Discovering product and coproduct Rules for Bases of ${\mathsf{\text{QSym}}}_F$ through Supercharacters

In this paper, we establish product and coproduct rules for three bases of the Hopf algebra ${\mathsf{\text{QSym}}}_F$ of quasisymmetric functions over $F$, with $F$ being either $\mathbb{C}(q,t)$ or $\mathbb{C}(q)$. These results are derived through the categorizations of ${\mathsf{\text{QSym}}}_{\mathbb{C}}$ obtained by utilizing the normal lattice supercharacter theories. Firstly, we deal with a basis $\{{\mathcal{D}}_{\alpha }(q,t)\mid \alpha \in \mathsf{\text{Comp}}\}$ of ${\mathsf{\text{QSym}}}_{\mathbb{C}(q,t)}$, where $\mathsf{\text{Comp}}$ denotes the set of all compositions. This basis is obtained from the direct sum of specific supercharacter function spaces and consists of superclass identifier functions. Upon appropriate specializations of $q$ and $t$, it yields notable bases of ${\mathsf{\text{QSym}}}_{\mathbb{C}}$ and ${\mathsf{\text{QSym}}}_{\mathbb{C}(q)}$, including enriched $q$-monomial quasisymmetric functions introduced by Grinberg and Vassilieva. Secondly, we deal with the basis $\{G_{\alpha }(q)\mid \alpha \in \mathsf{\text{Comp}}\}$ of ${\mathsf{\text{QSym}}}_{\mathbb{C}(q)}$, where $G_{\alpha }(q)$ represents the quasisymmetric Hall-Littlewood function introduced by Hivert. Our product rule is new, whereas our coproduct rule turns out to be equivalent to the existing coproduct rule of Hivert. Finally, we consider a basis $\{M_{\alpha }(q)\mid \alpha \in \mathsf{\text{Comp}}\}$ of ${\mathsf{\text{QSym}}}_{\mathbb{C}(q)}$, where $M_{\alpha }(q)$ is a $q$-analogue of the monomial quasisymmetric function.