A Root System Description of Pattern Avoidance with Applications to G/B

Pattern avoidance has been used to classify several notions for permutations and signed permutations. In this talk, we propose a new generalization of pattern avoidance which can be applied to all root systems and their Weyl groups. The main theorem shows that for any semisimple Lie group G and maximal Borel subgroup B, smooth Schubert varieties in G/B can be characterized by this new method with a very short list of patterns.

This is joint work with A. Postnikov.