Simple group theory vocabulary issue

[JoePhysicsNut]

I am reading about group theory in particle physics and I'm slightly confused about the word "representation".

Namely, it is sometimes said that the three lightest quarks form a representation of SU(3), or that the three colors do.

But at the same time, it is said that a group can be represented by a set of matrices, which operate on vectors that encode the flavor or color state.

I think it's the operators (ie matrices) that are the representation of a group, not the vectors that get operated on. How can both be a representation of a group when they're different things: operator vs the object that gets operated on?

 

[Bill_K]

Quite right, the same term representation is used to denote three things.

If we map a group G homomorphically on a group of operators D(G) in a vector space L, we say that the operator group D(G) is a representation of the group G in the representation space L. In particular, we assume the operators are linear. Then if we choose a basis in L, the linear operators of the representation can be described by their matrix representatives.

So the same word "representation" is applied to the mapping, the matrices and the vector space!

 

[JoePhysicsNut]

Quite right, the same term representation is used to denote three things.

If we map a group G homomorphically on a group of operators D(G) in a vector space L, we say that the operator group D(G) is a representation of the group G in the representation space L. In particular, we assume the operators are linear. Then if we choose a basis in L, the linear operators of the representation can be described by their matrix representatives.

So the same word "representation" is applied to the mapping, the matrices and the vector space!

Thanks! That cleared it up.