Descent-Cycling in Schubert Calculus

Abstract

We prove two lemmata about Schubert calculus on generalized flag manifolds $G/B$, and in the case of the ordinary flag manifold $\GL_n/B$ we interpret them combinatorially in terms of descents, and geometrically in terms of missing subspaces. One of them gives a symmetry of Schubert calculus that we christen descent-cycling. Computer experiment shows these two lemmata are surprisingly powerful: they already suffice to determine all of $\GL_n$ 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 also hold in equivariant ("double'') Schubert calculus for Kac--Moody groups $G$.