21 research outputs found
PBW-degenerated Demazure modules and Schubert varieties for triangular elements
We study certain faces of the normal polytope introduced by Feigin, Littelmann and the author whose lattice points parametrize a monomial basis of the PBW-degenerated of simple modules for sln+1. We show that lattice points in these faces parametrize monomial bases of PBW-degenerated Demazure modules associated to Weyl group elements satisfying a certain closure property, for example Kempf elements. These faces are again normal polytopes and their Minkowski sum is compatible with tensor products, which implies that we obtain flat degenerations of the corresponding Schubert varieties to PBW degenerated and toric varieties
PBW-degenerated Demazure modules and Schubert varieties for triangular elements
We study certain faces of the normal polytope introduced by Feigin,
Littelmann and the author whose lattice points parametrize a monomial basis of
the PBW-degenerated of simple modules for . We show that
lattice points in these faces parametrize monomial bases of PBW-degenerated
Demazure modules associated to Weyl group elements satisfying a certain closure
property, for example Kempf elements. These faces are again normal polytopes
and their Minkowski sum is compatible with tensor products, which implies that
we obtain flat degenerations of the corresponding Schubert varieties to PBW
degenerated and toric varieties.Comment: 17 page
Minuscule Schubert varieties: poset polytopes, PBW-degenerated demazure modules, and Kogan faces
We study a family of posets and the associated chain and order polytopes. We
identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple
of a fundamental weight. We show that the character of such a Kogan face equals to
the character of a Demazure module which occurs in the irreducible representation of sln+1
having highest weight multiple of fundamental weight and for any such Demazure module
there exists a corresponding poset and associated maximal Kogan face. We prove that
the chain polytope parametrizes a monomial basis of the associated PBW-graded Demazure
module and further, that the Demazure module is a favourable module, e.g. interesting geometric
properties are governed by combinatorics of convex polytopes. Thus, we obtain for
any minuscule Schubert variety a flat degeneration into a toric projective variety which is
projectively normal and arithmetically Cohen-Macaulay. We provide a necessary and sufficient
condition on the Weyl group element such that the toric variety associated to the chain
polytope and the toric variety associated to the order polytope are isomorphic
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 and monomial bases for Demazure modules in types A and C
We characterise the symplectic Weyl group elements such that the FFLV basis
is compatible with the PBW filtration on symplectic Demazure modules, extending
type A results by the second author. Surprisingly, the number of such elements
depends not on the type A or C of the Lie algebra but on the rank only.Comment: 20 page
Marked chain-order polytopes
We introduce in this paper the marked chain-order polytopes associated to a
marked poset, generalizing the marked chain polytopes and marked order
polytopes by putting them as extremal cases in an Ehrhart equivalent family.
Some combinatorial properties of these polytopes are studied. This work is
motivated by the framework of PBW degenerations in representation theory of Lie
algebras.Comment: 18 pages, title changed, the relation to string polytopes is remove