Rigidity of Schubert closures and invariance of Kazhdan–Lusztig polynomials

Abstract

Let [ e, w], [ e, w′] be intervals from the origin in two Coxeter groups W, W′. We prove that any poset isomorphism ϕ : [e,w]→[e,w′] preserves Kazhdan–Lusztig polynomials, in the sense that P ϕ( x), ϕ( y) = P x, y for any x⩽ y in [ e, w], in the case where one of the two groups W, W′ has the property that the Coxeter graph of each of its irreducible constituents is either a tree, or affine of type A ̃ n ; in particular, the result holds for all finite or affine Coxeter groups. We obtain this result as a consequence of an explicit description of all the poset isomorphisms from [ e, w] to [ e, w′].