Categorifications of ${\textsf {QSym}}$ using supercharacter theories and a new basis for ${\textsf {NSym}}_{\mathbb{C}(q,t)}$

Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, we consider a normal supercharacter theory $\mathcal{S}_n$ of $G_n$, where $G_n$ is the direct-product of $n-1$ copies of the cyclic group of order $\nu$. Then we endow $\bigoplus_{n \ge 0} \textsf{scf}(\mathcal{S}_n)$, the direct-product of supercharacter function spaces, with the Hopf algebra structure that is isomorphic to the Hopf algebra $\textsf{QSym}$ of quasisymmetric functions. Furthermore, we compute the structure constants of the Hopf algebra thus obtained for the basis consisting of superclass identifier functions. Using our categorifications, we study a new basis for the Hopf algebra $\textsf{NSym}_{\mathbb{C}(q,t)}$ of noncommutative symmetric functions over the rational function field $\mathbb{C}(q,t)$ in commuting variables $q$ and $t$, with an emphasis on the structure constants of $\textsf{NSym}_{\mathbb{C}(q,t)}$ for this basis. Some interesting applications are also obtained via the specializations of $q$ and $t$.