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? $\endgroup$ Commented Aug 20, 2018 at 18:49


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.