A generalization of Fulton's conjecture for arbitrary groups
We prove a generalization of Fulton's conjecture which relates intersection theory on an arbitrary flag variety to invariant theory.
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arXiv subjects
Nicolas Ressayre
Publications and source records attributed to Nicolas Ressayre.
Two generalizations of the PRV conjecture
Let G be a complex connected reductive group. The PRV conjecture, which was proved independently by S. Kumar and O. Mathieu in 1989, gives explicit irreducible submodules of the tensor product of two irreducible G-modules. This paper has three aims. First, we simplify the proof of the PRV conjecture, then we generalize it to other branching problems. Finally, we find other irreducible components of the tensor product of two irreducible G-modules that appear for "the same reason" as the PRV ones.
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Eigencones and the PRV conjecture
Let $G$ be a complex semisimple simply connected algebraic group. Given two irreducible representations $V_1$ and $V_2$ of $G$, we are interested in some components of $V_1\otimes V_2$. Consider two geometric realizations of $V_1$ and $V_2$ using the Borel-Weil-Bott theorem. Namely, for $i=1, 2$, let $\Li_i$ be a $G$-linearized line bundle on $G/B$ such that ${\rm H}^{q_i}(G/B,\Li_i)$ is isomorphic to $V_i$. Assume that the cup product $$ {\rm H}^{q_1}(G/B,\Li_1)\otimes {\rm H}^{q_2}(G/B,\Li_2)\longto {\rm H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2) $$ is non zero. Then, ${\rm H}^{q_1+q_2}(G/B,\Li_1\otimes\Li_2)$ is an irreducible component of $V_1\otimes V_2$; such a component is said to be {\it cohomological}. Solving a Dimitrov-Roth conjecture, we prove here that the cohomological components of $V_1\otimes V_2$ are exactly the PRV components of stable multiplicity one. Note that Dimitrov-Roth already obtained some particular cases. We also characterize these components in terms of the geometry of the Eigencone of $G$. Along the way, we prove that the structure coefficients of the Belkale-Kumar product on ${\rm H}^*(G/B,\ZZ)$ in the Schubert basis are zero or one.
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Branching Schubert calculus and the Belkale-Kumar product on cohomology
In 2006, Belkale and Kumar define a new product on the cohomology of flag varieties and use this new product to give an improved solution to the eigencone problem for complex reductive groups. In this paper, we give a generalization of the Belkale-Kumar product to the branching Schubert calculus setting. The study of Branching Schubert calculus attempts to understand the induced map on cohomology of an equivariant embedding of flag varieties. The main application of our work is a compact formulation of the solution to the branching eigencone problem.
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Spherical homogeneous spaces of minimal rank
Let $G$ be a complex connected reductive algebraic group and $G/B$ denote the flag variety of $G$. A $G$-homogeneous space $G/H$ is said to be {\it spherical} if $H$ acts on $G/B$ with finitely many orbits. A class of spherical homogeneous spaces containing the tori, the complete homogeneous spaces and the group $G$ (viewed as a $G\times G$-homogeneous space) has particularly nice proterties. Namely, the pair $(G,H)$ is called a {\it spherical pair of minimal rank} if there exists $x$ in $G/B$ such that the orbit $H.x$ of $x$ by $H$ is open in $G/B$ and the stabilizer $H_x$ of $x$ in $H$ contains a maximal torus of $H$. In this article, we study and classify the spherical pairs of minimal rank.
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GIT-cones and quivers
In this work, we improve results about GIT-cones associated to the action of any reductive group $G$ on a projective variety $X$. These results are applied to give a short proof of a Derksen-Weyman's Theorem which parametrizes bijectively the faces of a rational cone associated to any quiver without oriented cycle.
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Geometric Invariant Theory and Generalized Eigenvalue Problem
Let $H$ be a connected reductive subgroup of a complex connected reductive group $G$. Fix maximal tori and Borel subgroups of $H$ and $G$. Consider the pairs $(V,V')$ of irreducible representations of $H$ and $G$ such that $V$ is a submodule of $V'$. We are interested in the cone $LR(G,H)$ generated by the pairs of dominant weights of such a pair of representations. Our main result gives a minimal set of inequalities describing $LR(G,H)$ as a part of the dominant chamber. In way, we obtain results about the faces of the Dolgachev-Hu's $G$-ample cone and variations of this cone.
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A short geometric proof of a conjecture of Fulton
We give a new geometric proof of a conjecture of Fulton on the Littlewood-Richardson coefficients. This conjecture was firstly proved by Knutson, Tao and Woodward using the Honeycomb theory. A geometric proof was given by Belkale. Our proof is based on the geometry of Horn's cones.
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Multiplicative formulas in Cohomology of $G/P$ and in quiver representations
Consider a partial flag variety $X$ which is not a grassmaninan. Consider also its cohomology ring ${\rm H}^*(X,\ZZ)$ endowed with the base formed by the Poincaré dual classes of the Schubert varieties. In \cite{Richmond:recursion}, E. Richmond showed that some coefficient structure of the product in ${\rm H}^*(X,\ZZ)$ are products of two such coefficients for smaller flag varieties. Consider now a quiver without oriented cycle. If $α$ and $β$ denote two dimension-vectors, $α\circβ$ denotes the number of $α$-dimensional subrepresentations of a general $α+β$-dimensional representation. In \cite{DW:comb}, H. Derksen and J. Weyman expressed some numbers $α\circβ$ as products of two smaller such numbers. The aim of this work is to prove two generalisations of the two above results by the same way.
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Invariant deformations of orbit closures in $\mathfrak{sl}_n$
We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be affine spaces.
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Lattice Polytopes and Root Systems
Consider a lattice in a real finite dimensional vector space. Here, we are interested in the lattice polytopes, that is the convex hulls of finite subsets of the lattice. Consider the group $G$ of the affine real transformations which map the lattice onto itself. Replacing the group of euclidean motions by the group $G$ one can define the notion of regular lattice polytopes. More precisely, a lattice polytope is said to be regular if the subgroup of $G$ which preserves the polytope acts transitively on the set of its complete flags. Recently, Karpenkov obtained a classification of the regular lattice polytopes. Here we obtain this classification by a more conceptual method. Another difference is that Karpenkov uses in an essential way the classification of the euclidean regular polytopes, but we don't.
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About some faces of the generalized Littlewood-Richardson cone
Let G be a connected reductive algebraic group and H be a reductive closed and connected subgroup of G both defined on an algebraically closed field of characteristic zero. We consider the set C of the couple (x,y) of the dominant weights such that the irreducible H-module of highest weight x is a submodule of the G-module corresponding to y. This set identifies canonically with a semi-group finitely generated in a rational vector space. We consider the generalized Littlewood-Richardson cone C' generated by C. This cone is polyhedral. For any (x',y') in C', x' is dominant, and so, satisfies a finite number (rank of H) of linear inequalities. We ask if these inequalities induce faces of codimension one of C'. By Geometric Invariant Theory methods, we give geometric criteria that imply an affirmative answer. We also apply our results to several classical example in representation theory.
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About Knop's action of the Weyl group on the set of the set of orbits of a spherical subgroup in the flag manifold
Let $G$ be a complex connected reductive algebraic group. Let $G/B$ denote the flag variety of $G$. Let $H$ be an algebraic subgroup of $G$ such that the set ${\bf H}(G/B)$ of the $H$-orbits in $G/B$ is finite ; $H$ is said to be {\it spherical}. These orbits are of importance in representation theory and in the geometry of the $G$-equivariant embeddings of $G/H$. In 1995, F. Knop has defined an action of the Weyl group $W$ of $G$ on ${\bf H}(G/B)$. The aim of this note is to construct natural invariants separating the $W$-orbits of Knop's action.
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The GIT-equivalence for $G$-line bundles
Let $X$ be a projective variety with an action of a reductive group $G$. Each ample $G$-line bundle $L$ on $X$ defines an open subset $X^{\rm ss}(L)$ of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of $L'$s with fixed $X^{\rm ss}(L)$. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the $G$-ample cone. We also study the corresponding variations of quotients $X^{\rm ss}(L)//G$. This sharpens results of Thaddeus and Dolgachev-Hu.
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