On q-symmetric functions and q-quasisymmetric functions
In this paper, we construct the q-analogue of PoirierReutenauer algebras, related deeply with other q-combinatorial Hopf algebras. As an application, we use them to realize the odd Schur functions defined by Ellis, Khovanov, and Lauda, then naturally obtain the odd LittlewoodRichardson rule concerned by Ellis. Moreover, we construct the refinement of the odd Schur functions, called odd quasisymmetric Schur functions, parallel to the consideration by Haglund, Luoto, Mason, and van Willigenburg. All the q-Hopf algebras we discuss here provide the corresponding q-dual graded graphs.
littlewoodrichardson rule odd quasisymmetric schur function q-hopf algebra