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 $\Gamma =\Gamma (S,>)$. As a side effect we get a vector space basis ${\mathbb{B}}_{\Gamma }$ of $\mathbb{C}[G//U]$, 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.

35 pages, to appear in Adv. Math

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Cited by in corpus (2)

• Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums
• Tropical symplectic flag varieties: a Lie-theoretic approach