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 $G/\hskip -3.5pt/U$ a monoid $\Gamma=\Gamma(S,>)$. As a side effect we get a vector space basis $\mathbb B_{\Gamma}$ of $\mathbb C[G/\hskip -3.5pt/U]$, the elements being indexed by $\Gamma$. The basis $\mathbb B_{\Gamma}$ has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of Alexeev and Brion \cite{AB} to this more general setting, once one knows that the monoid $\Gamma$ is finitely generated and saturated.