Powersum bases in quasisymmetric functions and quasisymmetric functions in non-commuting variables

Anthony Lazzeroni¹

¹Hong Kong Baptist University

The electronic journal of combinatorics, Tome 30 (2023) no. 4

Résumé

We introduce a new powersum basis for the Hopf algebra of quasisymmetric functions that refines the powersum symmetric basis. Unlike the quasisymmetric powersums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum P basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.

Détail

DOI : 10.37236/11724

Classification : 05E05, 16T30
Mots-clés : Hopf algebra of quasisymmetric function, quasisymmetric monomial functions, Malvenuto-Reutenauer Hopf algebra of permutations