Higher Verlinde Categories: The Mixed Case

Over a field of characteristic $p>0$, the higher Verlinde categories ${\mathrm{V}\mathrm{e}\mathrm{r}}_{p^n}$ are obtained by taking the abelian envelope of quotients of the category of tilting modules for the algebraic group ${\mathrm{S}\mathrm{L}}_2$. These symmetric tensor categories have been introduced in arXiv:2003.10499 & arXiv:2003.10105, and their properties have been extensively studied in the former reference. In arXiv:2105.07724, the above construction for ${\mathrm{S}\mathrm{L}}_2$ has been generalized to Lusztig’s quantum group for ${\mathfrak{s}\mathfrak{l}}_2$ and root of unity $\zeta$, which produces the mixed higher Verlinde categories ${\mathrm{V}\mathrm{e}\mathrm{r}}_{p^{(n)}}^{\zeta }$. Inspired by the results of arXiv:2003.10499, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of ${\mathrm{V}\mathrm{e}\mathrm{r}}_{p^{(n)}}^{\zeta }$, construct a braided embedding ${\mathrm{V}\mathrm{e}\mathrm{r}}_{p^n}\hookrightarrow {\mathrm{V}\mathrm{e}\mathrm{r}}_{p^{(n+1)}}^{\zeta }$, compute the symmetric center of ${\mathrm{V}\mathrm{e}\mathrm{r}}_{p^{(n)}}^{\zeta }$, and identify its Grothendieck ring.