[whatshisface]

While we're talking about representations, there's something I've always wondered. Why are the objects that the maps which are the representations act on also called representations? Spinors don't act as the spinor group, S ⊂ Hom(Spinor,Spinor) does.

[JadeNB]

> While we're talking about representations, there's something I've always wondered. Why are the objects that the maps which are the representations act on also called representations? Spinors don't act as the spinor group, S ⊂ Hom(Spinor,Spinor) does.

Physicists and mathematicians speak differently, but I think that mathematicians usually avoid this language. For us, spinors are elements of the spinor representation, and, more generally, the things on which a representation acts are called generically "vectors in the representation", not representation themselves.

(That said, one will often see in math language like "let V be a representation of G", meaning more formally "let G \to GL(V) be a representation", which probably is the sort of abuse of language you mean.)

[BlackFingolfin]

Actually in math parlance, the map G \to GL(V) is the representation while V is the underlying module, and yeah its elements are vectors.

[cbreezyyall]

That's the 'regular representation' of a group. https://en.wikipedia.org/wiki/Regular_representation