paper), and
  • a formula for the coefficient of q^2 of P_{e,v}(q) where \ell(u, v) = 6 in which every term depends only on poset data.

I believe these formulas are known in some form, but the method of proof is likely to be novel. In any case, the troublesome terms in the above are all essentially coefficients of R-polynomials. If I revisit this project in the future, I will be focusing my attention on these coefficients, and my goal will be to find a poset-theoretic formula for P_{u,v}(q) in the length 5 case, the smallest-length case where (to my knowledge) combinatorial invariance is open.

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The new StackExchange math site has finally gone out of private beta. I would like this site to succeed, and I had forgotten how much fun it is to actually be able to answer questions, so I hope everything works out. Come on over!

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The Hecke algebra attached to a Coxeter system (W, S) is a deformation of the group algebra of W defined as follows. Take the free \mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]-module \mathcal{H}_W with basis T_w, w \in W, and impose the multiplicative relations

T_w T_s = T_{ws}

if \ell(sw) > \ell(w), and

T_w T_s = q T_{ws} + (q - 1) T_w

otherwise. (For now, ignore the square root of q.) Humphreys proves that these relations describe a unique associative algebra structure on \mathcal{H}_W with T_e 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 W is the Weyl group of an algebraic group G with Borel subgroup B, the above relations describe the algebra of functions on G(\mathbb{F}_q) which are bi-invariant with respect to the left and right actions of B(\mathbb{F}_q) under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group G, and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing G and B with \text{SL}_2(\mathbb{Q}) and \text{SL}_2(\mathbb{Z}) gives the Hecke operators in the theory of modular forms.

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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 V of dimension n is a chain V_0 \subset V_1 \subset ... \subset V_n of subspaces such that \dim V_i = i. The flag variety of G = \text{SL}(V) = \text{SL}_n is, for our purposes, the “space” of all maximal flags. \text{SL}_n acts on the flag variety in the obvious way, and the stabilizer of any particular flag is a Borel subgroup B. If e_1, ... e_n denotes a choice of ordered basis, one can define the standard flag 0, \text{span}(e_1), \text{span}(e_1, e_2), ..., whose stabilizer is the space of upper triangular matrices of determinant 1 with respect to the basis e_i. This is the standard Borel, and all other Borel subgroups are conjugate to it. Indeed, it’s not hard to see that \text{SL}_n acts transitively on the flag variety, so the flag variety can be identified with the homogeneous space G/B.

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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 structural property, the strong exchange condition, which completely characterizes Coxeter groups among groups generated by involutions. Today we will prove this property.

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