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We consider equivariant integrals on flag manifolds of type A. Using a computational method inspired by the theory of wall-crossing formulas by Takuro Mochizuki, we re-prove residue formulas for equivariant integrals given by Weber and Zielenkiewicz. As an application, we give the determinantal formula of the Grothendieck polynomial by properly setting K theory classes. [ABSTRACT FROM AUTHOR]

Flag manifolds, sets of nested sequences of linear subspaces with fixed dimensions, are rising in numerical analysis and statistics. The current optimization algorithms on flag manifolds are based on the exponential map and parallel transport, which are expensive to compute. In this paper we propose practical optimization methods on flag manifolds without the exponential map and parallel transport. Observing that flag manifolds have a similar homogeneous structure with Grassmann and Stiefel manifolds, we generalize some typical retractions and vector transports to flag manifolds, including the Cayley-type retraction and vector transport, the QR-based and polar-based retractions, the projection-type vector transport and the projection of the differentiated polar-based retraction as a vector transport. Theoretical properties and efficient implementations of the proposed retractions and vector transports are discussed. Then we establish Riemannian gradient and Riemannian conjugate gradient algorithms based on these retractions and vector transports. Numerical results on the problem of nonlinear eigenflags demonstrate that our algorithms have a great advantage in efficiency over the existing ones. [ABSTRACT FROM AUTHOR]

A smooth curve on a homogeneous manifold G/H is called a Riemannian equigeodesic if it is a homogeneous geodesic for any G-invariant Riemannian metric. The homogeneous manifold G/H is called Riemannian equigeodesic, if for any x ∈ G/H and any nonzero y ∈ Tx(G/H), there exists a Riemannian equigeodesic c(t) with c(0) = x and ċ(0) = y. These two notions can be naturally transferred to the Finsler setting, which provides the definitions for Finsler equigeodesics and Finsler equigeodesic spaces. We prove two classification theorems for Riemannian equigeodesic spaces and Finsler equigeodesic spaces, respectively. Firstly, a homogeneous manifold G/H with a connected simply connected quasi compact G and a connected H is Riemannian equigeodesic if and only if it can be decomposed as a product of Euclidean factors and compact strongly isotropy irreducible factors. Secondly, a homogeneous manifold G/H with a compact semisimple G is Finsler equigeodesic if and only if it can be locally decomposed as a product, in which each factor is Spin(7)/G2, G2/SU(3) or a symmetric space of compact type. These results imply that the symmetric space and the strongly isotropy irreducible space of compact type can be interpreted by equigeodesic properties. As an application, we classify the homogeneous manifold G/H with a compact semisimple G such that all the G-invariant Finsler metrics on G/H are Berwald. It suggests a new project in homogeneous Finsler geometry, i.e., to systematically study the homogeneous manifold G/H on which all the G-invariant Finsler metrics satisfy a certain geometric property. [ABSTRACT FROM AUTHOR]

We provide explicit formulas for the Levi-Civita connection and Riemannian Hessian for a Riemannian manifold that is a quotient of a manifold embedded in an inner product space with a non-constant metric function. Together with a classical formula for projection, this allows us to evaluate Riemannian gradient and Hessian for several families of metrics on classical manifolds, including a family of metrics on Stiefel manifolds connecting both the constant and canonical ambient metrics with closed-form geodesics. Using these formulas, we derive Riemannian optimization frameworks on quotients of Stiefel manifolds, including flag manifolds, and a new family of complete quotient metrics on the manifold of positive-semidefinite matrices of fixed rank, considered as a quotient of a product of Stiefel and positive-definite matrix manifold with affine-invariant metrics. The method is procedural, and in many instances, the Riemannian gradient and Hessian formulas could be derived by symbolic calculus. The method extends the list of potential metrics that could be used in manifold optimization and machine learning. [ABSTRACT FROM AUTHOR]

By applying Gröbner basis theory to spectral sequences algebras, we develop a new computational methodology applicable to any Leray-Serre spectral sequence for which the cohomology of the base space is the quotient of a finitely generated polynomial algebra. We demonstrate the procedure by deducing the cohomology of the free loop space of flag manifolds, presenting a significant extension over previous knowledge of the topology of free loop spaces. A complete flag manifold is the quotient of a Lie group by its maximal torus. The rank of a flag manifold is the dimension of the maximal torus of the Lie group. The rank 2 complete flag manifolds are SU(3)/T 2, Sp(2)/T 2, Spin(4)/T², Spin(5)/T² and G2/T². In this paper we calculate the cohomology of the free loop space of the rank 2 complete flag manifolds. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss an important discourse, as most literatures highlighted the dimension of partial flag in a certain manner, according to a fixed definition that does not include all possible cases. In the present paper the existence of two different cases of flags in the partial flag varieties of the groups SO(n,c) and SP(2n,c) is explained wholly. In addition to this we will proceed by calculating the dimensions for all the given cases after pertaining the signature of these flags that are studied. [ABSTRACT FROM AUTHOR]

A study is made of algebraic curves and surfaces in the flag manifold F = S U (3) / T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane P 2 , defined with the help of an anti-holomorphic involution j . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P 3 of the 4-dimensional sphere S 4 . Deformations of twistor fibers project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1 , 1) . These surfaces define orthogonal complex structures on specified dense open subsets of P 2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1 , 1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F . [ABSTRACT FROM AUTHOR]

We provide two closed-form geodesic formulas for a family of metrics on Stiefel manifolds recently introduced by Hüper, Markina and Silva Leite, reparameterized by two positive numbers, having both the embedded and canonical metrics as special cases. The closed-form formulas allow us to compute geodesics by matrix exponential in reduced dimension for low-rank Stiefel manifolds. We follow the approach of minimizing the square Frobenius distance between a geodesic ending point to a given point on the manifold to compute the logarithm map and geodesic distance between two endpoints, using Fréchet derivatives to compute the gradient of this objective function. We focus on two optimization methods, gradient descent and L-BFGS. This leads to a new framework to compute the geodesic distance for manifolds with known geodesic formula but no closed-form logarithm map. We show the approach works well for Stiefel as well as flag manifolds. The logarithm map could be used to compute the Riemannian center of mass for these manifolds equipped with the above metrics. The method to translate directional derivatives using Fréchet derivatives to a gradient could potentially be applied to other matrix equations. [ABSTRACT FROM AUTHOR]

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical pde; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too—principal component, canonical correlation, and correspondence analyses may all be viewed as methods for extracting flags from a data set. The main goal of this article is to develop the tools needed for optimizing over a set of flags, which is a smooth manifold called the flag manifold, and it contains the Grassmannian as the simplest special case. We will derive closed-form analytic expressions for various differential geometric objects required for Riemannian optimization algorithms on the flag manifold; introducing various systems of extrinsic coordinates that allow us to parameterize points, metrics, tangent spaces, geodesics, distances, parallel transports, gradients, Hessians in terms of matrices and matrix operations; and thereby permitting us to formulate steepest descent, conjugate gradient, and Newton algorithms on the flag manifold using only standard numerical linear algebra. [ABSTRACT FROM AUTHOR]

This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar CP n − 1 and Grassmannian models. They naturally arise in the description of continuum limits of spin chains, and their phase structure is sensitive to the values of the topological angles, which are determined by the representations of spins in the chain. Gapless phases can in certain cases be explained by the presence of discrete 't Hooft anomalies in the continuum theory. We also discuss integrable flag manifold sigma models, which provide a generalization of the theory of integrable models with symmetric target spaces. These models, as well as their deformations, have an alternative equivalent formulation as bosonic Gross–Neveu models, which proves useful for demonstrating that the deformed geometries are solutions of the renormalization group (Ricci flow) equations, as well as for the analysis of anomalies and for describing potential couplings to fermions. [ABSTRACT FROM AUTHOR]

We review known and state some new results on homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of homotopy nilpotency of homogenous spaces G/K for a Lie group G and its closed subgroup K < G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)) and Ω(Vn,m(K)) of Grassmann Gn,m(K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is shown.

Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data. [ABSTRACT FROM AUTHOR]

We review known and state some new results on the homotopy nilpotency and co-nilpotency of spaces. Next, we take up the systematic study of the homotopy nilpotency of homogeneous spaces G/K for a Lie group G and its closed subgroup K ă G. The homotopy nilpotency of the loop spaces Ω(Gn,m(K)), Ω(Fn;n1,...,nk (K)), and Ω(Vn,m(K)) of Grassmann Gn,m(K), flag Fn;n1,...,nk (K) and Stiefel Vn,m(K) manifolds for K = R, C, the field of reals or complex numbers and H, the skew R-algebra of quaternions is studied. [ABSTRACT FROM AUTHOR]

We prove that for any positive integers n1, n2,..., nk, there exists a real flag manifold F(1,..., 1, n1,n2, ..., nk) with cup-length equal to its dimension. Additionally, we give a necessary condition that an arbitrary real flag manifold needs to satisfy in order to have cup-length equal to its dimension. [ABSTRACT FROM AUTHOR]

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1.

In this article we introduce flag Bott manifolds of general Lie type as the total spaces of iterated flag bundles. They generalize the notion of flag Bott manifolds and generalized Bott manifolds, and admit nice torus actions. We calculate the torus equivariant cohomology rings of flag Bott manifolds of general Lie type. [ABSTRACT FROM AUTHOR]

We compute the cup-length of flag manifolds F(2, 2, n3) and F(1, 3, 2s+1 − 3) and the height of the third Stiefel–Whitney characteristic class of the canonical vector bundle over the Grassmann manifold F(4, n). [ABSTRACT FROM AUTHOR]

For a given Hilbert space H , consider the space of self-adjoint projections P (H) . In this paper we study the differentiable structure of a canonical sphere bundle over P (H) given by R = { (P , f) ∈ P (H) × H : P f = f , ‖ f ‖ = 1 }. We establish the smooth action on R of the group of unitary operators of H , and it thereby turns out that the connected components of R are homogeneous spaces. Then we study the metric structure of R by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into R by the natural action of the unitary group. Then we study the restricted bundle R 2 + given by considering only the projections in the restricted Grassmannian, locally modeled by Hilbert–Schmidt operators. Therefore we endow R 2 + with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi–Civita connection of this metric and establish a Hopf–Rinow theorem for R 2 + , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds. [ABSTRACT FROM AUTHOR]

We use geometry to give a short proof of an equivariant Pieri rule in the classical flag manifold. This rule is due to Robinson, who gave an algebraic proof. [ABSTRACT FROM AUTHOR]

If G is a compact connected Lie group and T is a maximal torus, we give a wedge decomposition of ΣG/T by identifying families of idempotents in cohomology. This is used to give new information on the self-maps of G/T. [ABSTRACT FROM AUTHOR]

We enumerate smooth and rationally smooth Schubert varieties in the classical finite types A , B , C , and D , extending Haiman's enumeration for type A . To do this enumeration, we introduce a notion of staircase diagrams on a graph. These combinatorial structures are collections of steps of irregular size, forming interconnected staircases over the given graph. Over a Dynkin–Coxeter graph, the set of “nearly-maximally labelled” staircase diagrams is in bijection with the set of Schubert varieties with a complete Billey–Postnikov (BP) decomposition. We can then use an earlier result of the authors showing that all finite-type rationally smooth Schubert varieties have a complete BP decomposition to finish the enumeration. [ABSTRACT FROM AUTHOR]

The coadjoint orbits of compact Lie groups each carry a canonical (positive definite) Kähler structure, famously used to realize the group's irreducible representations in holomorphic sections of appropriate line bundles (Borel-Weil theorem). Less studied are the (indefinite) invariant pseudo -Kähler structures they also admit, which can be used to realize the same representations in higher cohomology of the sections (Bott's theorem). Using "eigenflag" embeddings, we give a very explicit description of these metrics in the case of the unitary group. As a byproduct we show that U n / (U n 1 × ⋯ × U n k ) has exactly k ! invariant complex structures, a count which seems to have hitherto escaped attention. [ABSTRACT FROM AUTHOR]

We propose a potential function for the cohomology ring of partial flag manifolds. We prove a formula expressing integrals over partial flag manifolds by residues, which generalizes [E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, Topology, Physics (International Press, 1995), pp. 357-422]. Using this formula, we prove a Landweber-Stong type vanishing theorem for generalized complete intersections in flag manifolds, which serves as evidence for the version of Stolz conjecture [Q. Chen, F. Han and W. Zhang, Generalized Witten genus and vanishing theorems, J. Differential Geom. 88(1) (2011) 1-39]. [ABSTRACT FROM AUTHOR]