Involution on the components of a group algebra

Question

If $G$ is a finite group and $k$ a field, there is a canonical involution (ie an involutive anti-automorphism) $\sigma$ on $k[G]$ induced by $g\mapsto g^{-1}$. Given that the center of $k[G]$ has $(\sum_{x\in C}x)_C$ as a $k$-basis, where $C$ runs over conjugacy classes, then if each $x$ is conjugated to its inverse then $\sigma$ is the identity on the center. Thus it should induce an involution on each component of the semi-simple algebra $k[G]$ (I assume that the characteristic of $k$ is nice), which are central simple algebras over some finite extension of $k$.

This is in particular the case for $G=S_n$ since conjugacy classes are given by the shape of the canonical decomposition in cycles, which is left unchanged when taking the inverse.

My question is: are the resulting algebras with involution (in the sense of The Book of Involutions for instance) studied somewhere ? Is anything interesting known about them, even just their type (orthogonal or symplectic) ? Even for the split components where the involutions will (at least in the orthogonal case) correspond to quadratic forms defined (modulo a multiplicative scalar) on the corresponding irreducible representation of $S_n$, is there anything known ?

I couldn't find any reference to that anywhere, even though it seems like a fairly natural question to ask.

Answer 1

In the book "Quadratic and hermitian forms" (by Winfried Scharlau), there is an entire section (in the Chapter 8) on the subject. The involutions induced by $g\mapsto g^{-1}$ are called the canonical involutions of $k[G]$.

For the case where the base field $k$ is a real closed field, for instance $k=\mathbb{R}$, every simple component of $k[G]$ is invariant under the canonical involution. These simple component are isomorphic (as $k$-algebra with involution) to a full matrix algebra over $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H} $ (hamiltonian quaternions) endowed with their usual involution. For the case where $k$ is an arbitrary ordered field, there is also a result on the type of the involutions on simple components of $k[G]$.

In the survey paper (by David W. Lewis) an entire section is devoted to the subject and the relevant references are cited.