generalized Kronecker delta symbol

Let l and n be natural numbersMathworldPlanetmath such that 1ln. Further, let ik and jk be natural numbers in {1,,n} for all k in {1,,l}. Then the generalized Kronecker delta symbol, denoted by δj1jli1il, is zero if ir=is or jr=js for some rs, or if {i1,,il}{j1,,jl} as sets. If none of the above conditions are met, then δj1jli1il is defined as the sign of the permutationMathworldPlanetmath that maps i1il to j1jl.

From the definition, it follows that when l=1, the generalized Kronecker delta symbol reduces to the traditional delta symbol δji. Also, for l=n, we obtain

δj1jni1in = εi1inεj1jn,
δj1jn1n = εj1jn,

where εj1jn is the Levi-Civita permutation symbol.

For any l we can write the generalized delta function as a determinantMathworldPlanetmath of traditional delta symbols. Indeed, if S(l) is the permutation groupMathworldPlanetmath of l elements, then

δj1jli1il = τS(l)signτδj1iτ(1)δjliτ(l)
= det(δj1i1δj1ilδjli1δjlil).

The first equality follows since the sum one the first line has only one non-zero term; the term for which iτ(k)=jk. The second equality follows from the definition of the determinant.

Title generalized Kronecker delta symbol
Canonical name GeneralizedKroneckerDeltaSymbol
Date of creation 2013-03-22 13:31:38
Last modified on 2013-03-22 13:31:38
Owner matte (1858)
Last modified by matte (1858)
Numerical id 5
Author matte (1858)
Entry type Definition
Classification msc 15A99
Related topic LeviCivitaPermutationSymbol3