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

Li Guo, Jean Yves Thibon, Houyi Yu

Research output: Contribution to journalArticlepeer-review


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 weak 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.

Original languageEnglish (US)
Article number107341
JournalAdvances in Mathematics
StatePublished - Nov 18 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


  • Malvenuto-Reutenauer Hopf algebra
  • Quasi-shuffle product
  • Quasi-symmetric function
  • Signed permutation
  • Weak P-partition
  • Weak quasi-symmetric function


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