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

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.

41 pages

References in corpus (5)

• The A^{(1)}_M automata related to crystals of symmetric tensors
• Soliton Cellular Automata Associated With Crystal Bases
• Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions
• Combinatorial R matrices for a family of crystals : B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n} and D^{(2)}_{n+1} cases
• Factorization of Combinatorial R matrices and Associated Cellular Automata

Cited by in corpus (7)

• Integrable structure of box-ball systems: crystal, Bethe ansatz, ultradiscretization and tropical geometry
• Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection
• Variations of $q$-Garnier system
• Tropical R and Tau Functions
• Factorization, reduction and embedding in integrable cellular automata
• Similarity and Kirillov-Schilling-Shimozono bijection
• Analysis of a particle antiparticle description of a soliton cellular automaton