Frank Sottile

Combinatorial Hopf algebras and generalized Dehn–Sommerville relations

A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ : H→ k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H ,ζ ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized Dehn–Sommerville relations. We show that, for H = QSym, the generalized Dehn–Sommerville relations are the Bayer–Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the Malvenuto–Reutenauer Hopf algebra of permutations, the Loday– Ronco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of non-commutative symmetric functions.

Numerical Schubert Calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Grobner basis for the Plucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.

Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems

Smale’s α-theory uses estimates related to the convergence of Newton’s method to certify that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on α-theory to certify solutions of polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements algorithms that certify whether a given point corresponds to a real solution, and algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.

Tableau Switching

We define and characterizeswitching, an operation that takes two tableaux sharing a common border and “moves them through each other” giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood?Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schutzenbergers evacuation procedure to switching and in the process obtain further results concerning evacuation. We definereversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood?Richardson coefficients.

Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

We illuminate the relation between the Bruhat order on the symmetric group and structure constants (Littlewood-Richardson coefficients) for the cohomology of the flag manifold in terms of its basis of Schubert classes. Equivalently, the structure constants for the ring of polynomials in variables

Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro

x_1,x_2,...

The special Schubert calculus is real

in terms of its basis of Schubert polynomials. We use combinatorial, algebraic, and geometric methods, notably a study of intersections of Schubert varieties and maps between flag manifolds. We establish a number of new identities among these structure constants. This leads to formulas for some of these constants and new results on the enumeration of chains in the Bruhat order. A new graded partial order on the symmetric group which contains Youngs lattice arises from these investigations. We also derive formulas for certain specializations of Schubert polynomials.

Some remarks on real and complex output feedback

Boris and Michael Shapiro have a conjecture concerning the Schubert calculus and real enumerative geometry and which would give infinitely many families of zero-dimensional systems of real polynomials (including families of overdetermined systems)—all of whose solutions are real. It has connections to the pole placement problem in linear systemstheory and to totally positive matrices. We give compelling computational evidence for its validity, prove it for infinitely many families of enumerative problems, show how a simple version implies more general versions, and present a counterexample to a general version of their conjecture.

Enumerative geometry for the real Grassmannian of lines in projective space

We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real. Fulton asked how many solutions to a problem of enumerative geometry can be real, when that problem is one of counting geometric figures of some kind having specified position with respect to some general fixed figures [5]. For the problem of plane conics tangent to five general conics, the (surprising) answer is that all 3264 may be real [10]. Recently, Dietmaier has shown that all 40 positions of the Stewart platform in robotics may be real [2]. Similarly, given any problem of enumerating lines in projective space incident on some general fixed linear subspaces, there are real fixed subspaces such that each of the (finitely many) incident lines are real [13]. Other examples are shown in [12, 14], and the case of 462 4-planes meeting 12 general 3-planes in R is due to an heroic symbolic computation [4]. For any problem of enumerating p-planes having excess intersection with a collection of fixed planes, we show there is a choice of fixed planes osculating a rational normal curve at real points so that each of the resulting p-planes is real. This has implications for the problem of placing real poles in linear systems theory [1] and is a special case of a far-reaching conjecture of Shapiro and Shapiro [15]. Special Schubert conditions For background on the Grassmannian, Schubert cycles, and the Schubert calculus, see any of [8, 7, 6]. Let m, p ≥ 1 be integers. Let γ be a rational normal curve in R. For k > 0 and s ∈ γ, let Kk(s) be the k-plane osculating γ at s. For every integer a > 0, let τa(s) be the special Schubert cycle consisting of pplanes H in C which meet Km+1−a(s) nontrivially, and let τ(s) be the special Schubert cycle consisting of p-planes H in C meeting Km−1+a(s) improperly: dim H ∩Km−1+a(s) > a− 1. These cycles τa(s) and τ(s) each have codimension a and τ = τ1. Recall that the Grassmannian of p-planes in C has dimension Received by the editors December 20, 1998. 1991 Mathematics Subject Classification. Primary 14P99, 14N10, 14M15, 14Q20; Secondary 93B55.

Noncommutative Pieri Operators on Posets

We provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers. We exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real.