The nil Hecke ring and singularity of Schubert varieties

We give a criterion for smoothness of a point in any Schubert variety in any G/B in terms of the nil Hecke ring.Comment: AMSTE

Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions of sets of masks on cograssmannian permutations that can be used in Deodhar's formula for Kazhdan-Lusztig basis elements of the Iwahori-Hecke algebra. The constructions are respectively based on a formula of Lascoux-Schutzenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott-Samelson resolution. These constructions give distinct answers to a question of Deodhar.Comment: 43 page

Fracture fixation in the operative management of hip fractures (FAITH): an international, multicentre, randomised controlled trial.

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