Mirković-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal. (English) Zbl 1194.17008
The paper is concerned to the Mirković-Vilonen (MV for short) polytopes, defined by J. E. Anderson [Duke Math. J. 116, No. 3, 567–588 (2003; Zbl 1064.20047)] and described by J. Kamnitzer [Adv. Math. 215, No. 1, 66–93 (2007; Zbl 1134.14028)] as certain pseudo-Weyl polytopes. A necessary and sufficient condition for a MV polytope in a highest weight crystal to lie in a fixed Demazure crystal or in an opposite Demazure crystal is given in terms of the lengths of its edges. An explicit description as pseudo-Weyl polytopes is given for the extremal MV polytopes in a highest weight crystal. As a consequence, a polytopal condition is given for a MV polytope to lie in a fixed opposite Demazure crystal.
Reviewer: Sorin Dascalescu (Bucureşti)
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 14L35 | Classical groups (algebro-geometric aspects) |
| 20G10 | Cohomology theory for linear algebraic groups |
| 52B12 | Special polytopes (linear programming, centrally symmetric, etc.) |
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