Essential bases and toric degenerations arising from birational sequences

We present a new approach to construct T-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair (S,>) consisting of a birational sequence and a monomial order, we attach to the affine variety G//U a monoid Gamma = Gamma(S, >). As a side effect we get a vector space basis is B-Gamma of C[G//U], the elements being indexed by Gamma. The basis B-Gamma has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of Alexeev and Brion [1] to this more general setting, once one knows that the monoid Gamma is finitely generated and saturated. (C) 2017 Elsevier Inc. All rights reserved.