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- W2467391738 abstract "In this paper we use ideas from mirror symmetry to construct a `canonical point' in certain natural polytopes arising in the representation theory of SLn. Recall that to an irreducible representation of G = SLn(C) one can associate various polytopes whose integral points parameterize a basis for the representation, for example, the Gelfand-Zetlin polytope. Such polytopes can be constructed via the theory of geometric crystals by tropicalising a certain function. Moreover, the function involved coincides with the superpotential associated by mirror symmetry to the flag variety G/B. In mirror symmetry a special role is played by the critical points of the superpotential. Motivated by this, we study a tropical version of these critical points. Our first main theorem is existence and uniqueness of the tropical critical point associated to any fixed highest weight. Using this result we find a canonical point in each polytope. We then characterise the highest weights for which this tropical critical point is integral and therefore corresponds to a basis vector of the corresponding representation. We suggest an interpretation of the tropical critical point by constructing a special vector in the representation using Borel-Weil theory and conjecturing a correspondence between this vector and the tropical critical point." @default.
- W2467391738 created "2016-07-22" @default.
- W2467391738 creator A5054187927 @default.
- W2467391738 date "2016-06-22" @default.
- W2467391738 modified "2023-09-26" @default.
- W2467391738 title "Tropical critical points of the superpotential" @default.
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- W2467391738 hasPublicationYear "2016" @default.
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