Antipode Of The Malvenuto-Reutenauer Hopf Algebra

The Malvenuto-Reutenauer Hopf algebra(?)Sym of permutations is a connected,graded Hopf algebra.Recently,significant attention has been dedicated to developing its various antipode formulas.In 2017,Benedetti and Sagan predicted two cancellation-free antipode formulas for some permutations in(?)Sym.In this paper,we mainly study the antipode formulas of(?)Sym,and the main results are divided into two parts.Firstly,using the Hoffman-Ihara operators,we establish a linear operator form for the antipode of quasi-shuffle Hopf algebra with weight in Chapter 3.As an application,we obtain an antipode formula for the Malvenuto-Reutenauer Hopf algebra in terms of Hoffman-Ihara operators.Secondly,in Chapter 4,we give a cancellation-free antipode formula for any permutation,in the one-line notation,of the form ab1…(b-1)(b+1)…(a-1)(a+1)…n,which starts with the decreasing sequence ab and ends with the increasing sequence 1…(b-1)(b+1)…(a-1)(a+1)…n,where 1≤b<a≤n.Then,we confirm two conjectures about the cancellation-free antipode formulas of(?)Sym proposed by Benedetti and Sagan.