The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer

TL;DR

This paper introduces a new Hopf algebra HSym of signed permutations, connecting it with existing algebras of permutations and weak quasi-symmetric functions, and reveals their deep interrelations through a commutative diagram.

Contribution

It extends the framework of Hopf algebras by defining HSym for signed permutations and establishing surjective homomorphisms to existing algebras, unifying various combinatorial structures.

Findings

Defined the Hopf algebra HSym of signed permutations.

Established surjective Hopf algebra homomorphisms from HSym to SSym and RQSym.

Revealed the structural relationships among permutations, signed permutations, compositions, and weak compositions.

Abstract

This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed permutation defines a linear surjection from HSym to RQSym. The notion of signed $P$-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations.