[dapperdrake]

How come "representation of the spin group" is an insufficient starting point?

Spin group seems like they either have a specific group (from Algebra) in mind or that spins are at least defined by choosing a specific group (a set with a binary operation satisfying the group axioms/definition).

A "representation" also has a definition in Algebra with regards to groups. There are group homo-morphisms between two groups. This means you have a mapping that preserves group structure. Representation theory is about mapping groups into the set of matrices or a subset of matrices "with numbers in the matrices." Then there are group actions (don’t care for the name) - basically/conceptually a set of functions that behaves like a specific group under composition, but way more notation around that. Finally, category theory looks at "groups of groups" with the binary operation being homo-morphisms between the "inside/smaller/contained/internal" groups thus forming a larger "outside" group called a category. Because this involves talking about sets of sets you end up also needing the term "class" from set-theory.

[JadeNB]

It's not that "representation of the spin group" is undefined, but that there are too many of them for it to pin things down uniquely. (In this case, fortunately, it's not hard to say which representation it is (see https://news.ycombinator.com/item?id=43388052), but just saying "a representation" isn't enough.)

[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