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Please look away if you’re an expert in Representation Theory… unless you’re happy to offer a helping hand to some tyros.

To those of us – especially physical chemists and experimental physicists (rather than mathematicians or theoretical physicists) who are trying to make full sense of elementary group theory and especially representation theory, it can be unclear what is meant precisely, in natural language terms, by phrases such as

  • spinor representation”

To see why it doesn’t immediately make complete sense to some of us, even though we probably get most of the maths involved (there have been related queries on Google groups/Physics Forum etc, over the years):

If an irreducible representation of a group can be realised as the corresponding block-diagonal elements of matrices representing each operation of the group (for discrete groups), then a phrase like “spinor representation” suggests that the representation (itself) IS a spinor, which I think is not correct (while there are matrix spinors as elements of the minimal left ideal in M(2,C), only the left column is non-zero).

TO ANSWER THE QUESTION: Would it be correct, and pedagogically more effective for beginners (though totally unnecessary for mathematicians et al.):

  • In place of simply “spinor representation” to explain initially via something more like:

    “The representation of the rotation group when it acts on a spinor, is known for short as the spinor representation (with the spinor itself usually being a column-vector in a spinor space), as distinct from, say, a vector representation of the rotation group when acting on a 3-space vector” ?

NB This isn't a question about the maths, it's only about the natural language description that tends to confuse at initial study. This is a more elementary query than earlier answer on spinors and representation.

With appreciation for any expert willing to help clear up this terminological inexactitude (and any related ones you might know of…)

UPDATE: Pleased to see Vibert's comment, acknowledging that "in physics we use the term representation in a sloppier way than our mathematician friends. – Vibert May 26 '13 at 7:02", in a discussion of reps of the inhomogeneous Lorentz group.

(If anyone is down-voting, please be helpful and explain the reason/correct errors, so some of us can learn a little more.)

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  • $\begingroup$ Related: physics.stackexchange.com/q/41424/2451 $\endgroup$ – Qmechanic May 22 '17 at 13:23
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    $\begingroup$ Physicists usually only talk about the vector space being acted on, leaving the representation implicit. $\endgroup$ – Javier May 22 '17 at 13:48
  • $\begingroup$ Thanks, Javier. My concern is not with how experienced and knowledgeable physicists talk, but making it clear at the outset for a beginner. Is my suggested description incorrect? $\endgroup$ – iSeeker May 22 '17 at 13:55
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I think it makes perfect sense to refer to the representation of the rotation group that acts on spinors as the "spinor representation", the representation and the vector space upon which that representation acts are in 1 to 1 correspondence, students should feel comfortable talking about either one and using context to figure out if it is the vector space or representation being discussed.

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    $\begingroup$ Thanks for the clarification - once confirmed it does make sense, but not at first reading (as explained in the question) if studying alone. Incidentally, is it the vector space that "carries" a representation? $\endgroup$ – iSeeker May 22 '17 at 14:09
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    $\begingroup$ No need now for a 2nd reply to comment, C*A. I've just found, in Arnold Neumaier's FAQs, "the state space of the object carries a unitary representation". $\endgroup$ – iSeeker May 22 '17 at 16:16

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