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The tropical totally positive Grassmannians. (English) Zbl 1094.14048
The additive group \((\mathbb{R},+)\) together with the binary operation “min” is a semiring, called the tropical semiring. The geometric objects of tropical geometry are tropical varieties, polyhedral cell complexes that are contained in tropical affine spaces or in tropical projective spaces over the tropical semiring. The field \(K\) of Puiseux series with coefficients in \(\mathbb{C}\) carries the order valuation, which takes its values in the tropical semiring. The order valuation is used to associate a tropical variety, the tropicalization, with every algebraic variety \(V(I)\subseteq(K\setminus\{0 \})^n\). The authors introduce the positive part of the tropicalization of an affine variety and study the notion for the case of Grassmannians. Tropicalizations and their positive parts are closed subcomplexes of the Gröbner complex associated with the defining ideal \(I\). The positive parts of several tropicalized Grassmannians are closely related to a class of polytopes, called associahedra. The connection is described explicitly.

MSC:
14P99 Real algebraic and real-analytic geometry
16Y60 Semirings
14M15 Grassmannians, Schubert varieties, flag manifolds
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
12J10 Valued fields
52B70 Polyhedral manifolds
Software:
cdd; polymake
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References:
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