Kronecker delta
Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as:

Let S p {\displaystyle \mathrm {S} _{p}} be the symmetric group of degree p {\displaystyle p}, then:

Using anti-symmetrization:

In terms of a p × p {\displaystyle p\times p} determinant:

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:

where the caron, ˇ {\displaystyle {\check {}}}, indicates an index that is omitted from the sequence.

When p = n {\displaystyle p=n} (the dimension of the vector space), in terms of the Levi-Civita symbol:

More generally, for m = n − p {\displaystyle m=n-p}, using the Einstein summation convention:
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