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---
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title: Gauge group constants
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permalink: /Gauge_group_constants/
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---
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[Category:Calculations](/Category:Calculations "wikilink") SARAH supports not only chiral superfields in the fundamental representation but in any irreducible representation of *S**U*(*N*). In most cases, it is possible to fix the transformation properties of the chiral superfield by stating the corresponding dimension *D*. If the dimension is not unique, also the Dynkin labels are needed. For calculating kinetic terms and D-terms, it is necessary to derive from representation the corresponding generators. Furthermore, the eigenvalues *C*<sub>2</sub> of the quadratic Casimir for any irreducible representation *r*
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*T*<sup>*a*</sup>*T*<sup>*a*</sup>*ϕ*(*r*)=*C*<sub>2</sub>(*r*)*ϕ*(*r*)
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as well as the Dynkin index *I*
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*T**r*(*T*<sup>*a*</sup>*T*<sup>*b*</sup>)*ϕ*(*r*)=*I**δ*<sub>*a**b*</sub>*ϕ*(*r*)
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are needed for the calculation of the RGEs. All of that is derived by SARAH due to the technique of Young tableaux. The following steps are evolved:
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1. The corresponding Young tableaux fitting to the dimension *D* is calculated using the hook formula:
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$D = \\Pi_i \\frac{N + d_i}{h_i}$
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*d*<sub>*i*</sub> is the distance of the *i*. box to the left upper corner and *h*<sub>*i*</sub> is the hook of that box.
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2. The vector for the highest weight *Λ* in Dynkin basis is extracted from the tableaux.
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3. The fundamental weights for the given gauge group are calculated.
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4. The value of *C*<sub>2</sub>(*r*) is calculated using the Weyl formula
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*C*<sub>2</sub>(*r*)=(*Λ*, *Λ* + *ρ*).
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*ρ* is the Weyl vector.
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5. The Dynkin index *I*(*r*) is calculated from *C*<sub>2</sub>(*r*). For this step, the value for the fundamental representation is normalized to $\\frac{1}{2}$.
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$I(r) = C_2(r) \\frac{D(r)}{D(G)}$
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With *D*(*G*) as dimension of the adjoint representation.
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6. The number of co- and contra-variant indices is extracted from the Young tableaux. With this information, the generators are written as tensor product.
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The user can calculate this information independently from the model using the new command
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CheckIrrepSUN[Dim,N]
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`Dim` is the dimension of the irreducible representation and `N` is the dimension of the *S**U*(*N*) gauge group. The result is a vector containing the following information: (i) repeating the dimension of the field, (ii) number of covariant indices, (iii) number of contra-variant indices, (iv) value of the quadratic Casimir *C*<sub>2</sub>(*r*), (v) value of the Dynkin index *I*(*r*), (vi) Dynkin labels for the highest weight.
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##### Examples
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1. <span>**Fundamental representation**</span> The properties of a particle, transforming under the fundamental representation of *S**U*(3) are calculated via <span>CheckIrrepSUN\[3,3\]</span>. The output is the well known result
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{3, 1, 0, 4/3, 1/2, {1, 0}}
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2. <span>*'Adjoint representation **</span> The properties of a field transforming as <span>**24*'</span> of *S**U*(5) are calculated by <span>CheckIrrepSUN\[24,5\] </span>. The output will be
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{24, 1, 1, 5, 5, {1, 0, 0, 1}}
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3. <span>*'Different representations with the same dimension **</span> The <span>**<span>70</span>*'</span> under *S**U*(5) is not unique. Therefore, <span>CheckIrrepSUN\[{70, {0, 0, 0, 4}}, 5\] </span> returns
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{70, 0, 4, 72/5, 42, {0, 0, 0, 4}}
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while <span>CheckIrrepSUN\[{70, {2, 0, 0, 1}}, 5\] </span> leads to
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{70, 2, 1, 42/5, 49/2, {2, 0, 0, 1}}
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See also
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-------- |
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\ No newline at end of file |