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We investigate which real flag manifolds of the form have the $${\mathbb Z_2}$$ -cup-length equal to the dimension. We obtain a complete classification of such manifolds of the form $${F(1,\ldots,1,2,m)}$$ and $${F(1,\ldots,1,2,2,m)}$$ . Additionally, we provide an infinite family of manifolds $${F(1,\ldots,1,2,\ldots,2,m)}$$ which give the negative answer to a question from J. Korbaš and J. Lörinc []. [ABSTRACT FROM AUTHOR]

Starting from Borel's description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring of semi-complete flag manifolds Fk,m := F(1, . . . , 1,m), where 1 is repeated k times. This is used to estimate Farber's topological complexity of Fk,m when m approaches (from below) a 2-power. In particular, we get almost sharp estimates for F2,2e-1 which resemble the known situation for the real projective spaces F1,2e. Our results indicate that the agreement between the topological complexity and the immersion dimension of real projective spaces no longer holds for other flag manifolds. We also get corresponding results for the s-th higher topological complexity of these spaces, proving the surprising fact that, as s increases, our cohomological estimates become stronger. Indeed, we get a full description of the higher motion planning problem of some of these manifolds. As a byproduct, we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not). [ABSTRACT FROM AUTHOR]

Let G be a connected real semisimple Lie group and H a closed connected subgroup. Let P be a minimal parabolic subgroup of G. It is shown that H has an open orbit on the flag manifold G/P if and only if it has finitely many orbits on G/P. This confirms a conjecture by T. Matsuki. [ABSTRACT FROM AUTHOR]

Let X = G / P be a complex flag manifold and E → X be a G -homogeneous holomorphic vector bundle. Fix a U -invariant Kähler metric on X with U ⊆ G maximal compact. We study the sheaf of nearly holomorphic sections and show that the space of global nearly holomorphic sections in E coincides with the space of U -finite smooth sections in E . The degree of nearly holomorphic sections defines a U -invariant filtration on this space. Using sheaf cohomology, we determine in suitable cases the corresponding Hilbert series. It turns out that this is given in terms of Lusztig's q -analog of Kostant's weight multiplicity formula, and hence gives a new representation theoretic interpretation of these formulas. [ABSTRACT FROM AUTHOR]

Abstract: Given a finite set of subspaces of , perhaps of differing dimensions, we describe a flag of vector spaces (i.e. a nested sequence of vector spaces) that best represents the collection based on a natural optimization criterion and we present an algorithm for its computation. The utility of this flag representation lies in its ability to represent a collection of subspaces of differing dimensions. When the set of subspaces all have the same dimension d, the flag mean is related to several commonly used subspace representations. For instance, the d-dimensional subspace in the flag corresponds to the extrinsic manifold mean. When the set of subspaces is both well clustered and equidimensional of dimension d, then the d-dimensional component of the flag provides an approximation to the Karcher mean. An intermediate matrix used to construct the flag can also be used to recover the canonical components at the heart of Multiset Canonical Correlation Analysis. Two examples utilizing the Carnegie Mellon University Pose, Illumination, and Expression Database (CMU-PIE) serve as visual illustrations of the algorithm. [Copyright &y& Elsevier]

Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ cohomology of the real flag manifolds $F(1,1,n)$ and $F(1,2,n)$ are obtained. These are used to compute appropriate Stiefel–Whitney classes in order to establish some new nonembedding and nonimmersion results for the manifolds $F(1,2,n)$. [ABSTRACT FROM AUTHOR]

Abstract: In this paper an extended Corach–Porta–Recht decomposition theorem for Finsler symmetric spaces of semi-negative curvature in the context of reductive structures is proven. This decomposition theorem is applied to give a geometric description of the complexification of some infinite dimensional homogeneous spaces. [Copyright &y& Elsevier]

We construct Koppelman formulas on manifolds of flags in $${\mathbb{C}^N}$$ for forms with values in any holomorphic line bundle as well as in the tautological vector bundles and their duals. As an application we obtain new explicit proofs of some vanishing theorems of the Bott-Borel-Weil type by solving the corresponding $${\bar{\partial}}$$-equation. We also construct reproducing kernels for harmonic ( p, q)-forms in the case of Grassmannians. [ABSTRACT FROM AUTHOR]

We prove that any invariant strong Kähler structure with torsion (SKT structure) on a flag manifold M = G/K of a semi-simple compact Lie group G is Kähler. As an application we describe invariant generalized Kähler structures on M. [ABSTRACT FROM PUBLISHER]

Abstract: The quantized flag manifold, which is a -analogue of the ordinary flag manifold, is realized as a non-commutative scheme, and we can define the category of -modules on it using the framework of non-commutative algebraic geometry; however, when the parameter is a root of unity, Lusztig’s Frobenius morphism allows us to handle -modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold. In this paper, we will show that this sheaf of rings on the ordinary flag manifold is an Azumaya algebra over its center. We also show that its restriction to certain subsets are split Azumaya algebras. These are analogues of some results of Bezrukavnikov–Mirković–Rumynin on -modules on flag manifolds in positive characteristics. [Copyright &y& Elsevier]

Abstract: We investigate the orientability of a class of vector bundles over flag manifolds of real semi-simple Lie groups, which include the tangent bundle and also stable bundles of certain gradient flows. Closed formulas, in terms of roots, are provided. [Copyright &y& Elsevier]

Going back to Kirwan and others, there is an established theory that uses moment map techniques to study actions by complex reductive groups on Kähler manifolds. Work of P. Heinzner, G. Schwarz, and H. Stötzel has extended this theory to actions by real reductive groups. In this paper, we apply these techniques to actions of the real group SU(1,1) ⊂ SL(2, ℂ) on a certain complex manifold of dimension two. More precisely, because of the SU(2)-invariant hyperkähler structure on this manifold, we are able to study a family of actions which includes and 'interpolates' two well-known actions of SL(2, ℂ): the adjoint action on the orbit of a semisimple element of 픰픩(2, ℂ), and the action of SL(2, ℂ) on the cotangent bundle of the flag variety of SL(2, ℂ). [ABSTRACT FROM AUTHOR]

This paper studies reversibility of subsemigroups acting on homogeneous spaces. The reversor set of a subsemigroup is defined and it is related to the invariant control sets for semigroups acting on certain homogeneous spaces. Let G be a connected noncompact semi-simple Lie group with finite center. Let L be a subgroup of G. Assume that S is a subsemigroup of G with int S≠∅. The main result characterizes the reversibility of the S-action on G/ L in terms of the actions of S and L on the flag manifolds of G. [ABSTRACT FROM AUTHOR]

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties. [ABSTRACT FROM AUTHOR]

In this paper we study the evolution operator of a time-dependent Hamiltonian in the three level system. The evolution operator is based on SU(3) and its dimension is eight, so we obtain three complex Riccati differential equations interacting with one another (which have been obtained by Fujii and Oike) and two real phase equations. This is a canonical form of the evolution operator. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group over ℂ and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in ℂ P n are prime if n satisfies certain conditions. [ABSTRACT FROM AUTHOR]

A hermitian algebra is a unital associative ℂ-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ℝ . In the case of an algebra A endowed with a Mackey-complete, locally convex topology such that the set of invertible elements is open and the inversion mapping is continuous, we construct the smooth structures on the appropriate versions of flag manifolds. Then we prove that if such a locally convex algebra A is endowed with a continuous involution, then it is a hermitian algebra if and only if the natural action of all unitary groups Un.A/ on each flag manifold is transitive. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincaré polynomial. Our main technical tools are chordal graphs and perfect elimination orderings. [Copyright &y& Elsevier]

The Landau problem on the flag manifold F2 = SU(3)/U(1)×U(1) is analyzed from an algebraic point of view. The involved magnetic background is induced by two U(1) Abelian connections. In quantizing the theory, we show that the wave functions, of a nonrelativistic particle living on F2, are the SU(3) Wigner ${\mathcal D}$-functions satisfying two constraints. Using the F2 algebraic and geometrical structures, we derive the Landau Hamiltonian as well as its energy levels. The lowest Landau level (LLL) wave functions coincide with the coherent states for the mixed SU(3) representations. We discuss the quantum Hall effect for a filling factor ν = 1, where the obtained particle density is constant and finite for a strong magnetic field. In this limit, we also show that the system behaves like an incompressible fluid. We study the semiclassical properties of the system confined in LLL. These will be used to discuss the edge excitations and construct the corresponding Wess–Zumino–Witten action. [ABSTRACT FROM AUTHOR]

The Landau problem on the flag manifold F2 = SU(3)/U(1)×U(1) is analyzed from an algebraic point of view. The involved magnetic background is induced by two U(1) Abelian connections. In quantizing the theory, we show that the wave functions, of a nonrelativistic particle living on F2, are the SU(3) Wigner ${\mathcal D}$-functions satisfying two constraints. Using the F2 algebraic and geometrical structures, we derive the Landau Hamiltonian as well as its energy levels. The lowest Landau level (LLL) wave functions coincide with the coherent states for the mixed SU(3) representations. We discuss the quantum Hall effect for a filling factor ν = 1, where the obtained particle density is constant and finite for a strong magnetic field. In this limit, we also show that the system behaves like an incompressible fluid. We study the semiclassical properties of the system confined in LLL. These will be used to discuss the edge excitations and construct the corresponding Wess–Zumino–Witten action. [ABSTRACT FROM AUTHOR]

In this note we study the Ledger conditions on the families of flag manifold (M6 = SU(3)/SU(1) x SU(1) x SU(1), g(c1,c2,c3)), (M12 = Sp(3)/SU(2) x SU(2) x SU(2), g(c1,c2,c3)), constructed by N.R. Wallach in [14]. In both cases, we conclude that every member of the both families of Riemannian flag manifolds is a D'Atri space if and only if it is naturally reductive. Therefore, we finish the study of M6 made by D'Atri and Nickerson in [7]. Moreover, we correct and improve the result given by the author and A. M. Naveira in [3] about M12. [ABSTRACT FROM AUTHOR]

On any manifold, any nondegenerate symmetric 2-form (metric) and any nondegenerate skew-symmetric differential form ω can be reduced to a canonical form at any point but not in any neighborhood: the corresponding obstructions are the Riemannian tensor and dω. The obstructions to flatness (to reducibility to a canonical form) are well known for any G-structure, not only for Riemannian or almost symplectic structures. For a manifold with a nonholonomic structure (nonintegrable distribution), the general notions of flatness and obstructions to it, although of huge interest (e.g., in supergravity) were not known until recently, although particular cases have been known for more than a century (e.g., any contact structure is nonholonomically “flat”: it can always be reduced locally to a canonical form). We give a general definition of the nonholonomic analogues of the Riemann tensor and its conformally invariant analogue, the Weyl tensor, in terms of Lie algebra cohomology and quote Premet’s theorems describing these cohomologies. Using Premet’s theorems and the SuperLie package, we calculate the tensors for flag manifolds associated with each maximal parabolic subalgebra of each simple Lie algebra (and in several more cases) and also compute the obstructions to flatness of the G(2)-structure and its nonholonomic superanalogue. [ABSTRACT FROM AUTHOR]

The conditions for a parallel manipulator to be flagged can be simply expressed in terms of linear dependencies between the coordinates of its leg attachments, both on the base and on the platform. These dependencies permit to describe the manipulator singularities in terms of incidences between two flags (hence, the name ‘flagged’). Although these linear dependencies might look, at first glance, too restrictive, in this paper, the family of flagged manipulators is shown to contain large subfamilies of six-legged and three-legged manipulators. The main interest of flagged parallel manipulators is that their singularity loci admit a well-behaved decomposition with a unique topology irrespective of the metrics of each particular design. In this paper, this topology is formally derived and all the cells, in the configuration space of the platform, of dimension 6 (nonsingular) and dimension 5 (singular), together with their adjacencies, are worked out in detail. [ABSTRACT FROM AUTHOR]

Abstract: The problem of classifying all homogeneous complex supermanifolds whose reduction is the complex Grassmannian is studied. Under the assumption that the odd isotropy representation is irreducible and under certain restrictions on , we prove that the only non-split supermanifold of this sort is the Π-symmetric super-Grassmannian constructed by Yu.I. Manin. [Copyright &y& Elsevier]

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cn and Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group. [ABSTRACT FROM AUTHOR]

Abstract: Our main result offers a new (quite systematic) way of deriving bounds for the cup-length of Poincaré spaces over fields; we outline a general research program based on this result. For the oriented Grassmann manifolds, already a limited realization of the program leads, in many cases, to the exact values of the cup-length and to interesting information on the Lyusternik–Shnirel''man category. [Copyright &y& Elsevier]

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad(G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold. [ABSTRACT FROM AUTHOR]

We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope PM which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number $\mathcal{E}(M)$ of isolate invariant holomorphic Einstein metrics (up to homothety) on Mℂ = Gℂ/Hℂ. Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that $\mathcal{E}(M)\le \nu(M)$, where ν(M) is the integer volume of PM, and give conditions when the defect $d (M)= \nu (M)-\mathcal{E}(M)=0$. In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space Mγ = Gγ/HP which is a contraction of M and is associated with a proper face γ of PM. [ABSTRACT FROM AUTHOR]

One describes, using a detailed analysis of Atiyah-Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle. [ABSTRACT FROM AUTHOR]

Abstract: We consider Springer fibers and orbital varieties for . We show that the irreducible components of an intersection of components of Springer fiber are in bijection with the irreducible components of intersection of orbital varieties; moreover, the corresponding irreducible components in this correspondence have the same codimension. Finally we give a sufficient condition to have an intersection in codimension one. [Copyright &y& Elsevier]

Some in-parallel robots, such as the 3-2-1 and the 3/2 manipulators, have attracted attention because their forward kine- matics can be solved by three consecutive trilaterations. In this paper, we identify a class of these robots, which we call flagged manipulators, whose singularity loci admit a well-behaved decomposition, i.e., a stratification, derived from that of the flag manifold. Two remarkable properties must be highlighted. First, the decomposition has the same topology for all members in the class, irrespective of the metric details of each particular robot instance. Thus, we provide explicitly all the singular strata and their connectivity, which apply to all flagged manipulators without any tailoring. Second, the strata can be easily characterized geometrically, because it is possible to assign local coordinates to each stratum (in the configuration space of the manipulator) that correspond to uncoupled rotations and/or translations in the workspace. [ABSTRACT FROM AUTHOR]

In this paper the classification of maps from a simply connected spaceXto a flag manifoldG/Tis studied. As an application, the structure of the homotopy set for self-maps of flag manifolds is determined. [ABSTRACT FROM AUTHOR]

We prove new results about the vector field problem on the real flag manifolds O(n)/O(1)xO(1)xO(n-2). For some infinite families of these manifolds, we completely solve the vector field problem. [ABSTRACT FROM AUTHOR]

Given an orbit space M/Γ and an equivalence relation defined in it by means of the action of a group G, we obtain a miniversal deformation of an orbit through a miniversal deformation in M with regard to a suitable group action of G×Γ. We show some applications to the perturbations of m-tuples of subspaces and (C,A)-invariant subspaces. [Copyright &y& Elsevier]

In the article [Spa1], N. Spaltenstein has established a bijection between the irreducible components of Fχ, the space of full flags fixed by a nilpotent element χ ε M(n, k), where k is an algebraically closed field, and the standard tableaux associated to the Young diagram of χ. In this present work we determine, when χ is of hook type, for each irreducible component X of Fχ, the unique Schubert cell CX of the full flag manifold F = F (V) (where V is vector space of dimension n over k), such that X ∩ CX is a dense subspace in X. This result will allow us to optimize the computation of Fχ and when k = C is the complex field, to see that the graph resolution of the partition (2, 1, …, 1) of n is related to the Dynkin diagram of sl(n, C). [Copyright &y& Elsevier]

We define a new family of partial orders generalizing the weak order on Coxeter groups called T-orders, where T is a set of reflections determining the covers in this order. We show that the Grassmann and Lagrange orders on the Coxeter groups of type An and Bn introduced by Bergeron and Sottile are in fact T-orders. These partial orders were used to compute certain products in the cohomology ring of the flag manifolds associated to the complex Chevalley groups of these types. We exhibit T-orders generalizing these orders to partial orders for the Coxeter groups of type Dn, E6, and E7. [Copyright &y& Elsevier]

The singular cohomology of the Grassmann variety of k-planes in Cn has a basis {sν} indexed by partitions. The classical Pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m ∪ sλ = ∑ c1m, λμsμ,where 1m is a column of length m and s1m is the mth Chern class of the tautological bundle. Lascoux and Schutzenberger [C. R. Acad. Sci. Paris294 (1982), 447–450] formulated a generalization of Pieri's formula to the cohomology of the flag variety SLn(C)/B and briefly indicated an algebraic proof. (Manivel [Cours Spe´cialise´s 3 (1998)] provides details of this proof.) A geometric proof was given by Sottile [Ann. Inst. Fourier (Grenoble) 46 (1996), 89–110]. In this paper we state and prove a generalization of this Pieri-type formula for the T-equivariant cohomology of the flag variety. We use the algebraic description of the T-equivariant cohomology of the flag variety due to Kostant and Kumar [Adv. Math.62 (1986), 187–237] and Arabia [Bull. Soc. Math. France117 (1989), 129–165], and our new formula exposes an equality of certain structure Constants in this algebra. Our proof is an induction based on the original idea in Lascoux and Schu¨tzenberger. [Copyright &y& Elsevier]

We address the problem of distinguishing between different Schubert cells using vanishing patterns of generalized Plücker coordinates. [ABSTRACT FROM AUTHOR]

We describe the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. It follows from this and our previous results that for the De Concini-Kac type quantized enveloping algebra, where the parameter q is specialized to a root of unity whose order is a prime power, the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber. This gives a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra. [ABSTRACT FROM AUTHOR]

We continue the study of the variety X[M] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of X[M] that are projective spaces. When M=G/T is the manifold of complete flags of a compact simple Lie group G, we obtain our main results. The first one characterizes those subspaces of the tangent space T[T] (M), invariant by the torus action and which give rise to real projective spaces in X[M]. The other one is the following. Let $$\mathfrak{p}$$ be the tangent space of the inner symmetric space G/K at [K] . Then RP ( $$\mathfrak{p}$$ ) is maximal in X[M] if and only if π2(G/K) does not vanish. [ABSTRACT FROM AUTHOR]