Descent-Cycling in Schubert Calculus

CONTENTS We prove two lemmata about Schubert calculus on generalized 1. Background on Schubert Problems f|ag manifolds G/B, and in the case of the ordinary flag manifold 2. The Schubert Problems Graph and Its Structure for Small GLn/B we interpret them combinatorially in terms of descents, GLn(C) and geometrically in terms of missing subspaces. One of them 3. Proofs of the Lemmata gives a symmetry of Schubert calculus that we christen descent4. A Geometrical Interpretation cycling. Computer experiment shows these two lemmata are M k' R I surprisingly powerful: they already suffice to determine all of GLn Schubert calculus through n = 5, and 99.97%+ at n = 6. ^ ^ We use them to give a quick proof of Monk's rule. The lemmata Acknowledgements a ! s o h o | d j n e q u i v a r jant ("double") Schubert calculus for KacAddendum Moody groups G. References