The PBW filtration and convex polytopes in type B

PBW degenerations of Lie superalgebras and their typical representations

“…Otherwise we use the fact that ≺ is a monomial order. Now the same strategy as in the papers [2,10,11] shows that the set {x s : s ∈ S(λ)} spans S(n − 0 ) ⊗ Λ(n − 1 )/I(λ) and hence the quotient space gr V (λ); we omit the details. In order to finish the proof of Theorem 2 we are left to show that {x s : s ∈ S(λ)} is a linearly independent subset of gr V (λ) (this shows part (1) and part (2) of the theorem) and that 3)).…”

“…The goal of this paper is to make a first step towards understanding the structure of gr V (λ) provided that the structure of gr V g0 (λ) is known, where V g0 (λ) is the finite-dimensional irreducible g0module and gr V g0 (λ) is defined as in [10]. The structure of gr V g0 (λ) including a computation of a monomial basis parametrized by the lattice points of a convex polytope has been worked out in [10] for type A n , in [11] for type C n , in type B 3 in [2] and for type G 2 in [14].…”

“…We emphasize the importance of Proposition 2.10. In order to determine a PBW basis for gr V (λ) we only have to compute a PBW basis for the representations of the underlying simple Lie algebras (most of the cases are known in the literature; see for example [2,10,11,14]) and a PBW basis for gr V (λ 1 ), where λ 1 is a dominant integral weight for the simple Lie algebra g (1). With other words, we can lift a PBW basis of gr V g0 (λ) to a PBW basis of gr V (λ) provided that the structure of gr V (λ 1 ) is known.…”

“…In the remaining infinite Lie superalgebra series of type II, we have statements reducing the problem of finding monomial bases for V a (λ) to the computation of monomial bases for PBW degenerate modules for simple Lie algebras of type B. Since these monomial bases are not yet described in full generality (see [2] for partial results), we omit our reduction statements for now and refer to future publications.…”

PBW degenerations, quiver Grassmannians, and toric varieties

“…The algebraic and representation theoretic properties of the PBW filtration and the g a action in more general settings are considered in [9,10,16,19,50,51,55,69,47,64,75,76,98,106,107].…”