Some Applications of Combinatorial Hopf Algebras to Integro-Differential Equations and Symmetric Function Identities

Abstract

Hopf algebras built from combinatorial objects have found application both within combinatorics and, following the work of Connes and Kreimer, in quantum field theory. Despite the apparent gulf between these areas, the types of Hopf algebras that arise are very similar. We use Hopf algebra techniques to solve two problems, one coming from quantum field theory and one from algebraic combinatorics.

(1) Dyson–Schwinger equations are a formulation of the equations of motion of quantum field theory. From a mathematical perspective they are integro-differential equations which have a recursive, tree-like structure. In some cases, these equations are known to have solutions which can be written as combinatorial expansions over connected chord diagrams. We give a new expansion in terms of rooted trees equipped with a kind of decomposition we call a binary tubing. This is similar to the chord diagram expansion, but holds in greater generality, including to systems of Dyson–Schwinger equations and to Dyson–Schwinger equations in which insertion places are distinguished by different variables in the Mellin transform. Moreover we prove these results as a direct application of a purely Hopf-algebraic theorem characterizing maps from the Connes–Kreimer Hopf algebra of rooted trees (and variants thereof) to the Hopf algebra of univariate polynomials which arise from the universal property of the former.

(2) A pair of skew Ferrers shapes are said to be skew-equivalent if they admit the same number of semistandard Young tableaux of each weight, or in other words if the skew Schur functions they define are equal. McNamara and van Willigenburg conjectured necessary and sufficient combinatorial conditions for this to happen but were unable to prove either direction in complete generality. Using Hopf-algebraic techniques building on a partial result of Yeats, we prove sufficiency.