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Geometric Crystal and Tropical R for D^(1)_n

Atsuo Kuniba·Masato Okado·Taichiro Takagi·Yasuhiko Yamada
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Abstract

We construct a geometric crystal for the affine Lie algebra D^{(1)}_n in the sense of Berenstein and Kazhdan. Based on a matrix realization including a spectral parameter, we prove uniqueness and explicit form of the tropical R, the birational map that intertwines products of the geometric crystals. The tropical R commutes with geometric Kashiwara operators and satisfies the Yang-Baxter equation. It is subtraction-free and yields a piecewise linear formula of the combinatorial R for crystals upon ultradiscretization.

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Atsuo Kuniba, Masato Okado, Taichiro Takagi, Yasuhiko Yamada. 2002-08-30. Geometric Crystal and Tropical R for D^(1)_n. https://doi.org/10.1155/s1073792803209041

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