If it is considered as a type (1,1)
tensor, the Kronecker tensor can be written \delta^i_j with a
covariant index j and
contravariant index i: \delta^{i}_{j} = \begin{cases} 0 & (i \ne j), \\ 1 & (i = j). \end{cases} This tensor represents: • The identity mapping (or identity matrix), considered as a
linear mapping V\to V or V^*\to V^* • The
trace or
tensor contraction, considered as a mapping V^* \otimes V\to K • The map K\to V^*\otimes V, representing
scalar multiplication as a sum of
outer products. The '
or multi-index Kronecker delta' of order 2p is a type (p,p) tensor that is completely
antisymmetric in its p upper indices, and also in its p lower indices. Two definitions that differ by a factor of p! are in use. Below, the version is presented has nonzero components scaled to be \pm 1. The second version has nonzero components that are \pm 1/p!, with consequent changes scaling factors in formulae, such as the scaling factors of 1/p! in '''' below disappearing.
Definitions of the generalized Kronecker delta In terms of the indices, the generalized Kronecker delta is defined as: \delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} = \begin{cases} \phantom-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an even permutation of } \mu_1 \dots \mu_p \\ -1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an odd permutation of } \mu_1 \dots \mu_p \\ \phantom-0 & \quad \text{in all other cases}. \end{cases} Let \mathrm{S}_p be the
symmetric group of degree p, then: \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_1}_{\nu_{\sigma(1)}}\cdots\delta^{\mu_p}_{\nu_{\sigma(p)}} = \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_{\sigma(1)}}_{\nu_1}\cdots\delta^{\mu_{\sigma(p)}}_{\nu_p}. Using
anti-symmetrization: \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = p! \delta^{\mu_1}_{[ \nu_1} \dots \delta^{\mu_p}_{\nu_p ]} = p! \delta^{[ \mu_1}_{\nu_1} \dots \delta^{\mu_p ]}_{\nu_p}. In terms of a p\times p
determinant: \delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} = \begin{vmatrix} \delta^{\mu_1}_{\nu_1} & \cdots & \delta^{\mu_1}_{\nu_p} \\ \vdots & \ddots & \vdots \\ \delta^{\mu_p}_{\nu_1} & \cdots & \delta^{\mu_p}_{\nu_p} \end{vmatrix}. Using the
Laplace expansion (
Laplace's formula) of determinant, it may be defined
recursively: \begin{align} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} &= \sum_{k=1}^p (-1)^{p+k} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k} \dots \check\mu_p}_{\nu_1 \dots \check\nu_k \dots \nu_p} \\ &= \delta^{\mu_p}_{\nu_p} \delta^{\mu_1 \dots \mu_{p - 1}}_{\nu_1 \dots \nu_{p-1}} - \sum_{k=1}^{p-1} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k-1}\, \mu_k\, \mu_{k+1} \dots \mu_{p-1}}_{\nu_1 \dots \nu_{k-1}\, \nu_p\, \nu_{k+1} \dots \nu_{p-1}}, \end{align} where the caron, \check{}, indicates an index that is omitted from the sequence. When p=n (the dimension of the vector space), in terms of the
Levi-Civita symbol: \delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}\,. More generally, for m=n-p, using the
Einstein summation convention: \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \tfrac{1}{m!} \varepsilon^{\kappa_1 \dots \kappa_m \mu_1 \dots \mu_p}\varepsilon_{\kappa_1 \dots \kappa_m \nu_1 \dots \nu_p}\,.
Contractions of the generalized Kronecker delta Kronecker Delta contractions depend on the dimension of the space. For example, \delta^{\nu_1}_{\mu_1} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = (d-1) \delta^{\mu_2}_{\nu_2} , where is the dimension of the space. From this relation the full contracted delta is obtained as \delta^{\nu_1 \nu_2}_{\mu_1 \mu_2} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = 2d(d-1) . The generalization of the preceding formulas is \delta^{\nu_1 \dots \nu_n}_{\mu_1 \dots \mu_n} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = n! \frac{(d-p+n)!}{(d-p)!} \delta^{\mu_{n+1} \dots \mu_p}_{\nu_{n+1} \dots \nu_p} .
Properties of the generalized Kronecker delta The generalized Kronecker delta may be used for
anti-symmetrization: \begin{align} \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{\nu_1 \dots \nu_p} &= a^{[ \mu_1 \dots \mu_p ]} , \\ \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{\mu_1 \dots \mu_p} &= a_{[ \nu_1 \dots \nu_p ]} . \end{align} From the above equations and the properties of
anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta: \begin{align} \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{[ \nu_1 \dots \nu_p ]} &= a^{[ \mu_1 \dots \mu_p ]} , \\ \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{[ \mu_1 \dots \mu_p ]} &= a_{[ \nu_1 \dots \nu_p ]} , \\ \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} \delta^{\nu_1 \dots \nu_p}_{\kappa_1 \dots \kappa_p} &= \delta^{\mu_1 \dots \mu_p}_{\kappa_1 \dots \kappa_p} , \end{align} which are the generalized version of formulae written in ''''. The last formula is equivalent to the
Cauchy–Binet formula. Reducing the order via summation of the indices may be expressed by the identity \delta^{\mu_1 \dots \mu_s \, \mu_{s+1} \dots \mu_p}_{\nu_1 \dots \nu_s \, \mu_{s+1} \dots \mu_p} = \frac{(n-s)!}{(n-p)!} \delta^{\mu_1 \dots \mu_s}_{\nu_1 \dots \nu_s}. Using both the summation rule for the case p=n and the relation with the Levi-Civita symbol,
the summation rule of the Levi-Civita symbol is derived: \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \frac{1}{(n-p)!}\varepsilon^{\mu_1 \dots \mu_p \, \kappa_{p+1} \dots \kappa_n}\varepsilon_{\nu_1 \dots \nu_p \, \kappa_{p+1} \dots \kappa_n}. The 4D version of the last relation appears in Penrose's
spinor approach to general relativity that he later generalized, while he was developing Aitken's diagrams, to become part of the technique of
Penrose graphical notation. Also, this relation is extensively used in
S-duality theories, especially when written in the language of
differential forms and
Hodge duals. ==Integral representations==