SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties

Kiumars Kaveh

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arXiv · math/0309413

SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties

Kiumars Kaveh

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Abstract

Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope Δ(X). The polytope Δ(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope Δ(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.

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BibTeX RIS

Kiumars Kaveh. 2003-09-25. SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties. https://arxiv.org/abs/math/0309413

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