Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal

Details Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal Mirkovic-Vilonen polytopes lying in a Demazure crystal and an opposite Demazure crystal

2008-06-19

arxiv.org/abs/0806.3112v2

Mathematics

Quantum Algebra

Scientific paper

We give a necessary and sufficient condition for an MV polytope $P$ in a highest weight crystal to lie in an arbitrary fixed Demazure crystal (resp., opposite Demazure crystal), in terms of the lengths of edges along a path through the 1-skeleton of $P$ corresponding to a reduced word for the longest element of the Weyl group $W$. % Also, we give an explicit description as a pseudo-Weyl polytope for extremal MV polytopes in a highest weight crystal. % Finally, by combining the results above, we obtain a polytopal condition for an MV polytope $P$ to lie in an arbitrary fixed opposite Demazure crystal.

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