# Working with generalized Kronecker Deltas

Given a Manifold with 6 dimensions with a Minkowski metric, how to define a generalized Kronecker Delta in this manifold, e.g. $\delta_{abc}^{cde}$?

My goal is to simplify, for example, the following expression: $\delta_{a_{1}c_{1}d_{1}e_{1}}^{b_{1}c_{2}d_{2}e_{2}}\delta_{a_{2}c_{2}d_{2}e_{2}}^{b_{2}c_{3}d_{3}e_{3}}\delta_{a_{3}c_{3}d_{3}e_{3}}^{b_{3}c_{4}d_{4}e_{4}}\delta_{a_{4}c_{4}d_{4}e_{4}}^{b_{4}c_{1}d_{1}e_{1}}$

In 6 dimensions with a Minkowski metric and using Einstein summation convention. How would I do that using Mathematica?

• Remember that: $\delta_{a_{1}\cdots a_{n}}^{b_{1}\cdots b_{n}} = n!\delta_{[a_{1}}^{b_{1}}\cdots \delta_{a_{n}]}^{b_{n}}$. – Davi Rohe Aug 11 '16 at 19:34

I would define it like so:

KDelta[a_, b_, c_, a1_, a2_, a3_,] := Det[({
{KroneckerDelta[a, a1], KroneckerDelta[a, a2], KroneckerDelta[a, a3]},
{KroneckerDelta[b, a1], KroneckerDelta[b, a2], KroneckerDelta[b, a3]},
{KroneckerDelta[c, a1], KroneckerDelta[c, a2], KroneckerDelta[c, a3]}
})];


Then refer to it by means of

KDelta[a, b, c, a1, a2, a3]


To generalize Mark's answer, you can use Outer to construct the matrix.

Surprisingly, I don't think you actually want to use the built-in KroneckerDelta because in Einstein summation notation, KroneckerDelta[a,a] should evaluate to the dimension of the space a labels, but in Mma it's automatically 1. If not, you can always define δ=KroneckerDelta.

 kδ[upper_,lower_]/;Length[upper]==Length[lower]:=Det[Outer[δ,upper,lower]]


Then kδ[{a,b,c,d},{α,β,γ,ζ}] evaluates to

  δ[a, ζ] δ[b, γ] δ[c, β] δ[d, α]
- δ[a, γ] δ[b, ζ] δ[c, β] δ[d, α]
- δ[a, ζ] δ[b, β] δ[c, γ] δ[d, α]
+ δ[a, β] δ[b, ζ] δ[c, γ] δ[d, α]
+ δ[a, γ] δ[b, β] δ[c, ζ] δ[d, α]
- δ[a, β] δ[b, γ] δ[c, ζ] δ[d, α]
- δ[a, ζ] δ[b, γ] δ[c, α] δ[d, β]
+ δ[a, γ] δ[b, ζ] δ[c, α] δ[d, β]
+ δ[a, ζ] δ[b, α] δ[c, γ] δ[d, β]
- δ[a, α] δ[b, ζ] δ[c, γ] δ[d, β]
- δ[a, γ] δ[b, α] δ[c, ζ] δ[d, β]
+ δ[a, α] δ[b, γ] δ[c, ζ] δ[d, β]
+ δ[a, ζ] δ[b, β] δ[c, α] δ[d, γ]
- δ[a, β] δ[b, ζ] δ[c, α] δ[d, γ]
- δ[a, ζ] δ[b, α] δ[c, β] δ[d, γ]
+ δ[a, α] δ[b, ζ] δ[c, β] δ[d, γ]
+ δ[a, β] δ[b, α] δ[c, ζ] δ[d, γ]
- δ[a, α] δ[b, β] δ[c, ζ] δ[d, γ]
- δ[a, γ] δ[b, β] δ[c, α] δ[d, ζ]
+ δ[a, β] δ[b, γ] δ[c, α] δ[d, ζ]
+ δ[a, γ] δ[b, α] δ[c, β] δ[d, ζ]
- δ[a, α] δ[b, γ] δ[c, β] δ[d, ζ]
- δ[a, β] δ[b, α] δ[c, γ] δ[d, ζ]
+ δ[a, α] δ[b, β] δ[c, γ] δ[d, ζ]


You may also want something like einsteinSum,

Clear[einsteinSum];