Multiplicity-free products of Schubert divisors

Abstract

Let G/B be a flag variety over C, where G is a simple algebraic group with a simply laced Dynkin diagram, and B is a Borel subgroup. We say that the product of classes of Schubert divisors in the Chow ring is 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 the maximal possible degree (in the Chow ring) of a multiplicity free product of classes of Schubert divisors. PreliminariesWe denote the subset of positive roots by Φ + , and the set of simple roots by Π. We enumerate simple roots as in [1]. Denote the fundamental weight corresponding to a simple root α i by ̟ i . We choose the scalar multiplication on Φ so that the scalar square of each simple root is 2. The scalar product of two roots α and β is denoted by (α, β). Note that with this choice of scalar multiplication, we can use a simple formula for reflection: usually, we writeBut with our choice of scalar product, we can writeMore generally, for two arbitrary vectors v, w in the ambient space of Φ, w = 0, denoteThen, with our choice of the scalar product, we have (•, α) = •, α for each α ∈ Φ. We use the following Pieri formula:

The canonical dimension of a semisimple group and the unimodular degree of a root system

“…Observe that the bound for type A is sharp as it coincides with the maximal length of an element in W that is |Σ + |. By the main result of [Dv17] the above bounds are also sharp for types D and E.…”

“…As an application, we obtain upper bounds computed by Karpenko using the Pieri formula for the intersection of effective divisors. We also simplify Devyatov's computations in [Dv17], where such bounds were obtained for all simply-laced simply-connected simple groups using advanced combinatorial techniques. In addition, we get new bounds (e.g.…”