Computing some KL-polynomials for the poset of B ×B-orbits in group compactifications

In [3], Springer studies a poset V which depends on a Coxeter group W with generating set S. IfW is a finite Weyl group, then V indexes the B×Borbits in the wonderful compactification of the adjoint semi-simple group G with Weyl group W . In that case Springer shows for instance that one has positivity results for the analogues cx,v of Kazhdan-Lusztig polynomials. But the poset V is much more complicated/larger thanW , so that computing the cx,v by hand is only attractive when S has one element. Therefore we wrote Mathematica code to get access to some more examples. Once the code was available we could experiment with the analogues c x,v of inverse KazhdanLusztig polynomials and also with Coxeter groups that are not finite Weyl groups. In our experiments we saw positivity properties for the c x,v. And the other Coxeter groups behaved just like finite Weyl groups. As already mentioned, V is much bigger than W . For instance, when W is of type B3, the size |V | of V is already 7056, while |W | is only 48. Therefore we further assume W is so small that we do not have to worry about its size. Our reason to choose Mathematica is that Mathematica provides a powerful high level language with which we are familiar. Our primary task was to code new combinatorics reliably for very smallW . Speed was no issue at this prototype stage. Everything could be developed from scratch without any need for libraries of fast specialized tools. Thus the fact that Mathematica is quite ignorant about Coxeter groups was no obstacle at all. Let us now digress and recall that Mathematica is optimized for replacement rules based on pattern matching. We did find this attractive because it allows to use very simple and transparent code for reducing words in small