Ghislain Fourier
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Ghislain Fourier has a total of 15 co-authors. Here are the top co-authors by number of shared publications:
- Evgeny Feigin
School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel
6 shared publications - Peter Littelmann
Universität zu Köln
6 shared publications - Deniz Kus
Faculty of Mathematics, University of Bochum, Bochum, Germany
3 shared publications - Vyjayanthi Chari
University of California
3 shared publications - Alistair Savage
University of Ottawa
2 shared publications - Daisuke Sagaki
Department of Mathematics Faculty of Pure and Applied Sciences University of Tsukuba 1-1-1 Tennodai 305-8571 Tsukuba Ibaraki Japan
2 shared publications
Latest Publications
- 2 May 2016
- Published by Springer Nature in Transformation Groups
- Vol. 22 (2) , 321-352
FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS
- 2 May 2016
- Published by Springer Nature in Transformation Groups
- Vol. 22 (2) , 321-352
FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS
- 26 June 2015
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 18 (6) , 1481-1503
Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces
- 26 June 2015
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 18 (6) , 1481-1503
Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces
- Open Access
- 10 August 2014
- Published by Mathematical Sciences Publishers in Algebra & Number Theory
- Vol. 8 (4) , 933-961
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 \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ 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 $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is 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.Comment: 24 pages, updated reference- 10 August 2014
- Published by Mathematical Sciences Publishers in Algebra & Number Theory
- Vol. 8 (4) , 933-961
Open AccessPosets, tensor products and Schur positivity
Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ 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 $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is 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.Comment: 24 pages, updated reference - Open Access
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.Open AccessNew 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. - Open Access
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.
Open AccessGlobal 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.
- Open Access
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.Open AccessExtended 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. - Open Access
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.
Open AccessWeyl 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.
- Open Access
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 \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ 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 $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is 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.Open AccessPosets, Tensor Products and Schur positivity
Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda, k)/\sim$ be the corresponding poset of equivalence classes defined by the preorder. We show that if \lambda is a multiple of a fundamental weight (and k is general) or if k=2 (and \lambda is general), then $P(\lambda, k)/\sim$ coincides with the set of S_k-orbits in $P(\lambda,k)$, where S_k acts on $P(\lambda, k)$ 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 $P(\lambda, k)$, consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, \lambda, and k) the dimension of this tensor product increases along with the partial order. We also show that in the case when \lambda is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A_2 and k=2 (\lambda is 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. - 13 June 2013
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 365 (11) , 6037-6064
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 modules 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 previously known for untwisted current algebras and are new for all twisted types.
- 13 June 2013
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 365 (11) , 6037-6064
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 modules 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 previously known for untwisted current algebras and are new for all twisted types.
- 13 March 2013
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 17 (2) , 519-538
On Multigraded Generalizations of Kirillov–Reshetikhin Modules
- 13 March 2013
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 17 (2) , 519-538
On Multigraded Generalizations of Kirillov–Reshetikhin Modules