Correct classification of sesquilinear forms

[fingolfin]

Also clarify a comment about historic chouces.

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[chseeger]

Thank you for pointing this out. The current version in the manual is definitely incorrect.

In [BHR13, Theorem 1.5.13] (a slight variation of the Birkhoff-von Neumann theorem), the premise of this classification theorem is that " $\beta$ is a quasi-symmetric $\sigma$-sesquilinear form".

My original intention behind introducing $\tau$ was to state the classification theorem correctly without having to formally define "quasi-symmetric" (since its definition is not uniform in the literature and the term is only used in this specific place).

For context, the relevant definitions from [BHR13, Definition 1.5.8] are: A $\sigma$-sesquilinear form $\beta$ is

• quasi-symmetric if there exist $\lambda \in F^\times$ and $\tau \in {\rm Aut}(F)$ such that $\beta(v, u) = \lambda \beta(u, v)^\tau$ for all $u, v \in V$.
• reflexive if it satisfies the property that $\beta(u, v) = 0$ if and only if $\beta(v, u) = 0$.

The current version in the manual is clearly wrong because I forgot the argument swap $(u,v) \mapsto (v,u)$ in $\beta(v, u) = \lambda \beta(u, v)^\tau$. I think if we correct this, the statement should be accurate again?

Consequently, I think the proposed change is also not entirely correct, as it misses the argument swap as well (for instance, in the symplectic case outside characteristic 2, this would imply that $\beta = 0$).

As a side note, the more well-known version of the Birkhoff-von Neumann theorem requires the form to be "reflexive non-degenerate" instead of "quasi-symmetric". As an alternative, we could introduce the term "reflexive" and present that version instead ...

[fingolfin]

Indeed, I noticed the missing swap but then ultimately forgot to fix it sigh.

Anyway, I've rewritten it again to not claim a classification (though it is mentioned in the title), but just list the form types. That is simpler and avoids introducing notions like quasi-symmetric.

I've also re-arranged sections a bit

[chseeger]

Yes, I agree that avoiding a rigorous classification theorem and just listing the form types is much simpler and entirely sufficient.

[chseeger]

Thank you! Looks very good from my side.
(with a nitpick suggested change for a closing parenthesis)