Renormalization of quasisymmetric functions

Abstract: The algebra of quasisymmetric functions (QSym) has played a central role in multiple zeta values and a large class of combinatorial algebraic structures related to symmetric functions. A natural linear basis of QSym is the set of monomial quasisymmetric functions defined by compositions, that is, vectors of positive integers. Extending such a definition for weak compositions, that is, vectors of nonnegative integers, leads to divergent expressions. This phenomenon is closely related to the divergency of multiple zeta values with nonpositive integer arguments.

We apply the method of renormalization in the spirit of Connes and Kreimer to address the divergency, and realize weak composition quasisymmetric functions as power series. The resulting Hopf algebra has the Hopf algebra of quasisymmetric functions as both a Hopf subalgebra and a Hopf quotient algebra.

This is joint work with Houyi Yu and Bin Zhang.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)