Antipodes and involutions

TL;DR

This paper introduces sign-reversing involutions to derive cancellation-free formulas for antipodes in various Hopf algebras, simplifying calculations and revealing new explicit expressions.

Contribution

It applies involution techniques to obtain cancellation-free antipode formulas across multiple Hopf algebras, including some novel results.

Findings

Derived cancellation-free antipode formulas for multiple Hopf algebras

Reobtained known formulas for polynomial and quasisymmetric function Hopf algebras

Discovered new explicit antipode expressions in noncommutative symmetric functions and related algebras

Abstract

If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmertic functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation-free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto-Reutenauer and Porier-Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research.