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

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

2009-10-18

arxiv.org/abs/0910.3395v1

Mathematics

Representation Theory

24 pages, 3 figures

Scientific paper

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.

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