`Coxeter`

### Inverse Kazhdan-Lusztig polynomials

The program contains an implementation of the computation of inverse
Kazhdan-Lusztig polynomials. Recall that they are the unique family of
polynomials Q_{x,y} s.t. the matrix
((-1)^{l(y)-l(x)}Q_{y,x}) be the inverse of the matrix
(P_{x,y}); one may also define them combinatorially as the
generlized Stanley polynomials for the interval $[x,y]$, for the same
R-function as the ordinary ones (these being the generalized Stanley
polynomials associated to the *dual* poset of [x,y].)

This part of the program could certainly be improved; it is in any case a lot
slower than the computation of the ordinary polynomials. It has also been
checked much less thoroughly, mostly for lack of comparison material! For
finite groups, it is known that Q_{x,y} =
P_{w0y,w0x} for all x <= y in the group, where
w_{0} is the longest element in the group. I have checked that for
the groups A6, B5, F4 and H4, the inverse polynomials computed with the
algorithm of the program coincide with the
P_{w0y,w0x} computed with the algorithm for the
ordinary polynomials; this should be a rather convincing verification (of
both computations, actually.)

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