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We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. This Galois cohomology fits into an exact
Combinatorial Hopf algebras and generalized Dehn–Sommerville relations
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field $\Bbbk$ equipped with a character (multiplicative linear functional) $\zeta\colon{\mathcal H}\to \Bbbk$. We show that the
Structure of The Malvenuto-Reutenauer Hopf Algebra of Permutations (Extended Abstract)
We analyze the structure of the Malvenuto-Reutenauer Hopf algebra of permutations in detail. We give explicit formulas for its antipode, prove that it is a cofree coalgebra, determine its primitive
Structure of the Loday–Ronco Hopf algebra of trees
Abstract Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative
Tableau Switching: Algorithms and Applications
It is shown that switches can be performed in virtually any order without affecting the final outcome, and combinatorial proofs of various symmetries of Littlewood?Richardson coefficients are given.
Numerical Schubert Calculus
Numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus are developed, which are optimal in that generically no paths diverge.
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
Real Schubert Calculus: Polynomial Systems and a Conjecture of Shapiro and Shapiro
  • F. Sottile
  • Mathematics, Computer Science
    Exp. Math.
  • 24 April 1999
This work gives compelling computational evidence for its validity, proves it for infinitely many families of enumerative problems, and shows how a simple version implies more general versions, and presents a counterexample to a general version of their conjecture.
Pieri's formula for flag manifolds and Schubert polynomials
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Pieri's formula describes the intersection product of a Sc hubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational