TY - JOUR

T1 - The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer

AU - Guo, Li

AU - Thibon, Jean Yves

AU - Yu, Houyi

N1 - Funding Information:
This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11771190 and 11501467 ) and the Natural Science Foundation of Chongqing (Grant Nos. cstc2019jcyj-msxmX0435 and cstc2019jcyj-msxmX0432 ). The authors thank the anonymous referees for their detailed and helpful comments.
Publisher Copyright:
© 2020 Elsevier Inc.

PY - 2020/11/18

Y1 - 2020/11/18

N2 - 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.

AB - 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.

KW - Malvenuto-Reutenauer Hopf algebra

KW - Quasi-shuffle product

KW - Quasi-symmetric function

KW - Signed permutation

KW - Weak P-partition

KW - Weak quasi-symmetric function

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U2 - 10.1016/j.aim.2020.107341

DO - 10.1016/j.aim.2020.107341

M3 - Article

AN - SCOPUS:85088953506

VL - 374

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 107341

ER -