On the Natural Involution in Group Algebras of Finite Groups

Explore the mathematical conditions under which a semisimple rational algebra with an involution can be identified as a group algebra of a finite group, where the involution corresponds to the natural group inversion operation. Learn about computational investigations of invariants for the natural involution restricted to direct summands of rational group algebras, covering ATLAS groups up to the Harada–Norton group and numerous finite groups of Lie type. Discover surprising properties of adjoint involutions of invariant quadratic forms, including cases where quadratic extensions of character fields determined by discriminants are not Galois over the rationals for certain sporadic groups. Examine Richard Parker's conjecture that discriminants of invariant quadratic forms always have odd square classes, supported by proven results for solvable groups and some infinite series of finite groups of Lie type, while noting the absence of a complete structural explanation for this phenomenon.