A tightness property of relatively smooth permutations

Abstract

It is well known that many geometric properties of Schubert varieties of type A can be interpreted combinatorially. Given two permutations w,x∈ Sn we give a combinatorial consequence of the property that the smooth locus of the Schubert variety Xw contains the Schubert cell Yx. This is a necessary ingredient for the interpretation of recent representation-theoretic results of the author with M\'inguez in terms of identities of Kazhdan--Lusztig polynomials.

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