73 research outputs found
New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules
New graded modules for the current algebra of are
introduced. Relating these modules to the fusion product of simple
-modules and local Weyl modules of truncated current algebras
shows their expected impact on several outstanding conjectures. We further
generalize results on PBW filtrations of simple -modules and
use them to provide decomposition formulas for these new modules in important
cases.Comment: 23 page
Weyl modules and Levi subalgebras
For a simple complex Lie algebra of finite rank and classical type, we fix a
triangular decomposition and consider the simple Levi subalgebras associated to
closed subsets of roots. We study the restriction of global and local Weyl
modules of current algebras to this Levi subalgebra. We identify necessary and
sufficient conditions on a pair of a Levi subalgebra and a dominant integral
weight, such that the restricted module is a global (resp. a local) Weyl
module.Comment: 22 pages, final version, to appear in Jol
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
Extended partial order and applications to tensor products
We extend the preorder on k-tuples of dominant weights of a simple complex
Lie algebra g of classical type adding up to a fixed weight defined
by V. Chari, D. Sagaki and the author. We show that the induced extended
partial order on the equivalence classes has a unique minimal and a unique
maximal element. For k=2 we compute its size and determine the cover relation.
To each k-tuple we associate a tensor product of simple g-modules and we show
that for k=2 the dimension increases also along with the extended partial
order, generalizing a theorem proved in the aforementioned paper. We also show
that the tensor product associated to the maximal element has the biggest
dimension among all tuples for arbitrary k, indicating that this might be a
symplectic (resp. orthogonal) analogon of the row shuffle defined by Fomin et
al.
The extension of the partial order reduces the number of elements in the
cover relation and may facilitate the proof of an analogon of Schur positivity
along the partial order for symplectic and orthogonal types.Comment: 16 pages, final version, to appear in AJo
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
Tensor product structure of affine Demazure modules and limit constructions
Let \Lg be a simple complex Lie algebra, we denote by \Lhg the
corresponding affine Kac--Moody algebra. Let be the additional
fundamental weight of \Lhg. For a dominant integral \Lg--coweight
\lam^\vee, the Demazure submodule V_{-\lam^\vee}(m\Lam_0) is a
\Lg--module. For any partition of \lam^\vee=\sum_j \lam_j^\vee as a sum of
dominant integral \Lg--coweights, the Demazure module is (as \Lg--module)
isomorphic to \bigotimes_j V_{-\lam^\vee_j}(m\Lam_0). For the ``smallest''
case, \lam^\vee=\om^\vee a fundamental coweight, we provide for \Lg of
classical type a decomposition of V_{-\om^\vee}(m\Lam_0) into irreducible
\Lg--modules, so this can be viewed as a natural generalization of the
decomposition formulas in \cite{KMOTU} and \cite{Magyar}. A comparison with the
U_q(\Lg)--characters of certain finite dimensional U_q'(\Lhg)--modules
(Kirillov--Reshetikhin--modules) suggests furthermore that all quantized
Demazure modules V_{-\lam^\vee,q}(m\Lam_0) can be naturally endowed with the
structure of a U_q'(\Lhg)--module. Such a structure suggests also a
combinatorially interesting connection between the LS--path model for the
Demazure module and the LS--path model for certain U_q'(\Lhg)--modules in
\cite{NaitoSagaki}. For an integral dominant \Lhg--weight let
V(\Lam) be the corresponding irreducible \Lhg--representation. Using the
tensor product decomposition for Demazure modules, we give a description of the
\Lg--module structure of V(\Lam) as a semi-infinite tensor product of
finite dimensional \Lg--modules. The case of twisted affine Kac-Moody
algebras can be treated in the same way, some details are worked out in the
last section.Comment: 24 pages, in the current version we added the case of twisted affine
Kac--Moody algebra
PBW degenerations of Lie superalgebras and their typical representations
We introduce the PBW degeneration for basic classical Lie superalgebras and
construct for all type I, and exceptional Lie
superalgebras new monomial bases. These bases are parametrized by lattice
points in convex lattice polytopes, sharing useful properties such as the
integer decomposition property. This paper is the first step towards extending
the framework of PBW degenerations to the Lie superalgebra setting
Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions
We study finite dimensional representations of current algebras, loop
algebras and their quantized versions. For the current algebra of a simple Lie
algebra of type {\tt ADE}, we show that Kirillov-Reshetikhin modules and Weyl
modules are in fact all Demazure modules. As a consequence one obtains an
elementary proof of the dimension formula for Weyl modules for the current and
the loop algebra. Further, we show that the crystals of the Weyl and the
Demazure module are the same up to some additional label zero arrows for the
Weyl module.
For the current algebra \Lgc of an arbitrary simple Lie algebra, the fusion
product of Demazure modules of the same level turns out to be again a Demazure
module. As an application we construct the \Lgc-module structure of the
Kac-Moody algebra \Lhg-module V(\ell\Lam_0) as a semi-infinite fusion
product of finite dimensional \Lgc--modules
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