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

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

2002-08-30

arxiv.org/abs/math/0208239v1

Int. Math. Res. Notices 48 (2003) 2565-2620

Mathematics

Quantum Algebra

41 pages

Scientific paper

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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