`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 w₀ 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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