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Ghislain Fourier

Publications and source records attributed to Ghislain Fourier.

At least 55 records · Page 3Linked to original sources
arXiv2014

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.

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arXiv2014

Schur Positivity and Kirillov-Reshetikhin Modules

In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a tensor product. We show that there exists an embedding of the tensor products, with respect to the classical structure, along with the reverse dominance relation on the set of partitions.

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arXiv2014

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.

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arXiv2013

Global Weyl modules for equivariant map algebras

Equivariant map algebras are Lie algebras of algebraic maps from a scheme (or algebraic variety) to a target finite-dimensional Lie algebra (in the case of the current paper, we assume the latter is a simple Lie algebra) that are equivariant with respect to the action of a finite group. In the first part of this paper, we define global Weyl modules for equivariant map algebras satisfying a mild assumption. We then identify a commutative algebra A that acts naturally on the global Weyl modules, which leads to a Weyl functor from the category of A-modules to the category of modules for the equivariant map algebra in question. These definitions extend the ones previously given for generalized current algebras (i.e. untwisted map algebras) and twisted loop algebras. In the second part of the paper, we restrict our attention to equivariant map algebras where the group involved is abelian, acts on the target Lie algebra by diagram automorphisms, and freely on (the set of rational points of) the scheme. Under these additional assumptions, we prove that A is finitely generated and the global Weyl module is a finitely generated A-module. We also define local Weyl modules via the Weyl functor and prove that these coincide with the local Weyl modules defined directly in a previous paper. Finally, we show that A is the algebra of coinvariants of the analogous algebra in the untwisted case.

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arXiv2013

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 $λ$ 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.

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arXiv2013

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.

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arXiv2013

Posets, Tensor Products and Schur positivity

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight λ, we define a preorder on the set $P(λ, k)$ of k-tuples of dominant weights which add up to λ. Let $P(λ, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if λis a multiple of a fundamental weight (and k is general) or if k=2 (and λis general), then $P(λ, k)/\sim$ coincides with the set of S_k-orbits in $P(λ,k)$, where S_k acts on $P(λ, k)$ as the permutations of components. If g is of type A_n and k=2, we show that the S_2-orbit of the row shuffle defined by Fomin et al is the unique maximal element in the poset. Given an element of $P(λ, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, λ, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when λis a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (λis general), there exists an inclusion of tensor products of g-modules along with the partial order. In particular, if g is of type A_n, this means that the difference of the characters is Schur positive.

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arXiv2012

On multigraded generalizations of Kirillov-Reshetikhin modules

We study the category of Z^l-graded modules with finite-dimensional graded pieces for certain Z+^l-graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov-Reshetikhin modules and give a recursive formula for computing their graded characters.

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arXiv2012

PBW--filtration over $\bz$ and compatible bases for $V_\bz(\la)$ in type ${\tt A}_n$ and ${\tt C}_n$

We study the PBW-filtration on the highest weight representations $V(\la)$ of the Lie algebras of type ${\tt A}_{n}$ and ${\tt C}_{n}$. This filtration is induced by the standard degree filtration on $\U(\fn^-)$. In previous papers, the authors studied the filtration and the associated graded algebras and modules over the complex numbers. The aim of this paper is to present a proof of the results which holds over the integers and hence makes the whole construction available over any field.

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arXiv2011

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional representations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defining twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological properties and a tensor product property) extend to the more general setting considered in the current paper.

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arXiv2011

Global Weyl modules for the twisted loop algebra

We define global Weyl modules for twisted loop algebras and analyze their high- est weight spaces, which are in fact isomorphic to Laurent polynomial rings in finitely many variables. We are able to show that the global Weyl module is a free module of finite rank over these rings. Furthermore we prove, that there exist injective maps from the global Weyl modules for twisted loop algebras into a direct sum of global Weyl modules for untwisted loop algebras. Relations between local Weyl modules for twisted and untwisted generalized current algebras are known; we provide for the first time a relation on global Weyl modules.

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arXiv2011

Demazure modules and Weyl modules: The twisted current case

We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure module for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight. These results were known before for untwisted current algebras and are new for all twisted types.

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arXiv2011

Weyl modules for multiloop algebras

Global and local Weyl modules for the untwisted multiloop Lie algebras were defined by Chari, the first and the second author via homological properties. In this paper we extended the ideas to give a categorical definition of the Weyl modules for twisted multiloop algebras. Our methods led us to describe an identification of the finite--dimensional highest weight modules for twisted multiloop algebras with suitably chosen finite--dimensional highest weight modules for untwisted multiloop algebras.

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arXiv2010

PBW filtration and bases for symplectic Lie algebras

We study the PBW filtration on the highest weight representations $V(\la)$ of $\msp_{2n}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr V(\la)$ in terms of generators and relations. We also construct a basis of $gr V(\la)$. As an application we derive a graded combinatorial formula for the character of $V(\la)$ and obtain a new class of bases of the modules $V(\la)$.

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arXiv2010

PBW filtration and bases for irreducible modules in type $A_n$

We study the PBW filtration on the highest weight representations $V(\la)$ of $\msl_{n+1}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr V(\la)$ in terms of generators and relations. We also construct a basis of $gr V(\la)$. As an application we derive a graded combinatorial character formula for $V(\la)$, and we obtain a new class of bases of the modules $V(\la)$ conjectured by Vinberg in 2005.

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arXiv2009

A categorical approach to Weyl modules

Global and local Weyl Modules were introduced via generators and relations in the context of affine Lie algebras in a work by the first author and Pressley and were motivated by representations of quantum affine algebras. A more general case was considered by Feigin and Loktev by replacing the polynomial ring with the coordinate ring of an algebraic variety. We show that there is a natural definition of the local and global modules via homological properties. This characterization allows us to define the Weyl functor from the category of left modules of a commutative algebra to the category of modules for a simple Lie algebra. As an application we are able to understand the relationships of these functors to tensor products, generalizing previous results. Finally an analysis of the fundamental Weyl modules proves that the functors are not left exact in general, even for coordinate rings of affine varieties.

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arXiv2009

Kirillov--Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov--Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types D_n^(1), B_n^(1), A_{2n-1}^(2) we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type C_n^(1) we use a Dynkin diagram folding and for types A_{2n}^(2), D_{n+1}^(2) a similarity construction. We also show that for types C_n^(1) and D_{n+1}^(2) the analog of the Dynkin diagram automorphism exists on the level of crystals.

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