Automorphisms of generic Iwahori-Hecke algebras and integral group rings of finite Coxeter groups
Let \(G\) be a finite group and let \(Cl(G)\) be the set of conjugacy classes of \(G\). The group \(\Aut CT(G)\) of character table automorphisms is defined to be a bijection \(\tau\colon Cl(G)\to Cl(G)\) such that the induced map \(\tau\colon Z(\mathbb{Z} G)\to Z(\mathbb{Z} G)\) is a unitary ring automorphism which commutes with all power maps. The authors prove that if \((G,S)\) is a finite Coxeter system, then \(\Aut CT(G)\simeq\Aut(G)/\text{Inn}(G)\), the group of outer automorphisms of \(G\). Moreover, the authors prove that if \((G,S)\) is a finite Coxeter system and if \(H\) is a finite group such that \(\mathbb{Z} G\simeq\mathbb{Z} H\) as augmented algebras, then there is a unit \(u\) in \(\mathbb{Q} G\) such that \(uGu^{-1}=H\). The authors establish the structure of algebra automorphisms of any Hecke algebra. They prove that the automorphism group is a semidirect product of the central automorphisms by the direct product of the graph automorphisms and the group generated by the Alvis-Curtis duality. The article contains a wealth of information on automorphism groups of finite groups in general, of finite Coxeter groups and character table automorphisms. The heart of the proof of the above results is a detailed discussion of the standard reflection character of the Coxeter group. The authors prove by a lengthy case by case analysis that the reflection character is fixed by a character table automorphism if and only if it is induced by a graph automorphism. This reflection character takes the role of the trivial character in [\textit{F. Bleher}, Automorphismen von Gruppenringen und Blocktheorie (Dissertation Stuttgart 1996; Zbl 0858.20001)]\ where she used this character to establish analogous statements for certain blocks with cyclic defect groups. Work of Geck and Geck, Rouquier set up a theory similar to blocks with cyclic defect groups and using this and similar considerations as in the first author's dissertation, the authors establish the above statements.
scientific article; zbMATH DE number 1104118
scientific article