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arXiv · 1510.02295

Essential bases and toric degenerations arising from birational sequences

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

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Xin Fang, Ghislain Fourier, Peter Littelmann. 2017-03-13. Essential bases and toric degenerations arising from birational sequences. https://arxiv.org/abs/1510.02295

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