Monomial bases and PBW filtration in representation theory

In this thesis we study the Poincaré–Birkhoff–Witt (PBW) filtration on simple finite-dimensional modules of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on the universal enveloping algebra. For modules of certain rectangular highest weights we provide a new de- scription of the associated PBW-graded module in terms of generators and relations. We also construct a new basis parametrized by the lattice points of a normal polytope. If the Lie algebra is of type B3 we construct new bases of PBW-graded modules associated to simple modules of arbitrary highest weight. As an application we find that these modules are favourable modules, implying interesting geometric properties for the degenerate flag varieties. As a side product we state sufficient conditions on convex lattice 0,1-polytopes to be normal. We study the Hilbert–Poincaré polynomials for the associated PBW- graded modules of simple modules. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these degrees explicitly. We extend the framework of the PBW filtration to quantum groups and provide case independent constructions, such as giving a filtration on the negative part of the quantum group, such that the associated graded algebra becomes a q-commutative polynomial algebra. By taking the classical limit we obtain, in some cases new, monomial bases and monomial ideals of the associated graded modules

Degree cones and monomial bases of Lie algebras and quantum groups

We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}^-$. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an $\mathbb{N}$-filtration on any finite dimensional simple $\mathfrak{g}$-module. We prove for type $\tt{A}_n$, $\tt{C}_n$, $\tt{B}_3$, $\tt{D}_4$ and $\tt{G}_2$ that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.Comment: 26 pages, an inaccuracy correcte

The degree of the Hilbert-Poincar\'e polynomial of PBW-graded modules

In this note, we study the Hilbert-Poincar\'e polynomials for the PBW-graded of simple modules for a simple complex Lie algebra. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these degrees explicitly.Comment: 7 pages, updated references, improved exposition, journal versio

PBW Filtration: Feigin-Fourier-Littelmann Modules Via Hasse Diagrams

We study the PBW filtration on the irreducible highest weight representations of simple complex finite-dimensional Lie algebras. This filtration is induced by the standard degree filtration on the universal enveloping algebra. For certain rectangular weights we provide a new description of the associated graded module in terms of generators and relations. We also construct a basis parametrized by the integer points of a normal polytope. The main tool we use is the Hasse diagram defined via the standard partial order on the positive roots. As an application we conclude that all representations considered in this paper are Feigin-Fourier-Littelmann modules

The PBW filtration and convex polytopes in type B

We study the PBW filtration on irreducible finite-dimensional representations for the Lie algebra of type B-n. We prove in various cases, including all multiples of the adjoint representation and all irreducible finite-dimensional representations for 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. (C) 2018 Elsevier B.V. All rights reserved