Permutations with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\). With an appendix by Sara Billey and Jonathan Weed (Q1028863)

Let \(w \in S_n\) be a permutation, \(X_w\) the corresponding Schubert variety, and \(P_{id,w}(q)\) the corresponding Kazhdan-Lusztig polynomial. It is known that \(P_{id,w}(1)\) equals \(1\) if and only if \(X_w\) is smooth. (Since Kazhdan-Lusztig polynomials have non--negative integer coefficients and constant term \(1\), \(P_{id,w}(1)=1\) is equivalent to \(P_{id,w}(q)=1\)). In this paper, the author gives necessary and sufficient conditions for \(P_{id,w}(1)=2\): he shows that this is equivalent to a geometric condition, namely that the singular locus of \(X_w\) has exactly one irreducible component, plus a combinatorial one, namely that the permutation \(w\) avoids a list of six patterns. He further shows that when \(P_{id,w}(1)=2\), \(P_{id,w}(q)=1+q^h\) where \(h\) is computed combinatorially from \(w\). An appendix by Sara Billey and Jonathan Weed gives a characterization of \(P_{id,w}(1)=2\) purely in terms of pattern avoidance, the number of patterns used being \(66\).