Quadratic degenerations of odd-orthogonal Schubert varieties

This paper is the second in a series leading to a type B_n geometric Littlewood-Richardson rule. The rule will give an interpretation of the B_n Littlewood-Richardson numbers as an intersection of two odd-orthogonal Schubert varieties and will consider a sequence of linear and quadratic deformations of the intersection into a union of odd-orthogonal Schubert varieties. This paper describes the setup for the rule and specifically addresses results for quadratic deformations, including a proof that at each quadratic degeneration, the results occur with multiplicity one. This work is strongly influenced by Vakil’s [14].     

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.     

An algorithmic Littlewood-Richardson rule

We introduce a Littlewood-Richardson rule based on an algorithmic deformation of skew Young diagrams and present a bijection with the classical rule. The result is a direct combinatorial interpretation and proof of the geometric rule presented by Coskun (2000). We also present a corollary regarding the Specht modules of the intermediate diagrams.     

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.     

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.     

Young-Fibonacci insertion, tableauhedron and Kostka numbers

This work is first concerned with some properties of the Young-Fibonacci insertion algorithm and its relation with Fomin's growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebras associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are conveniently interpreted as saturated chains in the Young-Fibonacci lattice. Using our conventions, we give a simpler proof of a property of Killpatrick's evacuation algorithm for Fibonacci tableaux. It also appears that this evacuation is no longer needed in making Roby's and Fomin's constructions coincide. We provide the set of Young-Fibonacci tableaux of size n with a structure of graded poset called tableauhedron, induced by the weak order of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients of the transition matrix from the analogue of complete symmetric functions to analogue of the Schur functions in Okada's algebra associated to the Young-Fibonacci lattice. We prove a similar result relating usual Kostka numbers with four partial orders on Young tableaux, studied by Melnikov and Taskin.     

Root games on Grassmannians

We recall the root game, introduced in [8], which gives a fairly powerful sufficient condition for non-vanishing of Schubert calculus on a generalised flag manifold G/B. We show that it gives a necessary and sufficient rule for non-vanishing of Schubert calculus on Grassmannians. In particular, a Littlewood-Richardson number is non-zero if and only if it is possible to win the corresponding root game. More generally, the rule can be used to determine whether or not a product of several Schubert classes on Gr _l (ℂ ⁿ ) is non-zero in a manifestly symmetric way. Finally, we give a geometric interpretation of root games for Grassmannian Schubert problems. Research partially supported by an NSERC scholarship.     

A k-tableau characterization of k-Schur functions

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.     

Chains in the Bruhat order

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Littlewood-Richardson coefficients. In the second half of the paper, we study the classical flag manifold and discuss related combinatorial objects: flagged Schur polynomials, 312-avoiding permutations, generalized Gelfand-Tsetlin polytopes, the inverse Schubert-Kostka matrix, parking functions, and binary trees. A.P. was supported in part by National Science Foundation grant DMS-0201494 and by Alfred P. Sloan Foundation research fellowship. R.S. was supported in part by National Science Foundation grant DMS-9988459.     

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.     

On the set of imputations induced by the k-additive core

An extension to the classical notion of core is the notion of k-additive core, that is, the set of k-additive games which dominate a given game, where a k-additive game has its Möbius transform (or Harsanyi dividends) vanishing for subsets of more than k elements. Therefore, the 1-additive core coincides with the classical core. The advantages of the k-additive core is that it is never empty once k ? 2, and that it preserves the idea of coalitional rationality. However, it produces k-imputations, that is, imputations on individuals and coalitions of at most k individuals, instead of a classical imputation. Therefore one needs to derive a classical imputation from a k-order imputation by a so-called sharing rule. The paper investigates what set of imputations the k-additive core can produce from a given sharing rule.     

Maximal 0–1-fillings of moon polyominoes with restricted chain lengths and rc-graphs

We show that maximal 0–1-fillings of moon polynomials with restricted chain lengths can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0–1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino S with no north-east chains longer than k depends only on k and the multiset of column heights of S.     

Equivariant quantum Schubert calculus

We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients.     

Fuzzy inference to assess manufacturing process capability with imprecise data

Process capability indices provide numerical measures on whether a process conforms to the defined manufacturing capability prerequisite. These have been successfully applied by companies to compete with and to lead high-profit markets by evaluating the quality and productivity performance. The loss-based process capability index C_pm, sometimes called the Taguchi index, was proposed to measure process capability, wherein the output process measurements are precise. In the present study, we develop a realistic approach that allows the consideration of imprecise output data resulting from the measurements of the products quality. A general method combining the vector of fuzzy numbers to produce the membership function of fuzzy estimator of Taguchi index is introduced for further testing process capability. With the sampling distribution for the precise estimation of C_pm, two useful fuzzy inference criteria, the critical value and the fuzzy P-value, are proposed to assess the manufacturing process capability based on C_pm. The presented methodology takes into the consideration of a certain degree of imprecision on the sample data and leads to the three-decision rule with the four quadrants decision-making plot. The fuzzy inference procedure presented to assess process capability is a natural generalization of the traditional test, when the data are precise the proposed test is reduced to a classical test with a binary decision.     

Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood-Richardson rule. Given three Schubert varieties S₁, S₂, S₃ with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a,b,c with a+b+c=0 in the factor Rω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x,y,z are self-adjoint elements of such a factor and x+y+z=0, then there exist self-adjoint a,b,c∈Rω such that a+b+c=0 and a (respectively, b,c) has the same eigenvalue distribution as x (respectively, y,z). A (‘complete’) matricial form of this result is known to imply an affirmative answer to an embedding question formulated by Connes. The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of n×n complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds.     

On Schensted's construction and the multiplication of schur functions

In this paper, certain aspects of Schensted's construction are examined and a generalization is described. From this generalization, the Littlewood-Richardson rule for the multiplication of Schur functions is derived by purely combinatorial methods.     

Affine Stanley symmetric functions for classical types

We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the special linear and symplectic groups, respectively. For the odd orthogonal groups, a Hopf-algebra isomorphism is given, identifying (co)homology Schubert classes with symmetric functions. For the even orthogonal groups, we conjecture an approximate model of (co)homology via symmetric functions. In the process, we develop type B and type D non-commutative k-Schur functions as elements of the affine nilCoxeter algebra that model homology of the affine Grassmannian. Additionally, Pieri rules for multiplication by special Schubert classes in homology are given in both cases. Finally, we present a type-free interpretation of Pieri factors, used in the definition of noncommutative k-Schur functions or affine Stanley symmetric functions for any classical type.     

Global convergence properties of two modified BFGS-type methods

This article studies a modified BFGS algorithm for solving smooth unconstrained strongly convex minimization problem. The modified BFGS method is based on the new quasi-Newton equation B_k+1s_k=yk where y_k*, =yk + A_ks_k andA _k is a matrix. Wei, Li and Qi [WLQ] have proven that the average performance of two of those algorithms is better than that of the classical one. In this paper, we prove the global convergence of these algorithms associated to a general line search rule.     

A new perspective on k-triangulations

We connect k-triangulations of a convex n-gon to the theory of Schubert polynomials. We use this connection to prove that the simplicial complex with k-triangulations as facets is a vertex-decomposable triangulated sphere, and we give a new proof of the determinantal formula for the number of k-triangulations.     

Skew Schubert polynomials

We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron and Sottile in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs.