On the cancellation-free antipode formula for the Malvenuto-Reutenauer Hopf Algebra

TL;DR

This paper presents a new cancellation-free antipode formula for a specific class of permutations in the Malvenuto-Reutenauer Hopf algebra, confirming two prior conjectures and simplifying calculations.

Contribution

It introduces a novel cancellation-free antipode formula for permutations with a specific structure, advancing understanding of the algebra's combinatorial properties.

Findings

Provides a cancellation-free antipode formula for certain permutations

Confirms two conjectures by Benedetti and Sagan

Simplifies antipode calculations in the Hopf algebra

Abstract

For the Malvenuto-Reutenauer Hopf algebra of permutations, we provide a cancellation-free antipode formula for any permutation of the form $ab1\cdots(b-1)(b+1)\cdots(a-1)(a+1)\cdots n$, which starts with the decreasing sequence $ab$ and ends with the increasing sequence $1\cdots(b-1)(b+1)\cdots(a-1)(a+1)\cdots n$, where $1\leq b<a\leq n$. As a consequence, we confirm two conjectures posed by Carolina Benedetti and Bruce E. Sagan.