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

TL;DR

This paper investigates the PBW filtration on representations of type B Lie algebras, establishing a connection with convex polytopes to parametrize bases and derive combinatorial character formulas.

Contribution

It proves the existence of normal polytopes parametrizing bases for certain representations of type B Lie algebras, including all multiples of the adjoint and specific cases like B3.

Findings

Existence of normal polytopes for basis parametrization

Derived graded combinatorial character formulas

Identified classes of favourable modules

Abstract

We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional representations for $\tt B_3$, 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 favourable modules and graded combinatorial character formulas.