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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.

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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.

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  • $\begingroup$ Thanks, I can't believe I didn't think to check the Sharlau... Anyway, judging from the Lewis paper, very few is known for arbitrary fields (even though apparently the condition I gave for $G$ is the right one to ensure that there will be involutions of the first kind on each component), even in the case of the symmetric group. In case anyone stumbles upon this post, just note that I computed the quadratic form corresponding to the standard representation of $S_n$ over $\mathbb{Q}$ : it has matrix $Id+U$ where $U$ has $1$ everywhere. $\endgroup$ Feb 25 '15 at 12:44

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