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In the context of group theory (in my case, applications to physics), I frequently come across the phrase "the ${\bf N}$ of a group", for example "a ${\bf 24}$ of $SU(5)$" or "the ${\bf 1}$ of $SU(5)$" (the integer is usually typeset in bold).

My knowledge of group theory is pretty limited. I know the basics, like what properties constitute a group, and I'm familiar with simple cases that occur in physics (e.g. rotation groups $SO(2)$, $SO(3)$, the Lorentz group, $SU(2)$ with the Pauli matrices as a representation), but not much more. I've got a couple of related questions:

  • What is meant by "${\bf N}$ of a group"?
  • Is is just shorthand for an ${\bf N}$ representation? If so, what exactly is an ${\bf N}$ representation of a given group? :-)
  • How can I work out / write down such a representation concretely, like the Pauli matrices for $SU(2)$? I'd be grateful for a simple example.
  • What does it mean when something "transforms like the ${\bf N}$"?
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The dimension of SU(5) is 24. – MBN Oct 22 '12 at 14:45
    
"Standard thing" depends on who you're talking to. Physicists use jargon for talking about Lie groups that isn't standard among mathematicians and vice versa. – Qiaochu Yuan Oct 22 '12 at 14:47

OP wrote (v1):

What does "the ${\bf N}$ of a group" mean?

1) Physicists are referring to an irreducible representation (irrep) for whatever group $G$ we are talking about. The number ${\bf N}$ refers to the dimension of the irrep. The point is that irreps are so rare that irreps are often uniquely specified by their dimension (modulo isomorphisms). (This is not quite true in general, and physicists then start to decorate the bold-faced dimension symbol with other ornaments, e.g. ${\bf 3}$ and $\bar{\bf 3}$, or e.g. ${\bf 8}_v$ and ${\bf 8}_s$ and ${\bf 8}_c$, etc, to distinguish.)

2) By the way, concerning a group representation $\rho: G \to GL(V,\mathbb{F})$, where $G$ is a group, where $\mathbb{F}$ is a field (typically $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$), where $V$ is a $\mathbb{F}$-vector space, and where $\rho$ is a group homomorphism; be aware that physicists refer to both the map $\rho$ and the vector space $V$ as "a representation".

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From my observation, physics textbooks refer $GL(V, F)$ as the representation of $G$. Math textbooks refer $\rho$ (the mapping) or, $V$ as the representation. Reference: Wiki, Representation Theory: A First Course - William Fulton, Joe Harris [page 3]. – omephy Jul 22 at 13:13
    
@omephy: Which physics textbooks? – Qmechanic Jul 22 at 13:52
    
1. Matthew Robinson - Symmetry and the Standard Model (2011) [page 59] 2. Jakob Schwichtenberg - Physics from Symmetry (2015) [page 50]: "Although we define a representation as a map, most of the time we will call a set of matrices a representation." – omephy Jul 22 at 13:55
    
These are textbooks on group theory. Most physics literature identify the ${\bf N}$ of $G$ with the vector space $V$, not the map $\rho$ as they properly should, nor $GL(V,\mathbb{F})$. NB: If we discuss representation $\rho(g)$ of a specific group element $g\in G$, then $\rho(g)$ is typically represented by a $N\times N$ matrix in the physics literature. – Qmechanic 2 days ago

''the $N$ of a group $G$'' refers to an $N$-dimensional irreducible (projective) representation of the (typically semisimple) group $G$. A representation is a homomorphism $U$ from $G$ to the space of linear self-mappings of a vector space $V$ (in the projective case acting on the rays); it is irreducible if there is no basis in which all $U(g)$ are block triangular. The dimension of the representation is the dimension of $V$.

For example, the representation theory of $SO(3)$ implies that there is precisely one irreducible projective representation of every dimension $N$. The 2-dimensional representation is the spinor representation, the three-dimensional one the ordinary vector representation.

If an object $x$ transforms like an $N$ then $x$ is a generic element from an $N$-dimensional space with the representation $N$, and hence trasforms under a group element $g$ by means of $x\to U(g)x$. For example in case of $SO(3)$, if $x$ transforms as a $2$ then it is a spinor, if it transforms like a $3$ then it is a vector, etc.

In many cases, the dimension determines the representation up to isomorphism, hence the jargon. (Otherwise, representations may be called $N$ and $\overline N$, etc., to distinguish them.) For example, the dimension of SU(5) is 24, and the 24 characterizes the adjoint representation (which has dimension 24).

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Maybe you meant $so(3)$ instead of $SO(3)$. The Lie group $SO(3)$ has irreps of only odd dimension but it's Lie algebra $so(3) = su(2)$ has irreps of every dimension. – Eric Oct 22 '12 at 15:20
    
@Eric: Thanks. I edited the answer to make clear that I meant projective representations. These are the relevant ones in quantum mechanics. – Arnold Neumaier Oct 22 '12 at 16:38

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