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.     

Between two moments

In this short note, we draw attention to a relation between two Horn polytopes which is proved in [3] as a result, on the one hand, of a deep combinatorial result in [5] and, on the other hand, of a simple computation involving complex structures. This suggests an inequality between Littlewood-Richardson coefficients, which we prove using the symmetric characterization of these coefficients given in [1].     

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.     

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.     

The number of extremal components of a rigid measure

The Littlewood-Richardson rule can be expressed in terms of measures, and the fact that the Littlewood-Richardson coefficient is one amounts to a rigidity property of some measure. We show that the number of extremal components of such a rigid measure can be related to easily calculated geometric data. We recover, in particular, a characterization of those extremal measures whose (appropriately defined) duals are extremal as well. This result is instrumental in writing explicit solutions of Schubert intersection problems in the rigid case.     

The honeycomb model of tensor products I: Proof of the saturation conjecture

Recently Klyachko has given linear inequalities on triples of dominant weights of necessary for the corresponding Littlewood-Richardson coefficient to be positive. We show that these conditions are also sufficient, which was known as the saturation conjecture. In particular this proves Horn's conjecture from 1962, giving a recursive system of inequalities. Our principal tool is a new model of the Berenstein-Zelevinsky cone for computing Littlewood-Richardson coefficients, the honeycomb model. The saturation conjecture is a corollary of our main result, which is the existence of a particularly well-behaved honeycomb associated to regular triples .

     

A combinatorial proof of the reduction formula for Littlewood-Richardson coefficients

There are well-known reduction formulas for the universal Schubert coefficients defined on Grassmannians. These coefficients are also known as the Littlewood-Richardson coefficients in the theory of symmetric functions. We restate the reduction formulas combinatorially and provide a combinatorial proof for them.     

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.     

Bounds on the diagonal elements of a unitary matrix

It was shown by A. Horn that the diagonal elements of a unitary n×n matrix satisfy a set of linear inequalities (Theorem I). We give a simple proof of this result, and we show that the diagonal elements satisfy additional linear inequalities (Theorem II) if the matrix is also symmetric.     

Generalised stretched Littlewood-Richardson coefficients

The Littlewood-Richardson (LR) coefficient counts, among many other things, the LR tableaux of a given shape and a given content. We prove that the number of LR tableaux weakly increases if one adds to its shape and content the shape and the content of another LR tableau. We also investigate the behaviour of the number of LR tableaux, if one repeatedly adds to the shape another shape with either fixed or arbitrary content. This is a generalisation of the stretched LR coefficients, where one repeatedly adds the same shape and content to itself.     

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite-dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood-Richardson coefficients.     

Weyl and Lidski(i) Inequalities for General Hyperbolic Polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidski(i) inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.     

Weyl and Lidski? Inequalities for General Hyperbolic Polynomials

The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices, after a conjecture by A. Horn. Among them are the so-called Weyl and Lidskiǐ inequalities. An elementary proof of the latter for hyperbolic polynomials is given. This proof follows an idea from H. Weinberger and is free from representation theory and Schubert calculus arguments, as well as from hyperbolic partial differential equations theory.     

Analytical Linear Inequality Systems and Optimization

In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.     

Factorisation in the ring of exponential polynomials

We study factorisation in the ring of exponential polynomials and provide a proof of Ritt's factorisation theorem in modern notation and so generalised as to deal with polynomial coefficients as well as with several variables. We do this in the more general context of a group ring of a divisible torsion-free ordered abelian group over a unique factorisation domain.

     

Applications of a Laplace-Beltrami operator for Jack polynomials

An explicit computation is made for a Laplace-Beltrami type operator for Jack polynomials. As applications we obtain: combinatorial formula, determinantal formula and raising operator formula for Jack polynomials, as well as an iterative formula for the Littlewood-Richardson coefficients. One special case of our results implies Mimachi-Yamada’s result on Jack polynomials of rectangular shapes.     

Fourier Series in Weighted Lorentz Spaces

The Fourier coefficient map is considered as an operator from a weighted Lorentz space on the circle to a weighted Lorentz sequence space. For a large range of Lorentz indices, necessary and sufficient conditions on the weights are given for the map to be bounded. In addition, new direct analogues are given for known weighted Lorentz space inequalities for the Fourier transform. Applications are given that involve Fourier coefficients of functions in LogL and more general Lorentz–Zygmund spaces.     

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.