Representation and derived representation rings of Nakayama truncated algebras and a viewpoint under monoidal categories (Q1749215)

For a Hopf algebra \(H\), the representation (or Green) ring \(\mathsf{gr}(H)\) of \(H\) is the abelian group generated by the isomorphism classes \([V]\) of finite-dimensional \(H\)-modules modulo the relations \([V\oplus W]=[V]+[W]\). Its multiplication is given by \([V][W]=[V\otimes W]\). In particular, \(\mathsf{gr}(H)\) is the free abelian group with basis \(\{[V]\mid V \text{ indecomposable}\}\). More generally, for a Krull-Schmidt monoidal category \(\mathcal{C}\) (an additive monoidal category in which every object decomposes into a finite direct sum of indecomposable ones) one can define an analogue ring \(\mathsf{gr}(\mathcal{C})\) generated by isomorphism classes of objects modulo the same relations (e.g. \(\mathsf{gr}(H)=\mathsf{gr}({_H\mathsf{mod}})\)). The main aim of the present paper is to provide an explicit realization of the Green rings \(\mathsf{gr}(H), \mathsf{gr}(\mathsf{Ch}^{\mathsf{Sh}}(H)), \mathsf{gr}(\mathsf{D}^b(H))\) (the Green ring of \(H\), of its category of shifted complexes and of the bounded derived category of \({_H\mathsf{mod}}\)) as quotients of suitable polynomial rings in the particular case \(H=\Bbbk \mathcal{Z}_n/J^d\), the Nakayama truncated algebra with \(\mathsf{char}(\Bbbk)=p>0\) and \(d=p^m\leq n\) for some \(m> 0\). One distinguished feature of this work is that the polynomial description is obtained by using less generators than the number of iso-classes of indecomposables objects, at the price however of needing more relations. As a final remark, we point out that the case \(\mathsf{gr}(\mathsf{D}^b(H))\) has been dealed with only for \(d=2\) and the question has not been completely solved yet. Some particular structure constants seem difficult to determine, but the authors were able to restrict the possibilities to three options (see Theorem 5.9 (3)).