Stable Grothendieck rings of wreath product categories (Q2313408)

Let \(\mathcal{R}\) be a Hopf algebra and \(S_n\) the symmetric group on \(n\) symbols. It is well known that one can form the wreath product \(\mathcal{R}\wr S_{n}\), which is a Hopf algebra too. Therefore, we have a ring structure on the Grothendieck group of the category of modules over \(\mathcal{R}\wr S_{n}\). We denote this Grothendieck ring by \(\mathcal{G}_{n}(\mathcal{C})\) where \(\mathcal{C}=\mathcal{R}\)-mod. In the paper under review, the author finds that there is a stability of structure constants (through fixing distinguished generating sets called basic hooks). Therefore, one can define a limiting Grothendieck ring \(\mathcal{G}_{\infty}(\mathcal{C})\), which was shown to be the Grothendieck ring of the wreath product Deligne category \(S_{t}(\mathcal{C})\) [\textit{P. Deligne}, Stud. Math., Tata Inst. Fundam. Res. 19, 209--273 (2007; Zbl 1165.20300)]. The first main result of the paper is Theorem 8.8, in which the author describes the structure of \(\mathcal{G}_{\infty}(\mathcal{C})\) clearly. The author also gives a presentation of the ring and an expression for the distinguished basis arising from simple objects in the wreath product categories as polynomials in basic hooks. Finally, the author tells us how to generalize the result from a Hopf algebra to a tensor category.