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Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant,Macdonald,Shapiro and Steinberg

The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.

preprint2010arXivOpen access
Ersan AkyildizJames B. Carrell
math.COmath.AG
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Ersan AkyildizJames B. Carrell

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Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant,Macdonald,Shapiro and Steinberg | BZPEER | BZPEER