Orbital Varieties Research Articles

Computing fusion products of MV cycles using the Mirković–Vybornov isomorphism

The fusion of two Mirković–Vilonen cycles is a degeneration of their product, defined using the Beilinson–Drinfeld Grassmannian. In this paper, we put in place a conceptually elementary approach to computing this product in type A . We do so by transferring the problem to a fusion of generalized orbital varieties using the Mirković–Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of \mathbf{GL}_4 , confirming that all the cluster variables are contained in the Mirković–Vilonen basis.

Weierstrass sections for parabolic adjoint action in type A

The notion of “Weierstrass Section”, comes from Weierstrass canonical form for elliptic curves. In celebrated work Kostant (1963) [26] constructed such a section for the action of a semisimple Lie algebra on its dual using a principal s-triple. Actually it is enough to have an “adapted pair” and indeed the construction in Joseph and Shafrir (2010) [25] works rather well for the co-adjoint action of an algebraic, but not necessarily reductive Lie algebra.In the present work a Weierstrass section is constructed for the adjoint action of a parabolic subgroup P on the nilradical m of its Lie algebra in type A. The starting point is Richardson's theorem which asserts that m admits a dense P orbit (in all types).For type A an explicit generator of this dense orbit was given in Brüstle et al. (1999) [4]. It involves the joining boxes in the Young tableau associated to P.On the other hand Richardson's theorem implies the polynomiality of the (semi)-invariant subalgebra, where the generators themselves are given through ideals of definition of hypersurface orbital varieties Joseph and Melnikov (2003) [24]. (Again this relationship holds in all types.)In this adapted pairs seldom exist.Thus an entirely new construction is developed and this is mainly combinatorial.The initial step is the construction of Ringel et al. ([24]). In this the joining of boxes is significantly modified to take account of the explicit form of the generators of the invariant algebra suggested in Benlolo and Sanderson (2001) [3].Indications are given as to how one might extend this construction of a Weierstrass section to other types.

Fano manifolds and stability of tangent bundles

Abstract We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

Conormal varieties on the cominuscule Grassmannian-II

Let $X_w$ be a Schubert subvariety of a cominuscule Grassmannian $X$, and let $\mu:T^*X\rightarrow\mathcal N$ be the Springer map from the cotangent bundle of $X$ to the nilpotent cone $\mathcal N$. In this paper, we construct a resolution of singularities for the conormal variety $T^*_XX_w$ of $X_w$ in $X$. Further, for $X$ the usual or symplectic Grassmannian, we compute a system of equations defining $T^*_XX_w$ as a subvariety of the cotangent bundle $T^*X$ set-theoretically. This also yields a system of defining equations for the corresponding orbital varieties $\mu(T^*_XX_w)$. Inspired by the system of defining equations, we conjecture a type-independent equality, namely $T^*_XX_w=\pi^{-1}(X_w)\cap\mu^{-1}(\mu(T^*_XX_w))$. The set-theoretic version of this conjecture follows from this work and previous work for any cominuscule Grassmannian of type A, B, or C.

GENERALIZED ORBITAL VARIETIES FOR MIRKOVIĆ–VYBORNOV SLICES AS AFFINIZATIONS OF MIRKOVIĆ–VILONEN CYCLES

We show that generalized orbital varieties for Mirkovic-Vybornov slices can be indexed by semi-standard Young tableaux. We also check that the Mirkovic-Vybornov isomorphism sends generalized orbital varieties to (dense subsets of) Mirkovic-Vilonen cycles, such that the (combinatorial) Lusztig datum of a generalized orbital variety, which it inherits from its tableau, is equal to the (geometric) Lusztig datum of its MV cycle.

Nonlinear power-like iteration by polar decomposition and its application to tensor approximation

Low rank tensor approximation is an important subject with a wide range of applications. Most prevailing techniques for computing the low rank approximation in the Tucker format often first assemble relevant factors into matrices and then update by turns one factor matrix at a time. In order to improve two factor matrices simultaneously, a special system of nonlinear matrix equations over a certain product Stiefel manifold must be resolved at every update. The solution to the system consists of orbit varieties invariant under the orthogonal group action, which thus imposes challenges on its analysis. This paper proposes a scheme similar to the power method for subspace iterations except that the polar decomposition is used as the normalization process and that the iteration can be applied to both the orbits and the cross-sections. The notion of quotient manifold is employed to factor out the effect of orbital solutions. The dynamics of the iteration is completely characterized. An isometric isomorphism between the tangent spaces of two properly identified Riemannian manifolds is established to lend a hand to the proof of convergence.

On the existence of smooth orbital varieties in simple Lie algebras

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (non-exhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularities which arise as ‘exceptional orbit varieties’ of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements, and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a ‘complex geometry’ resulting from a transitive action of an appropriate algebraic group, yielding a compact ‘model submanifold’ for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2-torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the ‘stable range’ as the stable homotopy groups of the associated infinite-dimensional symmetric spaces. Lastly, we combine the preceding with a Theorem of Oka to obtain a class of ‘formal linear combinations’ of exceptional orbit hypersurfaces which have Milnor fibers that are homotopy equivalent to joins of the compact model submanifolds. It follows that Milnor fibers for all of these hypersurfaces are essentially never homotopy equivalent to bouquets of spheres (even allowing differing dimensions).

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularites which arise as exceptional orbit varieties of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a complex geometry resulting from a transitive action of an appropriate algebraic group, yielding a compact model submanifold for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2 torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. The cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the stable range as the stable homotopy groups of the associated infinite dimensional symmetric spaces. Applying a Theorem of Oka we obtain a class of formal linear combinations of exceptional orbit hypersurfaces which have Milnor fibers which are homotopy equivalent to joins of the compact model submanifolds.

Solvable groups, free divisors and nonisolated matrix singularities II: Vanishing topology

In this paper we use the results from the first part to compute the vanishing topology for matrix singularities based on certain spaces of matrices. We place the variety of singular matrices in a geometric configuration of free divisors which are the "exceptional orbit varieties" for repesentations of solvable groups. Because there are towers of representations for towers of solvable groups, the free divisors actually form a tower of free divisors $E_n$, and we give an inductive procedure for computing the vanishing topology of the matrix singularities. The inductive procedure we use is an extension of that introduced by L\^{e}-Greuel for computing the Milnor number of an ICIS. Instead of linear subspaces, we use free divisors arising from the geometric configuration and which correspond to subgroups of the solvable groups. Here the vanishing topology involves a singular version of the Milnor fiber; however, it still has the good connectivity properties and is homotopy equivalent to a bouquet of spheres, whose number is called the singular Milnor number. We give formulas for this singular Milnor number in terms of singular Milnor numbers of various free divisors on smooth subspaces, which can be computed as lengths of determinantal modules. In addition to being applied to symmetric, general and skew-symmetric matrix singularities, the results are also applied to Cohen--Macaulay singularities defined as 2 x 3 matrix singularities. We compute the Milnor number of isolated Cohen--Macaulay surface singularities of this type in $\mathbb{C}^4$ and the difference of Betti numbers of Milnor fibers for isolated Cohen--Macaulay 3--fold singularities of this type in $\mathbb{C}^5$.

The Brauer loop scheme and orbital varieties

A. Joseph invented multidegrees in Joseph (1984) to study orbital varieties, which are the components of an orbital scheme, itself constructed by intersecting a nilpotent orbit with a Borel subalgebra. Their multidegrees are known as Joseph polynomials, and these polynomials give a basis of a (Springer) representation of the Weyl group. In the case of the nilpotent orbit {M2=0}, the orbital varieties can be indexed by noncrossing chord diagrams in the disk.In this paper we study the normal cone to the orbital scheme inside this nilpotent orbit {M2=0}. This gives a better-motivated construction of the Brauer loop scheme we introduced in Knutson and Zinn-Justin (2007), whose components are indexed by all chord diagrams (now possibly with crossings) in the disk.The multidegrees of its components, the Brauer loop varieties, were shown to reproduce the ground state of the Brauer loop model in statistical mechanics (Di Francesco and Zinn-Justin, 2006). Here, we reformulate and slightly generalize these multidegrees in order to express them as solutions of the rational quantum Knizhnik–Zamolodchikov equation associated to the Brauer algebra. In particular, the vector of the multidegrees satisfies two sets of equations, corresponding to the ei and fi generators of the Brauer algebra. The proof of the analogous statement in Knutson and Zinn-Justin (2007) was slightly roundabout; we verified the fi equation using the geometry of multidegrees, and used algebraic results of Di Francesco and Zinn-Justin (2006) to show that it implied the ei equation. We describe here the geometric meaning of both ei and fi equations in our slightly extended setting.We also describe the corresponding actions at the level of orbital varieties: while only the ei equations make sense directly on the Joseph polynomials, the fi equations also appear if one introduces a broader class of varieties. We explain the connection of the latter with matrix Schubert varieties.

B-orbits of nilpotency order 2 and link patterns

Let Bn be the group of upper-triangular invertible n×n matrices and Xn be the set of strictly upper triangular n×n matrices of square zero seen as an algebraic variety. Bn acts on Xn by conjugation. In this paper we give first results on the geometry of orbits Xn/Bn in terms of link patterns.Further we apply this description to the computations of the closures of orbital varieties of nilpotency order 2 and their pairwise intersections. In particular, we connect our results on intersections to the combinatorics of meanders in Temperley–Lieb algebras and pairwise intersections of the components of a Springer fiber over a nilpotent element with two Jordan blocks.

Extended Joseph polynomials, quantized conformal blocks, and a [formula omitted]-Selberg type integral

We consider the tensor product V=(CN)⊗n of the vector representation of glN and its weight decomposition V=⊕λ=(λ1,…,λN)V[λ]. For λ=(λ1⩾⋯⩾λN), the trivial bundle V[λ]×Cn→Cn has a subbundle of q-conformal blocks at level ℓ, where ℓ=λ1−λN if λ1−λN>0 and ℓ=1 if λ1−λN=0. We construct a polynomial section Iλ(z1,…,zn,h) of the subbundle. The section is the main object of the paper. We identify the section with the generating function Jλ(z1,…,zn,h) of the extended Joseph polynomials of orbital varieties, defined in Di Francesco and Zinn-Justin (2005) [11] and Knutson and Zinn-Justin (2009) [12].For ℓ=1, we show that the subbundle of q-conformal blocks has rank 1 and Iλ(z1,…,zn,h) is flat with respect to the quantum Knizhnik–Zamolodchikov discrete connection.For N=2 and ℓ=1, we represent our polynomial as a multidimensional q-hypergeometric integral and obtain a q-Selberg type identity, which says that the integral is an explicit polynomial.

Smooth Orbital Varieties and Orbital Varieties with a Dense B-Orbit

The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of a reductive Lie algebra. Orbital varieties arise in representation theory as the geometric analogs of the enveloping algebra’s primitive ideals. They are also related to the irreducible components of the Springer fibers. In type A, the orbital varieties have a convenient parameterization by standard Young tableaux, in particular, one has a natural duality on the set of orbital varieties induced by the Young tableau transposition. In this paper, we provide several combinatorial criteria to guarantee that an orbital variety is smooth or contains a dense orbit for the action of the Borel subgroup B. We point out that, in all the cases that we can compute, the smooth orbital varieties and the orbital varieties which have a dense B-orbit correspond via the aforementioned duality.

Solvable group representations and free divisors whose complements are [formula omitted]ʼs

We apply previous results on the representations of solvable linear algebraic groups to construct a new class of free divisors whose complements are K ( π , 1 ) ʼs. These free divisors arise as the exceptional orbit varieties for a special class of “block representations” and have the structure of determinantal arrangements. Among these are the free divisors defined by conditions for the (modified) Cholesky-type factorizations of matrices, which contain the determinantal varieties of singular matrices of various types as components. These complements are proven to be homotopy tori, as are the Milnor fibers of these free divisors. The generators for the complex cohomology of each are given in terms of forms defined using the basic relative invariants of the group representation.

On geometric properties of orbital varieties in type A

The intersection between a nilpotent orbit O ⊂ gl n ( C ) and the Lie algebra Lie ( B ) ⊂ gl n ( C ) of a Borel subgroup B ⊂ GL n ( C ) is an equidimensional, quasi-affine algebraic variety. Its irreducible components are called orbital varieties. In this Note, we provide criteria to guarantee that an orbital variety is smooth or has a dense orbit for the adjoint action of B. In addition, we point out a possible relation between these two properties.

Some Characterizations of Singular Components of Springer Fibers in the Two-Column Case

Let u be a nilpotent endomorphism of a finite dimensional ℂ-vector space. The set \({\mathcal F}_u\) of u-stable complete flags is a projective algebraic variety called a Springer fiber. Its irreducible components are parameterized by a set of standard tableaux. We provide three characterizations of the singular components of \({\mathcal F}_u\) in the case u2 = 0. First, we give the combinatorial description of standard tableaux corresponding to singular components. Second, we prove that a component is singular if and only if its Poincare polynomial is not palindromic. Third, we show that a component is singular when it has too many intersections of codimension one with other components. Finally, relying on the second criterion, we infer that, for u general, whenever \({\mathcal F}_u\) has a singular component, it admits a component whose Poincare polynomial is not palindromic. This work relies on a previous criterion of singularity for components of \({\mathcal F}_u\) in the case u2 = 0 by the first author and on the description of the B-orbit decomposition of orbital varieties of nilpotent order two by the second author.

Codimension one intersections of the components of a Springer fiber for the two-column case

This paper is a subsequent paper of Melnikov and Pagnon: Reducibility of the intersections of components of a Springer fiber, Indag. Mathem. 19 (4) (2008) 611–631. Here we consider the irreducible components of a Springer fibre (or orbital varieties) for the two-column case in GL n (ℂ). We describe the intersection of two irreducible components, and specially give the necessary and sufficient condition for this intersection to be of codimension one. Since an orbital variety in the two-column case is a finite union of the Borel orbits, we solve the initial question by determining orbits of codimension one in the closure of a given orbit. We show that they are parameterized by a specific set of involutions called descendants, already introduced by the first author in a previous work. Applying this result we show that the codimension one intersection of two components is irreducible and provide the combinatorial description in terms of Young tableaux of the pairs of such components.

Reducibility of the intersections of components of a Springer fiber

The description of the intersections of components of a Springer fiber is a very complex problem. Up to now only two cases have been described completely. The complete picture for the hook case has been obtained by N. Spaltenstein and J.A. Vargas, and for two-row case by F.Y.C. Fung. They have shown in particular that the intersection of a pair of components of a Springer fiber is either irreducible or empty. In both cases all the components are non-singular and the irreducibility of the intersections is strongly related to the non-singularity. As it has been shown in J. Algebra 298 (2006) 1–4, a bijection between orbital varieties and components of the corresponding Springer fiber in GL n extends to a bijection between the irreducible components of the intersections of orbital varieties and the irreducible components of the intersections of components of Springer fiber preserving their codimensions. Here we use this bijection to compute the intersections of the irreducible components of Springer fibers for two-column case. In this case the components are in general singular. As we show the intersection of two components is non-empty. The main result of the paper is a necessary and sufficient condition for the intersection of two components of the Springer fiber to be irreducible in two-column case. The condition is purely combinatorial. As an application of this characterization, we give first examples of pairs of components with a reducible intersection having components of different dimensions.

The orbit structure of Dynkin curves

Let G be a simple algebraic group over an algebraically closed field k; assume that Char k is zero or good for G. Let \({\mathcal{B}}\) be the variety of Borel subgroups of G and let \(e \in {\rm Lie} G\) be nilpotent. There is a natural action of the centralizer CG(e) of e in G on the Springer fibre \({\mathcal{B}}_e = \{B' \in {\mathcal{B}} \mid e \in {\rm Lie} B'\}\) associated to e. In this paper we consider the case, where e lies in the subregular nilpotent orbit; in this case \({\mathcal{B}}_e\) is a Dynkin curve. We give a complete description of the CG(e)-orbits in \({\mathcal{B}}_e\) . In particular, we classify the irreducible components of \({\mathcal{B}}_e\) on which CG(e) acts with finitely many orbits. In an application we obtain a classification of all subregular orbital varieties admitting a finite number of B-orbits for B a fixed Borel subgroup of G.