We investigate the rigidity for the Hopf algebra QSym of quasisymmetric functions with respect to the monomial, the fundamental and the quasisymmetric Schur basis, respectively. By establishing some combinatorial properties of the posets of compositions arising from the analogous Pieri rules for quasisymmetric functions, we show that QSym is rigid as an algebra with respect to the quasisymmetric Schur basis, and rigid as a coalgebra with respect to the monomial and the quasisymmetric Schur basis, respectively. The natural actions of reversal, complement and transpose of the labelling compositions lead to some nontrivial graded (co)algebra automorphisms of QSym. We prove that the linear maps induced by the three actions are precisely the only nontrivial graded algebra automorphisms that take the fundamental basis into itself. Furthermore, the complement map on the labels gives the unique nontrivial graded coalgebra automorphism preserving the fundamental basis, while the reversal map on the labels gives the unique nontrivial graded algebra automorphism preserving the monomial basis. Therefore, QSym is rigid as a Hopf algebra with respect to the monomial and the quasisymmetric Schur basis.
@article{10_37236_8174,
author = {Wanwan Jia and Zhengpan Wang and Houyi Yu},
title = {Rigidity for the {Hopf} algebra of quasisymmetric functions},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {3},
doi = {10.37236/8174},
zbl = {1416.05287},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8174/}
}
TY - JOUR
AU - Wanwan Jia
AU - Zhengpan Wang
AU - Houyi Yu
TI - Rigidity for the Hopf algebra of quasisymmetric functions
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8174/
DO - 10.37236/8174
ID - 10_37236_8174
ER -
%0 Journal Article
%A Wanwan Jia
%A Zhengpan Wang
%A Houyi Yu
%T Rigidity for the Hopf algebra of quasisymmetric functions
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8174/
%R 10.37236/8174
%F 10_37236_8174
Wanwan Jia; Zhengpan Wang; Houyi Yu. Rigidity for the Hopf algebra of quasisymmetric functions. The electronic journal of combinatorics, Tome 26 (2019) no. 3. doi: 10.37236/8174
