We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified defini...

Die quantenlogik-basierte probabilistische Ähnlichkeitsanfragesprache QSQL2 soll vorgestellt werden. Dabei liegt das Hauptaugenmerk auf der Formulierung von Anfragen, welche “unsicher” sind, also nicht nur die traditionelle Boolesche Werte wahr und falsch annehmen können. QSQL2 kann Ungenauigkeiten sowohl auf Relationenebene als Eintrittswahrscheinlichkeiten, als auch auf Prädikatebene als Rele...

Complex phrases are also brought into the categorization, so that cutlery that is in the drawer and hot water are mass phrases while tree that is in the park and tall person are count phrases. I use ‘mass term’ and ‘count term’ to cover both nouns and more complex noun phrases. (As well, some theorists admit other grammatical categories into the count/mass realm: some verb phrases and some adje...

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have iden...

Kohnert’s algorithm for the generation of Schubert polynomials is derived from Monk’s rule for the multiplication of Schubert polynomials.

We prove an elegant combinatorial rule for the generation of Schubert polynomials based on box diagrams, which was conjectured by A. Kohnert. The main tools for the proof are (1) a recursive structure of Schubert polynomials and (2) a partial order on the set of box diagrams. As a byproduct we obtain (combinatorial) proofs for two other rules for the generation of Schubert polynomials based on ...

This paper begins with a description of the Schubert varieties of a Grassmannian variety Gr(k, n) over C. Following the technique of Ryan [3] for determining the singularity of flag manifolds we will then describe the Jacobian of a Schubert variety Xλ. This will be used to determine which Schubert varieties are nonsingular.

In this extended abstract, we give a complete description and enumeration of smooth and rationally smooth Schubert varieties in finite type. In particular, we show that rationally smooth Schubert varieties are in bijection with a new combinatorial data structure called staircase diagrams. Résumé. Dans ce résumé étendu, nous donnons une description complète et le dénombrement de variétés lisses ...

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to ŝu( ) are shown to b...

We illuminate the relation between the Bruhat order and structure constants for the polynomial ring 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 o...