# Physics Notes by Jakob Schwichtenberg

### Sidebar

group_theory:groups_invariants

# Group Invariants

One possibility to label representations is given by operators constructed from the generators known as Casimir operators. These are defined as those operators that commute with all generators. There is always a quadratic Casimir operator

$$C_2(r) = T^A T^A \, ,$$

where $T^A$ denotes the $d(r) \times d(r)$ matrices that represent the generators in the representation $r$. Another important label is the Dynkin index, which is defined as

$$T(r) \delta^{AB} = \text{Tr}(T^AT^B) \, .$$

The standard convention is that the fundamental representation has Dynkin index $\frac{1}{2}$. (Huge lists of Dynkin indices can be found in Slansky's famous paper. However the indices listed there must be divided by $2$ because the Slansky uses the non-standard convention that the fundamental representation has Dynkin index $1$. )

The Dynkin Index of a representation and the corresponding quadratic Casimir operator are related through

$$\label{eq:DynkinCasimirRelation} \frac{T(r)}{d(r)}= \frac{C_2(r)}{D},$$

where $D$ denotes the dimension of the adjoint representation, i.e. of the Lie algebra. For the adjoint representation we therefore have $T(\text{adjoint})=C_2(\text{adjoint})$.

The following tables list the quadratic Casimir operators and Dynkin indices for the most important representations.