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Crystal structure on the set of Lakshmibai–Seshadri paths of an arbitrary level‐zero shape
Author(s) -
Naito Satoshi,
Sagaki Daisuke
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm034
Subject(s) - mathematics , zero (linguistics) , lattice (music) , combinatorics , crystal (programming language) , affine transformation , pairwise comparison , crystal structure , pure mathematics , crystallography , physics , chemistry , philosophy , linguistics , computer science , programming language , statistics , acoustics
Let λ = ∑ i ∈ I 0 m i ϖ i, with m i ∈ℤ ⩾0 for i ∈ I 0 , be a level‐zero dominant integral weight for an affine Lie algebra ℊ over ℚ, where the ϖ i , i ∈ I 0 , are the level‐zero fundamental weights, and let B (λ) be the crystal of all Lakshmibai–Seshadri paths of shape λ. First, we give an explicit description of the decomposition of the crystal B (λ) into connected components and show that all the connected components are pairwise ‘isomorphic’ (up to a shift of weights). Second, we ‘realize’ the connected component B 0 (λ) of B (λ) containing the straight line π λ as a specified subcrystal of the affinizationB( λ )cl^(with weight lattice P ) of the crystal B( λ )cl ≃ ⊗ i ∈ I 0( B( ϖ i )cl)⊗ m i(with weight latticeP cl ≃ P / ( Q δ ∩ P ) , where δ is the null root of ℊ ), which we studied in a previous paper ( Int. Math. Res. Not. 2005 (2005) 815–840).