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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/ -3.5pt/U a monoid $Γ=Γ(S,>)$. As a side effect we get a vector space basis B_Γ of C[G/ -3.5pt/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 {AB} to this more general setting, once one knows that the monoid $Γ$ is finitely generated and saturated.
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Fang et al. (2015) studied this question.