The combinatorics of Bogoliubov's recursion in renormalization

We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters.Comment: improved version, 20 pages, CIRM 2006 workshop "Renormalization and Galois Theory", Org. F. Fauvet, J.-P. Rami

Crystal approach to affine Schubert calculus

We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-$A$ affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a $k$-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb C^n$ enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function $s_\lambda$ for all $|\lambda^\vee|< n$. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.Comment: 42 pages; version to appear in IMR