New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules

New graded modules for the current algebra of $\mathfrak{sl}_n$ are introduced. Relating these modules to the fusion product of simple $\mathfrak{sl}_n$-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 $\mathfrak{sl}_n$-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 $\lambda$ 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 $\mathfrak{sl}_{n+1}$. 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.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 $\Lambda_0$ 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 $\Lambda$ 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, $\mathfrak{osp}(1,2n)$ 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