The PBW filtration and convex polytopes in type $\tt B$

Essential signatures and canonical bases of irreducible representations of the group G 2

“…To prove the conjectures for D 4 we use the same method as in [1]. However we have a general approach which can be used to prove the conjectures for all Lie algebras of types A n , C n , B 3 , D 4 , G 2 .…”

“…Our goal is to select a basis of V (λ) from the set of all vectors v(σ). This problem was solved in [2], [3], [4], [1] for the algebras of types A n , C n , G 2 and B 3 . Let t Z ⊂ t be the coroot lattice, i.e., the lattice of vectors on which all weights take integer values.…”

“…In this paper, we solve the above problem for the Lie algebra of type D 4 . The basis vectors for D 4 are determined by a set of linear inequalities of the form (1), where a ij = 1, 2 or 3 (Proposition 4).…”