Higher Verlinde Categories: The Mixed Case

Over a field of characteristic $p>0$, the higher Verlinde categories Verpⁿ are obtained by taking the abelian envelope of a quotient of the category of tilting modules for the algebraic group SL₂. 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 SL₂ has been generalized to Lusztig's quantum group for sl₂ and root of unity ζ, which produces the mixed higher Verlinde categories Verζ_p⁽ⁿ⁾. 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 Verζ_p⁽ⁿ⁾, construct a braided embedding Verpⁿ Verζ_p⁽ⁿ⁺¹⁾, compute the symmetric center of Verζ_p⁽ⁿ⁾, and identify its Grothendieck ring.