Mirkovic-Vilonen cycles and polytopes
By Joel Kamnitzer. 2005
112 citations recorded by the source corpus.
We give an explicit description of the Mirkovie-Vilonen cycles on the affine Grassmannian for arbitrary complex reductive groups. We also give a combinatorial characterization of the MV polytopes. We prove that a polytope is an MV polytope if and only if it is a lattice polytope whose defining hyperplanes are parallel to those of the Weyl polytopes and whose 2-faces are rank 2 MV polytopes. As an application, we give a bijection between Lusztig's canonical basis and the set of MV polytopes.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/0412e2caa0d5dbfde36cb28cf6203134bfd5cd27
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
- Pierre Baumann
- Peter Tingley
- Peter J. McNamara
- Joel Kamnitzer
- Sabin Cautis
- David E. V. Rose
- Hoel Queffelec
- Aaron D. Lauda
In Mathematical Physics
- Pierre Baumann
- Peter Tingley
- Peter J. McNamara
- Joel Kamnitzer
- Sabin Cautis
- David E. V. Rose
- Hoel Queffelec
- Aaron D. Lauda
Citations in the corpus
- Metaplectic Whittaker functions and crystal bases (2009, background): “of the form e−α i1 (x1)...e−α iN (xN) where the xi have a fixed valuation, the exact formulae for which are given in [BG08]. There is a further description of the closures Ci m given in Theorem 4.5 of [Kam]. We will write B(λ) and B(−∞) for the abstract crystals reaslised by L(λ) and L(−∞) respective”
- Affine Mirković-Vilonen polytopes (2011, background): “26 constrains the lengths of the edges of this face; it is equivalent to the tropical Plücker relations of [36].”
- Preprojective algebras and MV polytopes (2010, methodology): “Following Appendix A in [10], we recall the notion of pseudo-Weyl polytope.”