Geometric and unipotent crystals II: From unipotent bicrystals to crystal bases

By Arkady Berenstein, David Kazhdan. 2006

109 citations recorded by the source corpus.

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for G^\vee-modules, where G^\vee is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to G^\vee-modules) are associated to G^\vee-modules directly, i.e., without quantum deformations. One of the main results of the present paper is an explicit construction of the crystal B_0 for the coordinate ring of the (Langlands dual) flag variety based on the positive unipotent bicrystal on the open Bruhat cell. Our general tropicalization procedure assigns to each strongly positive unipotent bicrystal a normal Kashiwara crystal B equipped with the multiplicity erasing homomorphism B--> B_0 and the combinatorial central charge B--> Z which is invariant under all crystal operators.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/79b8ae322680a64e06f6d14a023d4a0a3dc465c6

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In Geometry and Topology

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Citations in the corpus

On uniqueness of tensor products of irreducible categorifications (2013, unknown): “Note that the crystal of C is isomorphic to the crystal of V 1 ⊗ V 2 ⊗ . . . ⊗ V n by [BK07, 5.55] combined with [CR08, Proposition 5.20].”
Canonical bases for cluster algebras (2014, methodology): “Potentials were considered in the work of Goncharov and Shen, [GS13], which in turn built on work of Berenstein and Zelevinsky, [BZ01] and Berenstein and Kazhdan, [BK00], [BK07].”