Geodesic
On $q$-symmetric functions and $q$-quasisymmetric functions
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 2, pp. 323-364.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

In this paper, we construct the $q$-analogue of Poirier-Reutenauer algebras, related deeply with other $q$-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined by A. P. Ellis et al. [Int. Math. Res. Not. 2014, No. 4, 991--1062 (2014; Zbl 1356.20004)], then naturally obtain the odd Littlewood-Richardson rule concerned by A. P. Ellis [J. Algebr. Comb. 37, No. 4, 777--799 (2013; Zbl 1268.05229)]. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration by J. Haglund et al. [J. Comb. Theory, Ser. A 118, No. 2, 463--490 (2011; Zbl 1229.05270)]. All the $q$-Hopf algebras we discuss here provide the corresponding $q$-dual graded graphs.
Classification : 05E05, 05A30, 05E15
Keywords: $q$-Hopf algebra, odd quasisymmetric Schur function, Littlewood-Richardson rule 
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     author = {Li, Yunnan},
     title = {On $q$-symmetric functions and $q$-quasisymmetric functions},
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Li, Yunnan. On $q$-symmetric functions and $q$-quasisymmetric functions. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 2, pp. 323-364. http://geodesic.mathdoc.fr/item/JAC_2015__41_2_a7/