Character triples and equivalences over a group graded \(G\)-algebra (Q2204836)

The paper under review is motivated by recent reduction theorems for the Alperin-McKay conjecture and related conjectures in modular representation theory. Let \((\mathcal{K}, \mathcal{O},k)\) be a sufficiently large \(p\)-modular system where \(p\) is a prime, and let \(N\) be a normal subgroup of a finite group \(G\). Moreover, let \(b\) be a \(G\)-stable block idempotent of \(\mathcal{O}N\), and set \(A := \mathcal{O}Gb\), \(B := \mathcal{O}Nb\). Finally, let \(V\) be a \(G\)-stable simple module over \(\mathcal{K}B := \mathcal{K} \otimes_\mathcal{O} B\). The authors call the triple \((A,B,V)\) a module triple. This is a variant of the notion of a character triple widely used in character theory. Every module triple \((A,B,V)\) determines a twisted group algebra \(E(V) := \mathrm{End}_{\mathcal{K}A}(\mathcal{K}A \otimes_{\mathcal{K}B} V)^\mathrm{op}\) of \({\bar G} := G/N\) over \(\mathcal{K}\). Let \(G'\) be a subgroup of \(G\) such that \(G = G'N\) and \(C_G(N) \subseteq G'\), and let \(N' := G' \cap N\), so that \(G'/N'\) can be identified with \({\bar G}\). Moreover, let \(b'\) be a \(G'\)-stable block idempotent of \(\mathcal{O}N'\), and set \(A' := \mathcal{O}G'b'\), \(B' := \mathcal{O}N'b'\). Also, let \(V'\) be a \(G'\)-stable simple \(\mathcal{K}B'\)-module, so that \((A',B',V')\) is another module triple. The authors write \((A,B,V) \ge_\mathrm{c} (A',B',V')\) if there exists an isomorphism of \({\bar G}\)-graded algebras \(E(V) \cong E(V')\) which commutes with the canonical maps from \(\mathcal{K}C_G(N)\) into \(E(V)\) and \(E(V')\). This is a variant of an order relation on character triples defined by \textit{B. Späth} [in: Local representation theory and simple groups. Extended versions of short lecture courses given during a semester programme on ``Local representation theory and simple groups'' held at the Centre Interfacultaire Bernoulli of the EPF Lausanne, Switzerland, 2016. Zürich: European Mathematical Society (EMS). 23--61 (2018; Zbl 1430.20008)]. The authors prove that certain complexes of \((A,A')\)-bimodules (inducing what the authors call a \({\bar G}\)-graded Rickard equivalence over \(\mathcal{O}C_G(N)\) between \(A\) and \(A'\)) imply a relation \((A,B,V) \ge_\mathrm{c} (A',B',V')\) where \(V\) and \(V'\) are simple modules over \(\mathcal{K}B\) and \(\mathcal{K}B'\), respectively, corresponding under the equivalence. In order to prove this result, the paper develops a machinery of \({\bar G}\)-graded \({\bar G}\)-acted \(\mathcal{O}\)-algebras, of \({\bar G}\)-graded \(\mathcal{O}\)-algebras over a \({\bar G}\)-graded \({\bar G}\)-acted \(\mathcal{O}\)-algebra, and of \({\bar G}\)-graded bimodules over a \({\bar G}\)-graded \({\bar G}\)-acted \(\mathcal{O}\)-algebra; precise definitions can be found in the paper.