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LieART - Quick Start Tutorial
LieART provides functions to perform the most common tasks associated with irreducible representations (irreps) of the classical and exceptional Lie algebras.
This loads the package:
In[8]:=
<<LieART`
LieART 2.1.1
last revised 14 February 2025
Entering Irreducible Representations
Irreducible representations (irreps) are internally described by their Dynkin label with a combined head of Irrep and the Lie algebra.
irrep described by its algebraClass label |
Entering irreps by Dynkin label.
The follows the Dynkin classification of simple Lie algebras and can only be A, B, C, D for the classical algebras and E6, E7, E8, F4 and G2 for the exceptional algebras. The precise classical algebra is determined by the length of the Dynkin label.
algebraClass
Entering the of SU(5) by its Dynkin label and algebra class:
10
In[2]:=
Irrep[A][0,0,1,0]//FullForm
Out[2]//FullForm=
Irrep[A][0,0,1,0]
In[3]:=
Irrep[A][0,0,1,0]//StandardForm
Out[3]//StandardForm=
(0010)
In[4]:=
Irrep[A][0,0,1,0]
Out[4]=
10
The default output format type of LieART is . The associated user setting is overwritten for the front-end session LieART that is loaded in. For as default output format type evaluate: CurrentValue[$FrontEndSession,{CommonDefaultFormatTypes,"Output"}]=StandardForm
As an example for entering an irrep of an exceptional algebra, consider the of :
27
E
6
In[5]:=
Irrep[E6][1,0,0,0,0,0]
Out[5]=
27
Irreps may also be entered by their dimensional name. The package transforms the irrep into its Dynkin label. Since the algebra of an irrep of a classical Lie algebra becomes ambiguous with only the dimensional name, it has to be specified.
irrep entered by its algebra dimname |
Entering irreps by dimensional name.
Entering the of SU(5) by its dimensional name specifying the algebra by its Dynkin classification :
10
A
4
In[6]:=
Irrep[A4][Bar[10]]//InputForm
Out[6]//InputForm=
Irrep[A][0, 0, 1, 0]
The traditional name of the algebra SU(5) may also be used:
In[7]:=
Irrep[SU5][Bar[10]]//InputForm
Out[7]//InputForm=
Irrep[A][0, 0, 1, 0]
Irreps of product algebras like SU(3)⊗SU(2)⊗U(1) are specified by with the individual irreps of simple Lie algebras as arguments.
head of product irreps |
Product irreps.
The product irrep of SU(3)⊗SU(3):
(3,)
3
In[8]:=
ProductIrrep[Irrep[SU3][3],Irrep[SU3][Bar[3]]]
Out[8]=
(3,)
3
In[9]:=
%//InputForm
Out[9]//InputForm=
ProductIrrep[Irrep[A][1, 0], Irrep[A][0, 1]]
In[10]:=
ProductIrrep[Irrep[A][1,0],Irrep[A][0,1]]
Out[10]=
(3,)
3
Take for example the left-handed quark doublet in the Standard-Model gauge group SU(3)⊗SU(2)⊗U(1) (The U(1) charge is not typeset in bold face):
In[11]:=
ProductIrrep[Irrep[SU3][3],Irrep[SU2][2],Irrep[U1][1/3]]
Out[11]=
(3,2)(1/3)
In[12]:=
%//InputForm
Out[12]//InputForm=
ProductIrrep[Irrep[A][1, 0], Irrep[A][1], Irrep[U][1/3]]
Decomposing Tensor Products
decompose the tensor product of several irreps |
Tensor product decomposition.
Decompose the tensor product of SU(3):
3⊗
3
In[13]:=
DecomposeProduct[Irrep[SU3][3],Irrep[SU3][Bar[3]]]
Out[13]=
1+8
Decompose the tensor product of :
27⊗
27
E
6
In[14]:=
DecomposeProduct[Irrep[E6][27],Irrep[E6][Bar[27]]]
Out[14]=
1+78+650
Decompose the tensor product of SU(3):
3⊗3⊗3
In[15]:=
DecomposeProduct[Irrep[SU3][3],Irrep[SU3][3],Irrep[SU3][3]]
Out[15]=
1+2(8)+10
Decompose the tensor product 8⊗8 of SU(3):
In[16]:=
DecomposeProduct[Irrep[SU3][8],Irrep[SU3][8]]
Out[16]=
1+2(8)+10++27
10
In[17]:=
%//InputForm
Out[17]//InputForm=
IrrepPlus[Irrep[A][0, 0], IrrepTimes[2, Irrep[A][1, 1]], Irrep[A][3, 0], Irrep[A][0, 3], Irrep[A][2, 2]]
The multiplicity of irreps in the sum can be extracted with IrrepMultiplicity:
In[18]:=
IrrepMultiplicity[%,Irrep[SU3][8]]
Out[18]=
2
In[19]:=
%%//IrrepList
Out[19]=
{1,8,8,10,,27}
10
Decompose the tensor product of SU(4):
4⊗4⊗6⊗15
In[20]:=
DecomposeProduct[Irrep[SU4][4],Irrep[SU4][4],Irrep[SU4][6],Irrep[SU4][15]]
Out[20]=
2(1)+7(15)+4()+35+5(45)+3()+3(84)+2(175)+256
′
20
45
The Mathematica built-in command for products is replaced by for irreps as arguments. E.g., decompose the tensor product ⊗24⊗45 of SU(5):
10
In[21]:=
Irrep[SU5][Bar[10]]*Irrep[SU5][24]*Irrep[SU5][45]
Out[21]=
3()+6()+3()+5()+2()++6()+2()+++3()+2()++
5
45
50
70
105
′′
175
280
′
280
420
′
450
480
720
1120
2520
For powers of irreps the Mathematica built-in command may be used. E.g., decompose the tensor product of :
27⊗27⊗27
E
6
In[22]:=
Irrep[E6][27]^3
Out[22]=
1+2(78)+3(650)+2925+3003+2(5824)
Decompose tensor products of product irreps of SU(3)⊗SU(3)⊗SU(3):
(3,,1)⊗(,3,1)
3
3
In[23]:=
DecomposeProduct[ProductIrrep[Irrep[SU3][3],Irrep[SU3][Bar[3]],Irrep[SU3][1]],ProductIrrep[Irrep[SU3][Bar[3]],Irrep[SU3][3],Irrep[SU3][1]]]
Out[23]=
(1,1,1)+(8,1,1)+(1,8,1)+(8,8,1)
Decomposition to Subalgebras
Decompose irreps and product irreps.
Decompose the 16 of SO(10) to SU(5)⊗U(1):
Decompose the SU(3) irrep 3 in (24,3)(-3) of SU(5)⊗SU(3)⊗U(1) to SU(2)⊗U'(1), i.e., SU(5)⊗SU(3)⊗U(1)SU(5)⊗SU(2)⊗U'(1)⊗U(1):
The same decomposition as above displayed as branching rule:
Branching rules for all totally antisymmetric irreps, so-called basic irreps, of SU(6) to SU(3)⊗SU(3)⊗U(1):
Young Tableaux
Young tableaux.
Young tableau of the 720 of SU(5):
Display Young tableaux of SU(4) irreps with a maximum of one column per box count: