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.Comment: 10 pages, 7 figure

Equivariant Quantum Cohomology of the Grassmannian via the Rim Hook Rule

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of k-planes in complex n-space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo n, suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.Comment: 24 pages and 4 figures; typos corrected; final version to appear in Algebraic Combinatoric

A Combinatorial Derivation of the Racah-Speiser Algorithm for Gromov-Witten invariants

Using a finite-dimensional Clifford algebra a new combinatorial product formula for the small quantum cohomology ring of the complex Grassmannian is presented. In particular, Gromov-Witten invariants can be expressed through certain elements in the Clifford algebra, this leads to a q-deformation of the Racah-Speiser algorithm allowing for their computation in terms of Kostka numbers. The second main result is a simple and explicit combinatorial formula for projecting product expansions in the quantum cohomology ring onto the sl(n) Verlinde algebra. This projection is non-trivial and amounts to an identity between numbers of rational curves intersecting Schubert varieties and dimensions of moduli spaces of generalised theta-functions.Comment: 24 pages, 3 figure

Splines in geometry and topology

This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.Comment: 18 page