Multiplicity-Free Products of Schubert Divisors

Let $G/B$ be a flag variety over an arbitrary field, where $G$ is a semisimple split algebraic group with a simply laced Dynkin diagram, and $B$ is a Borel subgroup. We say that the product of several classes of Schubert divisors in the Chow ring is \emph{multiplicity-free} if it is possible to multiply it by a Schubert class (not necessarily of a divisor) and get the class of a point. In the present paper we find all possible degrees (in the Chow ring) of multiplicity-free products of classes of Schubert divisors. Also, given a product of several classes of Schubert divisors, we can decompose it into a linear combination of classes of Schubert varieties with (as was known before) nonnegative coefficients. We study the coefficients in this linear combination and provide a criterion detecting if such a coefficient equals 1, is greater than 1, or equals zero (i.e. a Schubert variety is not actually present in the linear combination).