Essential bases and toric degenerations arising from birational sequences

Abstract 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 Γ = Γ ( S , > ) . As a side effect we get a vector space basis B Γ of C [ G / / U ] , the elements being indexed by Γ. The basis B Γ 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 Γ is finitely generated and saturated.