# Differences between realizations and representation of a group

I am studying an introduction to group representation theory on my relativity class' lecture notes. I've previously learned in other classes and also on the Wikipedia article that a representation $T$ of a group $G$ on a space $V$ is a group homomorphism between $G$ and the automorphism of $V$:$$T:G\rightarrow Aut(V)$$ if $V$ is a vector space then the representation will be $$T:G\rightarrow GL(V)$$ and $T$ will be the linear representation of $G$ At a certain point the author of these lecture notes says that

Note that by representation of a group one actually means the set $V$ of objects $D(g)$ acts upon, where $D(G)$ is a realization of the group $G$. For example, for the Lorentz transformations, the matrices $\Lambda$ such that $\Lambda^T\eta\Lambda=\eta$ are the realization of the Lorentz transformation on the vectorial representation $V=\{v^{\mu}\}$

Instead on the Wikipedia article says:

Some people use realization for the general notion and reserve the term representation for the special case of linear representations.

I've also found here that what i call representation is for someone (probably also for the author of the notes) the realization. So i'm a little bit confused. Is there something wrong with my definition or i'm missing something?

• I would say that using the term realization for an arbitrary (linear) representation seems odd, as the representation need not be faithful. – Tobias Kildetoft May 25 '16 at 10:59
• @TobiasKildetoft What do you mean by this? What definition would you give for realization and what do you mean for "representation need not be faithful" ? Thank you for your time! – pier94 May 25 '16 at 11:15
• By not faithful, I mean that $T$ need not be injective. By a realization I would mean a collection of objects with a given multiplication (such as invertible matrices) which is isomorphic to the group. – Tobias Kildetoft May 25 '16 at 11:19
• So for you the representation is the map itself (that not need to be injective) between the group and the set of the automorphisms of the space and the realization is the set of the elements itself of the image of a map that need to be faithful this time with the group? – pier94 May 25 '16 at 11:27
• I don't usually distinguish between the map and the space with a given action. I usually do not use the term realization at all. – Tobias Kildetoft May 25 '16 at 11:29