Subject : Request for Formula Regarding Prefactor After Symmetrization for Wedge - Valued Tensors​
​​
​ I am using the Symmetrize function in Wolfram Mathematica for wedge - valued tensors, specifically in operations involving full inner antisymmetry.I would like to confirm the standard prefactor applied after symmetrization for tensors such as Ω for example
In[]:=
SeedRandom@0;​​B=RandomInteger[{-1,1},{3,2,2}]
Out[]=
{{{1,1},{0,-1}},{{-1,1},{-1,-1}},{{1,-1},{0,0}}}
i
(
Ω
μν
)
j
=
d
ν
j
(
A
μ
)
i
+2
j
(
A
μ
)
i

f
j
(
A
ν
)
i
​​f=#.#2-#2.# &acts on inner{i,j}​​used prefactor here (1+1)!
In[]:=
n=Length@B;​​X=Subscript[x,#]&/@Range@n;​​A0=Array[Subscript[a,##]@@X&,{n,n}];​​A=A0.B;​​Ω=(1+1)!Symmetrize[​​Transpose[Grad[A,X]+2A.Transpose[A,{3,1,2}],{1,3,4,2}],​​Antisymmetric@{1,2}];​​{TensorSymmetry@Ω,TensorDimensions@Ω}
Out[]=
{Antisymmetric[{1,2}],{3,3,2,2}}