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 
![[u, v]](https://s0.wp.com/latex.php?latex=%5Bu%2C+v%5D&bg=ffffff&fg=333333&s=0&c=20201002)
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 
- a formula for the coefficient of
- a formula for the coefficient of
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
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 
is a deformation of the group algebra of 
defined as follows. Take the free ![\mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5Bq%5E%7B+%5Cfrac%7B1%7D%7B2%7D+%7D%2C+q%5E%7B+-+%5Cfrac%7B1%7D%7B2%7D+%7D%5D&bg=ffffff&fg=333333&s=0&c=20201002)
-module 
with basis 
, and impose the multiplicative relations
, and 
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.)
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 
gives the Hecke operators in the theory of modular forms.
of dimension 
is a chain 
of subspaces such that 
. The flag variety of 
is, for our purposes, the “space” of all maximal flags. 
acts on the flag variety in the obvious way, and the stabilizer of any particular flag is a 
denotes a choice of ordered basis, one can define the standard flag 
, whose stabilizer is the space of upper triangular matrices of determinant 
with respect to the basis 
. This is the standard Borel, and all other Borel subgroups are conjugate to it. Indeed, it’s not hard to see that 
.
Annoying Precision 