A pair of dual Hopf algebras on permutations

Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra. Keywords: dual Hopf algebra; global descent; concatenation product; draw coproduct; antipode Mathematics Subject Classification: 05A05, 08C20, 16T05