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The PBW filtration and convex polytopes in type B. (English) Zbl 06934116
Summary: We study the PBW filtration on irreducible finite-dimensional representations for the Lie algebra of type \(\mathtt{B}_n\). We prove in various cases, including all multiples of the adjoint representation and all irreducible finite-dimensional representations for \(\mathtt{B}_3\), that there exists a normal polytope such that the lattice points of this polytope parametrize a basis of the corresponding associated graded space. As a consequence we obtain several classes of examples for favourable modules and graded combinatorial character formulas.

MSC:
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14M15 Grassmannians, Schubert varieties, flag manifolds
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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