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I'm looking for examples of non-trivial Kazhdan-Lusztig polynomials, specifically in the case where the Coxeter system is a Weyl group.

For example, the simplest polynomial with non-trivial $q$-coefficient is $p_{tsut,e}(q) = 1 + q$ in type $A3.$ Where can we find the first non-trivial coefficient of $q^2$, and $q^3...$ etc.

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    $\begingroup$ The non-triviality of the KL polynomials comes from the non-smoothness of the corresponding Schubert varieties, so you're kind of asking when do Schubert varieties have singularities. This is controlled to a large degree by the theory of permutation pattern containment. $\endgroup$ Sep 12, 2018 at 14:58

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Already in the case of finite symmetric groups, one can find any polynomial with non-negative integral coefficients and constant term 1 as KL polynomial for some pair of group elements. See the paper by Patrick Polo here. But specific examples take a little more work. Check the old tables for Weyl groups (and affine Weyl groups) on Mark Goresky's webpage at IAS here.

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