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Let us fix a positive integer $\nu>1$. For each positive integer $n>1$, weconsider 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 $\nu$. Thenwe endow $\bigoplus_{n \ge 0} \textsf{scf}(\mathcal{S}_n)$, the direct-productof supercharacter function spaces, with the Hopf algebra structure that isisomorphic to the Hopf algebra $\textsf{QSym}$ of quasisymmetric functions.Furthermore, we compute the structure constants of the Hopf algebra thusobtained for the basis consisting of superclass identifier functions. Using ourcategorifications, we study a new basis for the Hopf algebra$\textsf{NSym}_{\mathbb{C}(q,t)}$ of noncommutative symmetric functions overthe 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 applicationsare also obtained via the specializations of $q$ and $t$.

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