{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:31Z","timestamp":1753893811942,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.\u00a0By establishing some combinatorial properties of the posets of compositions arising from the analogous Pieri rules for quasisymmetric functions,\u00a0we show that QSym is rigid as an algebra with respect to the quasisymmetric Schur basis,\u00a0and rigid as a coalgebra with respect to the monomial and the quasisymmetric Schur basis, respectively.\u00a0The natural actions of reversal, complement and transpose of the labelling compositions lead to some nontrivial graded (co)algebra automorphisms of QSym.\u00a0We prove that the linear maps induced by the three actions are precisely the only nontrivial\u00a0graded algebra automorphisms that take the fundamental basis into itself. Furthermore,\u00a0the complement map on the labels gives the unique nontrivial graded coalgebra automorphism preserving the fundamental basis, while the\u00a0reversal map on the labels gives the unique nontrivial graded algebra automorphism preserving the monomial basis.\u00a0Therefore, QSym is rigid as a Hopf algebra with respect to the monomial and the quasisymmetric Schur basis.<\/jats:p>","DOI":"10.37236\/8174","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:06:14Z","timestamp":1578639974000},"source":"Crossref","is-referenced-by-count":0,"title":["Rigidity for the Hopf Algebra of Quasisymmetric Functions"],"prefix":"10.37236","volume":"26","author":[{"given":"Wanwan","family":"Jia","sequence":"first","affiliation":[]},{"given":"Zhengpan","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Houyi","family":"Yu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,7,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p4\/7865","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p4\/7865","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:11:46Z","timestamp":1579234306000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i3p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,5]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/8174","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,7,5]]},"article-number":"P3.4"}}