PlanetMath: generalized Kronecker delta symbol

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generalized Kronecker delta symbol

(Definition)

Let and be natural numbers such that $1\le l \le n$ . Further, let and be natural numbers in $\{1,\cdots, n\}$ for all in $\{1,\cdots, l\}$ . Then the generalized Kronecker delta symbol, denoted by $\delta_{j_1\cdots j_l}\!\!\!\!\!\!\!\!\!\!^{i_1\cdots i_l}$ , is zero if or for some $r\neq s$ , or if $\{i_1,\cdots, i_l\} \neq \{j_1,\cdots, j_l\}$ as sets. If none of the above conditions are met, then $\delta_{j_1\cdots j_l}\!\!\!\!\!\!\!\!\!\!^{i_1\cdots i_l}$ is defined as the sign of the permutation that maps $i_1\cdots i_l$ to $j_1\cdots j_l$ .

From the definition, it follows that when , the generalized Kronecker delta symbol reduces to the traditional delta symbol $\delta^i_j$ . Also, for , we obtain

$\displaystyle \delta_{j_1\cdots j_n}\!\!\!\!\!\!\!\!\!\!\!\!^{i_1\cdots \,i_n}$	$\displaystyle =$	$\displaystyle \varepsilon^{i_1\cdots i_n}\varepsilon_{j_1\cdots j_n},$
$\displaystyle \delta_{j_1\cdots j_n}\!\!\!\!\!\!\!\!\!\!\!\!^{1\cdots \,n}$	$\displaystyle =$	$\displaystyle \varepsilon_{j_1\cdots j_n},$

where $\varepsilon_{j_1\cdots j_n}$ is the Levi-Civita permutation symbol.

For any we can write the generalized delta function as a determinant of traditional delta symbols. Indeed, if is the permutation group of elements, then

$\displaystyle \delta_{j_1\cdots j_l}\!\!\!\!\!\!\!\!\!\!\!^{i_1\cdots i_l}$	$\displaystyle =$	$\displaystyle \sum_{\tau\in S(l)}$ sign $\displaystyle \, \tau\, \delta^{i_{\tau(1)}}_{j_1}\cdots \delta^{i_{\tau(l)}}_{j_l}$
	$\displaystyle =$	$\displaystyle \det \left( \begin {array}{lll} \delta^{i_1}_{j_1} & \cdots & \de... ...vdots \ \delta^{i_1}_{j_l} & \cdots & \delta^{i_l}_{j_l} \end{array} \right).$

The first equality follows since the sum one the first line has only one non-zero term; the term for which $i_{\tau(k)} = j_k$ . The second equality follows from the definition of the determinant.

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See Also: Levi-Civita permutation symbol

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Cross-references: term, line, sum, equality, permutation group, determinant, delta function, Levi-Civita permutation symbol, maps, permutation, natural numbers
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This is version 2 of generalized Kronecker delta symbol, born on 2003-03-22, modified 2003-03-23.
Object id is 4119, canonical name is GeneralizedKroneckerDeltaSymbol.
Accessed 20261 times total.

Classification:

AMS MSC:

15A99 (Linear and multilinear algebra; matrix theory :: Miscellaneous topics)

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