When we have a bilinear symmetric/ bilinear anti-symmetric/hermitian form $b$ on a real/complex vector space $V$, one can consider the group of invertible matrices $A \in GL(V)$ which respect $b$, that is $$b(Av, Aw) = b(v,w).$$ We get respectively the orthogonal, symplectic and unitary groups, denoted $O(V,b), Sp(V,b), U(V,b)$ respectively.

If we consider instead the corresponding similitudes, i.e. invertible matrices $A \in GL(V)$ satisfying $$\exists \lambda \in k^{\times},\; b(Av, Aw) = \lambda \cdot b(v,w)$$ (with $k=\Bbb R$ or $\Bbb C$), we obtain the subgroups usually denoted $$GO(V,b), GSp(V,b), GU(V,b)$$ respectively.

My question is to know what the letter $G$ stands for, in the notation of these three subgroups.

While the $G$ in $GL(V)$ denotes the "general" linear group, I don't think this has the same origin for the similitude group. Neither the French nor the German translations of similitude begin with "g". Notice that writing $SO(V,b)$ would be confusing, because we already have this notation for "special orthogonal" group.

Thank you!

  • 4
    $\begingroup$ I do think the notation is parallel to the $G$ in $GL_n$... but, indeed, $GSp_n$ is never pronounced "general symplectic group", but "symplectic similitude group". A slightly rare case that mathematical notation is pronounced much differently than written? $\endgroup$ Aug 20 '18 at 18:49

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