Editing Kronecker delta (section)

==Generalizations==
If it is considered as a type <math>(1,1)</math> [[tensor]], the Kronecker tensor can be written <math>\delta^i_j</math> with a [[covariance and contravariance of vectors|covariant]] index <math>j</math> and [[Covariance and contravariance of vectors|contravariant]] index <math>i</math>:
<math display="block">\delta^{i}_{j} = \begin{cases} 0 & (i \ne j), \\ 1 & (i = j). \end{cases}</math>

This tensor represents:
* The identity mapping (or identity matrix), considered as a [[linear mapping]] <math>V\to V</math> or <math>V^*\to V^*</math>
* The [[trace (linear algebra)|trace]] or [[tensor contraction]], considered as a mapping <math>V^* \otimes V\to K</math>
* The map <math>K\to V^*\otimes V</math>, representing [[scalar multiplication]] as a sum of [[outer product]]s.

The '''{{visible anchor|generalized Kronecker delta}}''' or '''multi-index Kronecker delta''' of order <math>2p</math> is a type <math>(p,p)</math> tensor that is completely [[antisymmetric tensor|antisymmetric]] in its <math>p</math> upper indices, and also in its <math>p</math> lower indices.

Two definitions that differ by a factor of <math>p!</math> are in use. Below, the version is presented has nonzero components scaled to be <math>\pm 1</math>. The second version has nonzero components that are <math>\pm 1/p!</math>, with consequent changes scaling factors in formulae, such as the scaling factors of <math>1/p!</math> in ''{{section link||Properties of the generalized Kronecker delta}}'' below disappearing<!--This is worded awkwardly-->.<ref>{{cite web| url=http://people.physics.tamu.edu/pope/geom-group.pdf| first=Christopher|last=Pope| date=2008| title=Geometry and Group Theory}}</ref>

=== Definitions of the generalized Kronecker delta ===
In terms of the indices, the generalized Kronecker delta is defined as:<ref>{{cite book|first=Theodore|last=Frankel|title=The Geometry of Physics: An Introduction|edition=3rd|date=2012|publisher=Cambridge University Press|isbn=9781107602601}}</ref><ref>{{cite book|first=D. C.|last=Agarwal|title=Tensor Calculus and Riemannian Geometry|edition=22nd|date=2007|publisher=Krishna Prakashan Media}}{{ISBN missing}}</ref>
<math display="block">\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}</math>

Let <math>\mathrm{S}_p</math> be the [[symmetric group]] of degree <math>p</math>, then:
<math display="block">\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}. </math>

Using [[Antisymmetric tensor#Notation|anti-symmetrization]]:
<math display="block">\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}.</math>

In terms of a <math>p\times p</math> [[determinant]]:<ref>{{cite book |first1=David |last1=Lovelock |first2=Hanno |last2=Rund |title=Tensors, Differential Forms, and Variational Principles |publisher=Courier Dover Publications |year=1989 |isbn=0-486-65840-6 }}</ref>
<math display="block">\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}.</math>

Using the [[Laplace expansion]] ([[Determinant#Laplace's expansion and the adjugate matrix|Laplace's formula]]) of determinant, it may be defined [[Recursion|recursively]]:<ref>A recursive definition requires a first case, which may be taken as {{math|1=''δ'' = 1}} for {{math|1=''p'' = 0}}, or alternatively {{math|1=''δ''{{su|p=''μ''|b=''ν''|lh=0.9em}} = ''δ''{{su|p=''μ''|b=''ν''|lh=0.9em}}}} for {{math|1=''p'' = 1}} (generalized delta in terms of standard delta).</ref>
<math display="block">\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}</math>
where the caron, <math>\check{}</math>, indicates an index that is omitted from the sequence.

When <math>p=n</math> (the dimension of the vector space), in terms of the [[Levi-Civita symbol]]:
<math display="block">\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}\,.</math>
More generally, for <math>m=n-p</math>, using the [[Einstein summation convention]]:
<math display="block">\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}\,.</math>

=== Contractions of the generalized Kronecker delta ===
Kronecker Delta contractions depend on the dimension of the space. For example,
<math display="block">\delta^{\nu_1}_{\mu_1} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = (d-1) \delta^{\mu_2}_{\nu_2} ,</math>
where {{mvar|d}} is the dimension of the space. From this relation the full contracted delta is obtained as
<math display="block">\delta^{\nu_1 \nu_2}_{\mu_1 \mu_2} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = 2d(d-1) .</math>
The generalization of the preceding formulas is{{cn|date=January 2023}}
<math display="block">\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} .</math>

=== Properties of the generalized Kronecker delta ===
The generalized Kronecker delta may be used for [[Antisymmetric tensor#Notation|anti-symmetrization]]:
<math display="block">\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}</math>

From the above equations and the properties of [[anti-symmetric tensor]]s, we can derive the properties of the generalized Kronecker delta:
<math display="block">\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}</math>
which are the generalized version of formulae written in ''{{section link||Properties}}''. The last formula is equivalent to the [[Cauchy–Binet formula]].

Reducing the order via summation of the indices may be expressed by the identity<ref>{{cite book |first=Sadri |last=Hassani |title=Mathematical Methods: For Students of Physics and Related Fields |edition=2nd |publisher=Springer-Verlag |year=2008 |isbn=978-0-387-09503-5 }}</ref>
<math display="block"> \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}.</math>

Using both the summation rule for the case <math>p=n</math> and the relation with the Levi-Civita symbol, [[Levi-Civita symbol#n dimensions|the summation rule of the Levi-Civita symbol]] is derived:
<math display="block">\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}.</math>
The 4D version of the last relation appears in Penrose's [[Mathematics of general relativity#Spinor formalism|spinor approach to general relativity]]<ref>{{Cite journal|last=Penrose|first=Roger|date=June 1960|title=A spinor approach to general relativity |url=https://linkinghub.elsevier.com/retrieve/pii/000349166090021X|journal=Annals of Physics|language=en| volume=10|issue=2 |pages=171–201|doi=10.1016/0003-4916(60)90021-X|bibcode=1960AnPhy..10..171P}}</ref> that he later generalized, while he was developing Aitken's diagrams,<ref>{{Cite book|last=Aitken|first=Alexander Craig|title=Determinants and Matrices|publisher=Oliver and Boyd|year=1958|location=UK}}</ref> to become part of the technique of [[Penrose graphical notation]].<ref>[[Roger Penrose]], "Applications of negative dimensional tensors," in ''Combinatorial Mathematics and its Applications'', Academic Press (1971).</ref> Also, this relation is extensively used in [[S-duality]] theories, especially when written in the language of [[Differential form|differential forms]] and [[Hodge star operator#Duality|Hodge duals]].