The recursive nature of cominuscule Schubert calculus

The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood-Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood-Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.     

A combinatorial rule for (co)minuscule Schubert calculus

We prove a root system uniform, concise combinatorial rule for Schubert calculus of minuscule and cominuscule flag manifolds G/P (the latter are also known as compact Hermitian symmetric spaces). We connect this geometry to the poset combinatorics of Proctor, thereby giving a generalization of Schützenberger's jeu de taquin formulation of the Littlewood-Richardson rule that computes the intersection numbers of Grassmannian Schubert varieties. Our proof introduces cominuscule recursions, a general technique to relate the numbers for different Lie types.     

Smooth and palindromic Schubert varieties in affine Grassmannians

We completely determine the smooth and palindromic Schubert varieties in affine Grassmannians, in all Lie types. We show that an affine Schubert variety is smooth if and only if it is a closed parabolic orbit. In particular, there are only finitely many smooth affine Schubert varieties in a given Lie type. An affine Schubert variety is palindromic if and only if it is a closed parabolic orbit, a chain, one of an infinite family of “spiral” varieties in type A, or a certain 9-dimensional singular variety in type B ₃. In particular, except in type A there are only finitely many palindromic affine Schubert varieties in a fixed Lie type. Moreover, in types D and E an affine Schubert variety is smooth if and only if it is palindromic; in all other types there are singular palindromics. The proofs are for the most part combinatorial. The main tool is a variant of Mozes’ numbers game, which we use to analyze the Bruhat order on the coroot lattice. In the proof of the smoothness theorem we also use Chevalley’s cup product formula.     

A partial Horn recursion in the cohomology of flag varieties

Horn recursion is a term used to describe when non-vanishing products of Schubert classes in the cohomology of complex flag varieties are characterized by inequalities parameterized by similar non-vanishing products in the cohomology of “smaller” flag varieties. We consider the type A partial flag variety and find that its cohomology exhibits a Horn recursion on a certain deformation of the cup product defined by Belkale and Kumar (Invent. Math. 166:185–228, 2006). We also show that if a product of Schubert classes is non-vanishing on this deformation, then the associated structure constant can be written in terms of structure constants coming from induced Grassmannians.     

Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S₃-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.     

Geometric cycles with local coefficients and the cohomology of arithmetic subgroups of the exceptional group <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

We study the cohomology of a compact locally symmetric space attached to an arithmetic subgroup of a rational form of a group of type G ₂ with values in a finite dimensional irreducible representation E of G ₂. By constructing suitable geometric cycles and parallel sections of the bundle [(E)\tilde]{\tilde{E}} we prove non-vanishing results for this cohomology. We prove all possible non-vanishing results compatible with the known vanishing theorems regarding unitary representations with non-zero cohomology in the case of the short fundamental weight of G ₂. A decisive tool in our approach is a formula for the intersection numbers with local coefficients of two geometric cycles.     

Lower bounds for real solutions to sparse polynomial systems

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through the sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.     

The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants

We give conditions on a curve class that guarantee the vanishing of the structure constants of the small quantum cohomology of partial flag varieties F(k ₁, ..., k _r ; n) for that class. We show that many of the structure constants of the quantum cohomology of flag varieties can be computed from the image of the evaluation morphism. In fact, we show that a certain class of these structure constants are equal to the ordinary intersection of Schubert cycles in a related flag variety. We obtain a positive, geometric rule for computing these invariants (see Coskun in A Littlewood–Richardson rule for partial flag varieties, preprint). Our study also reveals a remarkable periodicity property of the ordinary Schubert structure constants of partial flag varieties.     

Restriction varieties and geometric branching rules

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO(n) in SU(n), the asymptotic of the restrictions of representations of SL(n) to SO(n) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.     

PSPACE complexity of modal logic <Emphasis Type="Italic">K D</Emphasis>45<Subscript>n</Subscript>

We study sequent calculus for multi-modal logic K D45_n and its complexity. We introduce a loop-check free sequent calculus. Loop-check is eliminated by using the marked modal operator □_i^•, which is used as an alternative to sequents with histories ([8], [3], [5]). All inference rules are invertible or semi-invertible. To get this, we use or branches beside common and branches. We prove the equivalence between known sequent calculus and our newly introduced efficient sequent calculus. We concentrate on the complexity analysis of the introduced sequent calculus for multi-modal logic K D45_n. We prove that the space complexity of the given calculus is polynomial (O(l ³)). We show the maximum height of the constructed derivation tree that leads to the reduction of the time and space complexity. We present a decision algorithm for multi-modal logic K D45_n and some nontrivial examples to improve the introduced loop-check free sequent calculus.     

Approximation by Conic Splines

We show that the complexity of a parabolic or conic spline approximating a sufficiently smooth curve with non-vanishing curvature to within Hausdorff distance ɛ is c ₁ɛ^−1/4 + O(1), if the spline consists of parabolic arcs, and c ₂ɛ^−1/5 + O(1), if it is composed of general conic arcs of varying type. The constants c ₁ and c ₂ are expressed in the Euclidean and affine curvature of the curve. We also show that the Hausdorff distance between a curve and an optimal conic arc tangent at its endpoints is increasing with its arc length, provided the affine curvature along the arc is monotone. This property yields a simple bisection algorithm for the computation of an optimal parabolic or conic spline. The research of SG and GV was partially supported by grant 6413 of the European Commission to the IST-2002 FET-Open project Algorithms for Complex Shapes in the Sixth Framework Program.     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Infinitary calculus for a restricted first-order linear temporal logic without contraction on quantified formulas

An infinitary calculus for a restricted fragment of the first-order linear temporal logic is considered. We prove that for this fragment one can construct the infinitary calculusG _Lω ^* without contraction on predicate formulas. The calculusG _Lω ^* possesses the following properties: (1) the succedent rule for the existential quantifier is included into the corresponding axiom; (2) the premise of the antecedent rule for the universal quantifier does not contain a duplicate of the main formula. The soundness and completness ofG _Lω ^* are also proved. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 378–397, July–September, 1999. Translated by R. Lapinskas     

Kazhdan-Lusztig Polynomials for 321-Hexagon-Avoiding Permutations

In (Deodhar, Geom. Dedicata, 36(1) (1990), 95–119), Deodhar proposes a combinatorial framework for determining the Kazhdan-Lusztig polynomials P _x , _w in the case where W is any Coxeter group. We explicitly describe the combinatorics in the case where (the symmetric group on n letters) and the permutation w is 321-hexagon-avoiding. Our formula can be expressed in terms of a simple statistic on all subexpressions of any fixed reduced expression for w. As a consequence of our results on Kazhdan-Lusztig polynomials, we show that the Poincaré polynomial of the intersection cohomology of the Schubert variety corresponding to w is (1+q) ^l(w) if and only if w is 321-hexagon-avoiding. We also give a sufficient condition for the Schubert variety X _w to have a small resolution. We conclude with a simple method for completely determining the singular locus of X _w when w is 321-hexagon-avoiding. The results extend easily to those Weyl groups whose Coxeter graphs have no branch points (B C _n , F ₄, G ₂).     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

Equivariant Pieri Rule for the homology of the affine Grassmannian

An explicit rule is given for the product of the degree two class with an arbitrary Schubert class in the torus-equivariant homology of the affine Grassmannian. In addition a Pieri rule (the Schubert expansion of the product of a special Schubert class with an arbitrary one) is established for the equivariant homology of the affine Grassmannians of SL _n and a similar formula is conjectured for Sp _2n and SO _2n+1. For SL _n the formula is explicit and positive. By a theorem of Peterson these compute certain products of Schubert classes in the torus-equivariant quantum cohomology of flag varieties. The SL _n Pieri rule is used in our recent definition of k-double Schur functions and affine double Schur functions.     

Schubert Patches Degenerate to Subword Complexes

We study the intersections of general Schubert varieties X _w with permuted big cells, and give an inductive degeneration of each such “Schubert patch” to a Stanley–Reisner scheme. Similar results had been known for Schubert patches in various types of Grassmannians. We maintain reducedness using the results of Knutson [Kn07] on automatically reduced degenerations, or through more standard cohomology-vanishing arguments. The underlying simplicial complex of the Stanley–Reisner scheme is a subword complex, as introduced for slightly di_erent purposes in Knutson–Miller [KnM05], and is homeomorphic to a ball. This gives a new proof of the Andersen–Jantzen–Soergel/Billey and Graham/Willems formulae for restrictions of equivariant Schubert classes to fixed points.     

When Ramification points meet

Summary This paper contains three applications of the technique of limit series (our [1986]) to the theory of ramification of linear series on smooth curves, and curves of compact type, overC.Let {L _t t|<}, be a family of linear series on a smooth family of smooth curves {C _t }, and letp ₁(t),p ₂(t)C _t be sections of the family which coincide (only) att=0. Setp=p ₁())=p ₂(0)C ₀.We first give a condition related to the Schubert calculus which must be satisfied by the ramification series and the . We then take up the converse problem: In what ways can a given ramification point arise as a limit? We show that if the ramification point isdimensionally proper in the sense of our [1986], then families of every kind allowed by the Schubert calculus condition can actually be constructed. Finally, we prove that dimensional propriety is in a strong sense an open condition, so that ramification points constructed as above are again dimensionally proper.In the body of the paper we work not with pairs of points, as above, but with arbitrary finite collections of points approaching (possibly) several points of the limit curve. Further, by their nature, the results are valid for families of curves of compact type.     

Algebra Forms with $$d^{N} = 0$$ on Quantum Plane. Generalized Clifford Algebra Approach

We construct a q-analog of exterior calculus with a differential d satisfying d ^N = 0, where N ≥ 2 and q is a primitive Nth root of unity, on a noncommutative space and introduce a notion of a q-differential k-form. A noncommutative space we consider is a reduced quantum plane. Our construction of a q-analog of exterior calculus is based on a generalized Clifford algebra with four generators and on a graded q-differential algebra. We study the structure of the algebra of q-differential forms on a reduced quantum plane and show that the first order calculus induced by the differential d is a coordinate calculus. The explicit formulae for partial derivatives of this first order calculus are found.