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Building upon the application of flags to network coding introduced by Liebhold, Nebe, and Vazquez-Castro, we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding., Updated acknowledgements

Let $X$ be a complex manifold, $(E,h)\to X$ be a rank $r$ holomorphic hermitian vector bundle, and $\rho$ be a sequence of dimensions $0 = \rho_0 < \rho_1 < \cdots < \rho_m = r$. Let $Q_{\rho,j}$, $j=1,\dots,m$, be the tautological line bundles over the (possibly incomplete) flag bundle $\mathbb{F}_{\rho}(E) \to X$ associated to $\rho$, endowed with the natural metrics induced by that of $E$, with Chern curvatures $\Xi_{\rho,j}$. We show that the universal Gysin formula \textsl{\`{a} la} Darondeau--Pragacz for the push-forward of a homogeneous polynomial in the Chern classes of the $Q_{\rho,j}$'s also hold pointwise at the level of the Chern forms $\Xi_{\rho,j}$ in this hermitianized situation. As an application, we show the positivity of several polynomials in the Chern forms of a Griffiths (semi)positive vector bundle not previously known, thus giving some new evidences towards a conjecture by Griffiths, which in turn can be seen as a pointwise hermitianized version of the Fulton--Lazarsfeld Theorem on numerically positive polynomials for ample vector bundles., Comment: 24 pages, no figures, comments are very welcome! v3: several minor corrections, the main application is now stated for strongly positive forms

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of $I$-invariant projective vector fields. The sub-algebra of the $C$-projective vector fields, leaving the $H$-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature., 25 pages

We consider Hessenberg varieties in the flag variety of $GL_n(\mathbb{C})$ with the property that the corresponding Hessenberg function defines an ad-nilpotent ideal. Each such Hessenberg variety is contained in a Springer fiber. We extend a theorem of Tymoczko to this setting, showing that these varieties have an affine paving obtained by intersecting with Schubert cells. Our method of proof constructs an an affine paving for each Springer fiber that restricts to an affine paving of the Hessenberg variety. We use the combinatorial properties of this paving to prove that Hessenberg varieties of this kind are connected., Comment: 21 pages

We prove that if X is a Grassmannian of type A, then the Schubert basis of the (small) quantum cohomology ring QH(X) is the only homogeneous deformation of the Schubert basis of the ordinary cohomology ring of X that multiplies with non-negative structure constants. This implies that the (three point, genus zero) Gromov-Witten invariants of X are uniquely determined by Witten's presentation of QH(X) and the fact that they are non-negative. We conjecture that the same is true for any flag variety X = G/P of simply laced Lie type. For the variety GL(n)/B of complete flags, this conjecture is equivalent to Fomin, Gelfand, and Postnikov's conjecture that the quantum Schubert polynomials of type A are uniquely determined by positivity properties. Our proof for Grassmannians answers a question of Fulton., 16 pages

In this article, we investigate $2$-$(v,k,\lambda)$ designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$., Comment: arXiv admin note: text overlap with arXiv:1904.10518

The problem of extensions of the canonical Lee–Poisson–Kirillov–Kostant symplectic structure of coadjoint orbit of the complex general linear group is considered. The introduced method uses the concept of flag coordinates and does not depend on the Jordan structure of the matrices that form the orbit. The principal bundle associated with the fibration of the orbit over the Grassmanian of flags is constructed.

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $$(M=G/H,g)$$ whose geodesics are orbits of one-parameter subgroups of G. The corresponding metric g is called a geodesic orbit metric. We study the geodesic orbit spaces of the form (G/H, g), such that G is one of the compact classical Lie groups $${{\,\mathrm{SO}\,}}(n)$$ , $$\mathrm{U}(n)$$ , and H is a diagonally embedded product $$H_1\times \cdots \times H_s$$ , where $$H_j$$ is of the same type as G. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces (G/H, g) with H semisimple.

In their study of the equivariant K-theory of the generalized flag varieties G/P, where G is a complex semisimple Lie group, and P is a parabolic subgroup of G, Lenart and Postnikov introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand–Tsetlin pattern model for type A.

We introduce the notion of Darboux transformations for the strict KP hierarchy. We previously showed that solutions of this integrable hierarchy can be constructed from a flag variety $$ \mathcal{F} (1)$$ . Here, we describe which two points in this flag variety are connected by such a transformation. Moreover, we present a closed form of the operators that realize this transformation and describe their geometric characteristics. We show which of these Darboux transformations map solutions of the strict $$n$$ -KdV hierarchy to other solutions of this reduction of the strict KP hierarchy.

In 1987, Huw Davies proved that, for a flag-transitive point-imprimitive $2$-$(v,k,\lambda)$ design, both the block-size $k$ and the number $v$ of points are bounded by functions of $\lambda$, but he did not make these bounds explicit. In this paper we derive explicit polynomial functions of $\lambda$ bounding $k$ and $v$. For $\lambda\leq 4$ we obtain a list of `numerically feasible' parameter sets $v, k, \lambda$ together with the number of parts and part-size of an invariant point-partition and the size of a nontrivial block-part intersection. Moreover from these parameter sets we determine all examples with fewer than $100$ points. There are exactly eleven such examples, and for one of these designs, a flag-regular, point-imprimitive $2-(36,8,4)$ design with automorphism group ${\rm Sym}(6)$, there seems to be no construction previously available in the literature.

Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmetric matrices. This geometry implies that these polynomials are positive combinations of monomials in the variables $x_i + x_j$, and we give explicit formulas of this kind as sums over new objects called involution pipe dreams. Our formulas are analogues of the Billey-Jockusch-Stanley formula for Schubert polynomials. In Knutson and Miller's approach to matrix Schubert varieties, pipe dream formulas reflect Gr\"obner degenerations of the ideals of those varieties, and we conjecturally identify analogous degenerations in our setting., Comment: 36 pages, 2 figures; v2: reordered Sections 4 and 5, added references and examples; v3: some corrections, added exposition; v4: minor corrections; v5: fixed typo in equation (6.3), final version

Let $${\mathcal {D}}$$ be a 2- $$(v,k,\lambda )$$ symmetric design with k and $$\lambda $$ prime powers. If $${\mathcal {D}}$$ admits a flag-transitive, point-imprimitive automorphism group G, we show that both k and $$\lambda $$ must be powers of 2. Moreover, there exists an integer m such that either (a) $${\mathcal {D}}$$ has parameters $$(v,k,\lambda )=(2^{2m+2}-1, 2^{2m+1}, 2^{2m})$$ , and G preserves a partition of the points into $$2^{m+1}+1$$ classes of size $$2^{m+1}-1$$ , or (b) $${\mathcal {D}}$$ has parameters $$(v,k,\lambda )=((2^{2m-1}-2^m+1)(2^{m-1}+1),\ 2^{2m-1},\ 2^m)$$ , and G preserves a partition of the points into $$2^{2m-1}-2^m+1$$ classes of size $$2^{m-1}+1$$ .

Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.

We endow the disc $$D=\{(x_1,x_2)\in {\mathbb {R}}^2: x_1^2+x_2^2

Let $$\Gamma $$ be a finite simplicial graph such that the flag complex on $$\Gamma $$ is a 2-dimensional triangulated disk. We show that with some assumptions, the Dehn function of the associated Bestvina–Brady group is either quadratic, cubic, or quartic. Furthermore, we can identify the Dehn function from the defining graph $$\Gamma $$ .

Consider the affine space consisting of pairs of matrices ( A , B ) (A,B) of fixed size, and its closed subvariety given by the rank conditions rank ⁡ A ≤ a , rank ⁡ B ≤ b \operatorname {rank} A \leq a, \, \operatorname {rank} B \leq b , and rank ⁡ ( A ⋅ B ) ≤ c \operatorname {rank} (A\cdot B) \leq c , for three non-negative integers a , b , c a,b,c . These varieties are precisely the orbit closures of representations for the equioriented A 3 \mathbb {A}_3 quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of two Schur functors of tautological bundles on a 2-step flag variety. We provide several techniques for the determination of these groups, which is of independent interest.

We investigate the nondegeneracy of higher order Levi forms on weakly nondegenerate homogeneous $CR$ manifolds. Improving previous results, we prove that general orbits of real forms in complex flag manifolds have order less or equal $3$ and the compact ones less or equal~$2$. Finally we construct by Lee extensions weakly nondegenerate $CR$ vector bundles with arbitrary orders of nondegeneracy., Comment: 29 pages

Let $${\mathbb {K}}={\mathbb {R}},\,{\mathbb {C}}$$ K = R , C , the field of reals or complex numbers and $${\mathbb {H}}$$ H , the skew $${\mathbb {R}}$$ R -algebra of quaternions. We study the homotopy nilpotency of the loop spaces $$\Omega (G_{n,m}({\mathbb {K}}))$$ Ω ( G n , m ( K ) ) , $$\Omega (F_{n;n_1,\ldots ,n_k}({\mathbb {K}}))$$ Ω ( F n ; n 1 , … , n k ( K ) ) , and $$\Omega (V_{n,m}({\mathbb {K}}))$$ Ω ( V n , m ( K ) ) of Grassmann $$G_{n,m}({\mathbb {K}})$$ G n , m ( K ) , flag $$F_{n;n_1,\ldots ,n_k}({\mathbb {K}})$$ F n ; n 1 , … , n k ( K ) and Stiefel $$V_{n,m}({\mathbb {K}})$$ V n , m ( K ) manifolds. Additionally, homotopy nilpotency classes of p-localized $$\Omega (G^+_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( G n , m + ( K ) ( p ) ) and $$\Omega (V_{n,m}({\mathbb {K}})_{(p)})$$ Ω ( V n , m ( K ) ( p ) ) for certain primes p are estimated, where $$G^+_{n,m}({\mathbb {K}})_{(p)}$$ G n , m + ( K ) ( p ) is the oriented Grassmann manifolds. Further, the homotopy nilpotency classes of loop spaces of localized homogeneous spaces given as quotients of exceptional Lie groups are investigated as well.

An $$(n+1)$$ -toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group. Toroids are generalisations of maps on the torus to higher dimensions and also provide examples of abstract polytopes. Equivelar toroids are those that are induced by regular tessellations. In this paper we present a classification of equivelar $$(n+1)$$ -toroids with at most n flag-orbits; in particular, we discuss a classification of 2-orbit toroids of arbitrary dimension.

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal{R}_K$, to be a one-relator group; and for the Pontryagin algebra $H_*(\Omega \mathcal{Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterisation of these properties. For $RC_K'$, it is given by a condition on the homology group $H_2(\mathcal{R}_K)$, whereas for $H_*(\Omega \mathcal{Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal{Z}_K$., Comment: 16 pages, published version

We first classify left invariant Douglas $(\alpha , \beta )$-metrics on the Heisenberg group $H_{2n+1}$ of dimension $2n + 1$ and its extension i.e., oscillator group. Then we explicitly give the flag curvature formulas and geodesic vectors for these spaces, when equipped with these metrics. We also explicitly obtain $S$-curvature formulas of left invariant Randers metrics of Douglas type on these spaces and obtain a comparison on geometry of these spaces, when equipped with left invariant Douglas $(\alpha , \beta )$-metrics. More exactly, we show that although the results concerning bi-invariant Douglas $(\alpha ,\beta )$-metrics on these spaces are similar, several results concerning left invariant Douglas $(\alpha ,\beta )$-metrics on these spaces are different. For example we prove that the existence of left-invariant Matsumoto, Kropina and Randers metrics of Berwald type on oscillator groups can not extend to Heisenberg groups. Also we prove that oscillator groups have always vanishing $S$-curvature, whereas this can not occur on Heisenberg groups. Moreover, we prove that there exist new geodesic vectors on oscillator groups which can not extend to the Heisenberg groups.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. The Beilinson-Bernstein localization theorem establishes an equivalence of the category of $\mathfrak{g}$-modules of a fixed infinitesimal character and a category of modules over a twisted sheaf of differential operators on the flag variety of $\mathfrak{g}$. In this expository paper, we give four detailed examples of this theorem when $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$. Specifically, we describe the $\mathcal{D}$-modules associated to finite-dimensional irreducible $\mathfrak{g}$-modules, Verma modules, Whittaker modules, discrete series representations of $SL(2,\mathbb{R})$, and principal series representations of $SL(2,\mathbb{R})$.

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample line bundle on $X$. Given a toric degeneration of $(X,L)$ satisfying some natural technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $\mathcal{B}_s$ of $H^0(X,L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of a moment map associated to $X$ and its toric degeneration. The theory of Newton-Okounkov bodies and its associated toric degenerations shows that the technical hypotheses mentioned above hold in some generality. Our results significantly generalize previous results in geometric quantization which prove "independence of polarization" between Kahler quantizations and real polarizations. As an example, in the case of general flag varieties $X=G/B$ and for certain choices of $\lambda$, our result geometrically constructs a continuous degeneration of the (dual) canonical basis of $V_{\lambda}^*$ to a collection of dirac delta functions supported at the Bohr-Sommerfeld fibres corresponding exactly to the lattice points of a Littelmann-Berenstein-Zelevinsky string polytope $\Delta_{\underline{w}_0}(\lambda) \cap \mathbb{Z}^{\dim(G/B)}$.

In this two-part paper, we introduce a $$p$$ -adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when $$q$$ is not a root of unity and $$\vert q-1\vert

A special case of the geometric Langlands correspondence is given by the relationship between solutions of the Bethe ansatz equations for the Gaudin model and opers-connections on the projective line with extra structure. In this paper, we describe a deformation of this correspondence for $${{\,\mathrm{\mathrm {SL}}\,}}(N)$$ . We introduce a difference equation version of opers called q-opers and prove a q-Langlands correspondence between nondegenerate solutions of the Bethe ansatz equations for the XXZ model and nondegenerate twisted q-opers with regular singularities on the projective line. We show that the quantum/classical duality between the XXZ spin chain and the trigonometric Ruijsenaars–Schneider model may be viewed as a special case of the q-Langlands correspondence. We also describe an application of q-opers to the equivariant quantum K-theory of the cotangent bundles to partial flag varieties.

For an arbitrary complex (partial) flag manifold F, a Grobner basis for the ideal which (in the Borel picture) determines the cohomology algebra H ⁎ ( F ; Z ) is obtained. This Grobner basis is used to derive multiplication rules for a convenient additive basis of H ⁎ ( F ; Z ) given in terms of Chern classes of the canonical bundles over F. The analogous Grobner bases related to the mod 2 cohomology of real flag manifolds are also presented.

Let X be a smooth algebraic variety over an arbitrary field. Let φ be the canonical surjective homomorphism of the Chow ring of X onto the ring associated with the Chow filtration on the Grothendieck ring K ( X ) . We remark that φ is injective if and only if the connective K-theory CK ( X ) coincides with the terms of the Chow filtration on K ( X ) . As a consequence, CK ( X ) turns out to be computed for numerous flag varieties (under semisimple algebraic groups) for which the injectivity of φ had already been established. This especially applies to the so-called generic flag varieties X of many different types, identifying for them CK ( X ) with the terms of the explicit Chern filtration on K ( X ) . Besides, for arbitrary X, we compare CK ( X ) with the fibered product of the Chow ring of X and the graded ring formed by the terms of the Chow filtration on K ( X ) .

This paper is a review of results on the structure of homogeneous ind-varieties G/P of the ind-groups G = GL∞(ℂ), SL∞(ℂ), SO∞(ℂ), and Sp∞(ℂ), subject to the condition that G/P is the inductive limit of compact homogeneous spaces Gn/Pn. In this case, the subgroup P ⊂ G is a splitting parabolic subgroup of G and the ind-variety G/P admits a “flag realization.” Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains $$ \mathcal{C} $$ of subspaces in the natural representation V of G satisfying a certain condition; roughly speaking, for each nonzero vector 𝜐 of V , there exist the largest space in $$ \mathcal{C} $$ , which does not contain 𝜐, and the smallest space in $$ \mathcal{C} $$ , which contains v. We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form G/P for splitting parabolic ind-subgroups P ⊂ G. Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian X, we give a purely algebraic-geometric construction of X. Further topics discussed are the Bott–Borel–Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of G/P for arbitrary splitting parabolic ind-subgroups P ⊂ G, as well as the orbits of real forms on G/P for G = SL∞(ℂ).

For a reductive group $G$, Steinberg established a map from the Weyl group to the set of nilpotent $G$-orbits by using moment maps on double flag varieties. In particular, in the case of the general linear group, it provides a geometric interpretation of the Robinson-Schensted correspondence between permutations and pairs of standard tableaux of the same shape. We extend Steinberg's approach to the case of a symmetric pair $(G,K)$ to obtain two different maps, namely a \emph{generalized Steinberg map} and an \emph{exotic moment map}. Although the framework is general, in this paper we focus on the pair $(G,K) = (\mathrm{GL}_{2n}(\mathbb{C}), \mathrm{GL}_n(\mathbb{C}) \times \mathrm{GL}_n(\mathbb{C}))$. Then the generalized Steinberg map is a map from \emph{partial} permutations to the pairs of nilpotent orbits in $ \mathfrak{gl}_n(\mathbb{C}) $. It involves a generalization of the classical Robinson--Schensted correspondence to the case of partial permutations. The other map, the exotic moment map, establishes a combinatorial map from the set of partial permutations to that of signed Young diagrams, i.e., the set of nilpotent $K$-orbits in the Cartan space $(\mathrm{Lie}(G)/\mathrm{Lie}(K))^* $. We explain the geometric background of the theory and combinatorial algorithms which produce the above mentioned maps., Comment: 51 pages, 3 figures. Minor modifications. Accepted in the International Mathematics Research Notices

Let V be a finite dimensional complex vector space, and for each positive integer n let S n ( V ) denote the vector space whose elements are the complex-valued symmetric n-linear functions on V × V × ⋯ × V (n-copies). If f ∈ S 1 ( V ) , that is, f is a linear functional on V, then we define the symmetric n-linear function f n by f n ( x 1 , x 2 , … , x n ) = ∏ i = 1 n f ( x i ) for all x 1 , x 2 , … , x n ∈ V . It is well known that the elements f n , known as polar elements, span S n ( V ) . Our focus is the simple generation of sets of linearly independent polar elements of given order n. A basis for S n ( V ) consisting entirely of polar elements is a polar basis. We introduce several new structures that are closely connected to the problem of generating independent sets of polar elements. In particular, we introduce the notion of independent multistructure of order n, n ≥ 0 , and depth p, p ≥ 1 , and show how such structures generate sets of linearly independent polar elements of cardinality ( n + p p ) . When p = dim ⁡ ( V ) we obtain a basis for S n ( V ) consisting entirely of polar elements. We also introduce the dual concept of flag structure of order n and depth p. Both of these concepts evolved from a consideration of subspace filters which were defined previously in [6] .

Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note, we prove that arrangements that admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end, we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary, we give a new proof that irreducible arrangements with a generic hyperplane are totally non-free. In another application, we show that an irreducible multiarrangement of rank $3$ with at least two locally heavy hyperplanes is not free., Comment: 16 pages, 1 figure. arXiv admin note: text overlap with arXiv:1603.05803

Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits and we show that each $Z$-orbit has a structure of algebraic affine bundle. In particular, in type $A$, we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type $A$, we show that the Bruhat order of the $Z$-orbits can be described in this way., 42 pages, 2 figures

We address the problem of automatic detection and labelling of far-field activity patterns from target trajectories using vector field abstractions that are estimated by multiple communicating agents in a Kalman filter (KF) framework. We propose a novel 2D application of the diffusion KF. The proposed approach yields multiple vector field abstractions at the same spatial regions by allowing the estimator agents to create a library of states. We compute internal consistency measures to assess the estimated vector fields, and establish thresholds that signal low performance estimates. Experimental results on synthetic and real data sets show that the proposed approach correctly detects activity patterns from training trajectories and, using new test trajectories one at a time, either matches them to previously detected activity patterns or detects new activity patterns that are added to the agents library. Moreover, the internal consistency measures correctly flag the low-performer vector field abstractions.

The non-Riemannian quantity H was introduced by Akbar-Zadeh to characterization of Finsler metrics of constant flag curvature. In this paper, we study two important subclasses of Finsler metrics in the class of so-called (α,β)-metrics, which are defined by F=αϕ(s), s=β/α, where α is a Riemannian metric and β is a closed 1-form on a manifold. We prove that every polynomial metric of degree k≥3 and exponential metric has almost vanishing H-curvature if and only if H=0. In this case, F reduces to a Berwald metric. Then we prove that every Einstein polynomial metric of degree k≥3 and exponential metric satisfies H=0. In this case, F is a Berwald metric.

The current state of the art in supervised descriptive pattern mining is very good in automatically finding subsets of the dataset at hand that are exceptional in some sense. The most common form, subgroup discovery, generally finds subgroups where a single target variable has an unusual distribution. Exceptional model mining (EMM) typically finds subgroups where a pair of target variables display an unusual interaction. What these methods have in common is that one specific exceptionality is enough to flag up a subgroup as exceptional. This, however, naturally leads to the question: can we also find multiple instances of exceptional behaviour simultaneously in the same subgroup? This paper provides a first, affirmative answer to that question in the form of the SPEC (Subsets of Pairwise Exceptional Correlations) model class for EMM. Given a set of predefined numeric target variables, SPEC will flag up subgroups as interesting if multiple target pairs display an unusual rank correlation. This is a fundamental extension of the EMM toolbox, which comes with additional algorithmic challenges. To address these challenges, we provide a series of algorithmic solutions whose strengths/flaws are empirically analysed.

We study the truncation functors and show the existence of projective cover with a finite Verma flag of each irreducible module in parabolic BGG category O over infinite rank Lie algebra of types a , b , c , d . Moreover, O is a Koszul category. As a consequence, the corresponding parabolic BGG category O ‾ over infinite rank Lie superalgebra of types a , b , c , d through the super duality is also a Koszul category.

We establish a multiplication formula for a tridiagonal standard basis element in the idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$ arising from the geometry of affine partial flag varieties of type $C$. We apply this formula to obtain the stabilization algebras $\dot{\mathbf K}^{\mathfrak{c}}_n$, $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$, $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \imath}_{\eta}$, which are idempotented coideal subalgebras of quantum affine $\mathfrak{gl}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras $\dot{\mathbf K}^{\jmath \imath}_{\mathfrak{n}}$ and $\dot{\mathbf K}^{\imath \jmath}_{\mathfrak{n}}$ with compatible monomial, standard and canonical bases., Comment: 39 pages, fixed some typos, to appear in Journal of Pure and Applied Algebra

Let \( \mathfrak{g} \) be a complex semisimple Lie algebra. Although the quantum group \( {U}_{\hslash \mathfrak{g}} \) is known to be isomorphic, as an algebra, to the undeformed enveloping algebra \( U\mathfrak{g}\left\llbracket \hslash \right\rrbracket \), no such isomorphism is known when \( \mathfrak{g}\ne \mathfrak{s}{\mathfrak{l}}_2 \). In this paper, we construct an explicit isomorphism for \( \mathfrak{g}=\mathfrak{s}{\mathfrak{l}}_n \), for every n ≥ 2, which preserves the standard flag of type A. We conjecture that this isomorphism quantizes the Poisson diffeomorphism of Alekseev and Meinrenken [2].

For a right-invariant control system on a flag manifold $${\mathbb{F}_{\Theta}}$$ of a real semisimple Lie group, we relate the $$\mathfrak{a}$$ -Lyapunov exponents to the Lyapunov exponents of the system over regular points. Moreover, we adapt the concept of partial hyperbolicity from the theory of smooth dynamical systems to control-affine systems, and we completely characterize the partially hyperbolic chain control sets on $${\mathbb{F}_{\Theta}}$$ .

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $${\mathfrak {g}}$$ , we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of $${\mathscr {S}}({\mathfrak {g}})$$ on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in $${\mathfrak {g}}\simeq {\mathfrak {g}}^{*}$$ .

Let [ R n , k ] n , k ≥ 0 be an array of nonnegative numbers satisfying the recurrence relation R n , k = ( a 1 n + a 2 k + a 3 ) R n − 1 , k + ( b 1 n + b 2 k + b 3 ) R n − 1 , k − 1 + ( c 1 n + c 2 k + c 3 ) R n − 1 , k − 2 with R 0 , 0 = 1 and R n , k = 0 unless 0 ≤ k ≤ n . In this paper, we first prove that the array [ R n , k ] n , k ≥ 0 can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of [ R n , k ] n , k ≥ 0 . Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type B , and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong q -log-convexity of some generating functions.

In Weil (2001) formulae were proved for stationary Boolean models Z in ℝd with convex or polyconvex grains, which express the densities (specific mean values) of mixed volumes of Z in terms of related mean values of the underlying Poisson particle process X. These formulae were then used to show that in dimensions 2 and 3 the densities of mixed volumes of Z determine the intensity γ of X. For d = 4, a corresponding result was also stated, but the proof given was incomplete, since in the formula for the density of the Euler characteristic V̅0(Z) of Z a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result has recently been proved by Hug, Rataj, and Weil (2013), (2018) and is based on flag measures of the convex bodies involved. Here, we show that such flag representations not only lead to a correct derivation of the four-dimensional result, but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter we make use of Alesker’s representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the nonstationary case.

We prove a Chevalley formula for anti-dominant weights in the torus-equivariant K-group of semi-infinite flag manifolds, which is described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or equivalently, quantum Lakshmibai-Seshadri paths); in contrast to the Chevalley formula for dominant weights in our previous paper [17] , the formula for anti-dominant weights has a significant finiteness property. Based on geometric results established in [17] , our proof is representation-theoretic, and the Chevalley formula for anti-dominant weights follows from a certain identity for the graded characters of Demazure submodules of a level-zero extremal weight module over a quantum affine algebra; in the proof of this identity, we make use of the (combinatorial) standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, and also a string property of Demazure-like subsets of the set of semi-infinite Lakshmibai-Seshadri paths of a fixed shape, which gives an explicit realization of the crystal basis of a level-zero extremal weight module.

We propose the Lie-algebraic interpretation of poly-analytic functions in $L_2(\C,d\mu)$, with the Gaussian measure $d\mu$, based on a flag structure formed by the representation spaces of the $\mathfrak{sl}(2)$-algebra realized by differential operators in $z$ and $\bar z$. Following the pattern of the one-dimensional situation, we define poly-Fock spaces in $d$ complex variables in a Lie-algebraic way, as the invariant spaces for the action of generators of a certain Lie algebra. In addition to the basic case of the algebra $\mathfrak{sl}(d+1)$, we consider also the family of algebras $\mathfrak{sl}(m_1+1) \otimes \ldots \otimes \mathfrak{sl}(m_n+1)$ for tuples $\mathbf{m} = (m_1,m_2,\ldots,m_n)$ of positive integers whose sum is equal to $d$., Comment: 19 pages

Schubert polynomials form a basis of all polynomials and appear in the study of cohomology rings of flag manifolds. The vanishing problem for Schubert polynomials asks if a coefficient of a Schubert polynomial is zero. We give a tableau criterion to solve this problem, from which we deduce the first polynomial time algorithm. These results are obtained from new characterizations of the Schubitope, a generalization of the permutahedron defined for any subset of the n x n grid. In contrast, we show that computing these coefficients explicitly is #P-complete., 30 pages. This paper was split off from 1810.10361v1, with a new title and abstract. That earlier preprint has been replaced by a conference proceedings version 1810.10361v2, with a different title and abstract; it contains complexity discussion and a related conjecture not found here. To appear in Advances in Math

Kato's exotic nilpotent cone was introduced as a substitute for the ordinary nilpotent cone of type C with cleaner properties. The geometric Robinson-Schensted correspondence is obtained by parametrizing the irreducible components of the Steinberg variety (the conormal variety for the action of a semisimple group on two copies of its flag variety); in type A the bijection coincides with the classical Robinson-Schensted algorithm for the symmetric group. Here we give a combinatorial description of the bijection obtained by using the exotic nilpotent cone instead of ordinary type C nilpotent cone in the geometric Robinson-Schensted correspondence; we refer this as the "exotic Robinson-Schensted bijection". This is interesting from a combinatorial perspective, and not a naive extension of the type A Robinson-Schensted bijection., Comment: 33 pages

In this paper, we study $$\Xi $$ -curvature and H-curvature of a special class of Finsler metrics called general $$(\alpha ,\beta )$$ -metrics. We prove that every general $$(\alpha ,\beta )$$ -metric of almost vanishing H-curvature is of almost vanishing $$ \Xi $$ -curvature under certain conditions. Moreover, we study such Finsler metric with vanishing $$\Xi $$ -curvature and its interaction to the flag curvature.

Let g be a complex simple Lie algebra and let g˜ be the corresponding affine Kac-Moody Lie algebra. Fix a positive integer l. Let R1(G) be the free Abelian group generated by the set of integrable highest weight irreducible g˜-modules of level l. Then there is a fusion product in R1(G) making it into a commutative and associative algebra. The definition of the product is in terms of the dimension of a certain space of vacua. This chapter gives a new definition of a fusion product in terms of the Euler-Poincare characteristic of certain homogeneous vector bundles on the generalised affine flag variety X. A comparison of the two fusion products led us to define a certain chain-complex F^ whose terms are finite-dimensional G-modules. The differentials of this complex are highly non-trivial and are obtained by considering the BGG resolution for the affine Kac-Moody algebra g˜.