A new approach to Glauberman's correspondence.
This interesting paper generalizes the Glauberman character correspondence of finite group characters to twisted group algebras. Special cases of this result include the Glauberman correspondence and correspondences similar to those obtained by other authors. Let \(E\) be a finite group with a normal subgroup \(G\) such that \(F=E/G\) is cyclic. Let \(\mathcal A\) be a twisted group algebra of \(E\) over a field \(\mathcal F\) of characteristic zero that splits \({\mathcal A}[H]\) for each \(H\leq E\). For any such \(H\leq E\), the set \(\text{Ch}({\mathcal A}[H])\) of all virtual \(\mathcal F\)-characters of \({\mathcal A}[H]\) is a free \(\mathbb{Z}\)-module with basis \(\text{Irr}({\mathcal A}[H])\). The dual \(\widehat F\) of all linear \(\mathcal F\)-characters of \(F=E/G\) permutes \(\text{Irr}({\mathcal A}[H])\) and so \(\text{Ch}({\mathcal A}[H])\) becomes a \(\mathbb{Z} F\)-module and \(\widehat F\) preserves the natural inner product \((\phi,\psi)_{{\mathcal A}[H]}\in\mathbb{Z}\) of characters \(\phi,\psi\in\text{Ch}({\mathcal A}[H])\). A \(\mathbb{Z}\widehat F\)-submodule \(\text{Ch}({\mathcal A}[E]| E_0)\) of \(\text{Ch}({\mathcal A}[E])\) is defined, let \(\mu=\min\{(\theta,\theta)_{{\mathcal A}[E]}\mid\theta\in\text{Ch}({\mathcal A}[E]| E_0)\}\) and set \(\min\text{Ch}({\mathcal A}[E]| E_0)=\{\theta\in\text{Ch}({\mathcal A}[E]| E_0)\mid(\theta,\theta)=\mu\}\). Clearly \(\min\text{Ch}({\mathcal A}[E]| E_0)\) is \(\widehat F\)-stable. Two characters \(\phi,\psi\in\min\text{Ch}({\mathcal A}[E]| E_0)\) are ``connected'' if \((\phi,\lambda\psi)_{{\mathcal A}[E]}\neq 0\) for some \(\lambda\in\widehat F\). The fundamental result (Theorem 4.7) of this paper proves that this connectivity is an equivalence relation and that its equivalence classes biject with \(\text{Irr}^E({\mathcal A}[G])\), the \(E\)-stable characters of \(\text{Irr}({\mathcal A}[G])\). Hence any isomorphism preserving the relevant structures induces a bijection of characters. Sections 6-10 of this paper develop applications of these results that include the Glauberman correspondence, Kawanaka's bijection, the cases \(G=\text{SL}(2,2^n)\) with odd integer \(n\geq 1\), the Suzuki groups case and relative correspondences.
scientific article; zbMATH DE number 2037078