Geodesic
Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
Adam Doliwa1
1University of Warmia and Mazury
The electronic journal of combinatorics, Tome 28 (2021) no. 2
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We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the set of sentences over alphabet $A$ (the set of colours). We present also its graded dual algebra $\mathrm{QSym}_A$ of coloured quasi-symmetric functions together with its realization in terms of power series in partially commutative variables. We provide formulas expressing multiplication, comultiplication and the antipode for these Hopf algebras in various bases — the corresponding generalizations of the complete homogeneous, elementary, ribbon Schur and power sum bases of $\mathrm{NSym}$, and the monomial and fundamental bases of $\mathrm{QSym}$. We study also certain distinguished series of trees in the setting of restricted duals to Hopf algebras.
DOI : 10.37236/10184
Classification : 05E05, 16T30, 05C25, 06A07
Mots-clés : graded dual algebra, coloured quasi-symmetric functions, Schur functions

Adam Doliwa  1

1 University of Warmia and Mazury 
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     author = {Adam Doliwa},
     title = {Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables},
     journal = {The electronic journal of combinatorics},
     year = {2021},
     volume = {28},
     number = {2},
     doi = {10.37236/10184},
     zbl = {1466.05218},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10184/}
}
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Adam Doliwa. Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables. The electronic journal of combinatorics, Tome 28 (2021) no. 2. doi: 10.37236/10184

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