# generalized Kronecker delta symbol

Let $l$ and $n$ be natural numbers such that $1\leq l\leq n$. Further, let $i_{k}$ and $j_{k}$ be natural numbers in $\{1,\cdots,n\}$ for all $k$ 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 $i_{r}=i_{s}$ or $j_{r}=j_{s}$ 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 $l=1$, the generalized Kronecker delta symbol reduces to the traditional delta symbol $\delta^{i}_{j}$. Also, for $l=n$, 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 $l$ we can write the generalized delta function as a determinant of traditional delta symbols. Indeed, if $S(l)$ is the permutation group of $l$ elements, then

 $\displaystyle\delta_{j_{1}\cdots j_{l}}\!\!\!\!\!\!\!\!\!\!\!^{i_{1}\cdots i_{% l}}$ $\displaystyle=$ $\displaystyle\sum_{\tau\in S(l)}\mbox{sign}\,\tau\,\delta^{i_{\tau(1)}}_{j_{1}% }\cdots\delta^{i_{\tau(l)}}_{j_{l}}$ $\displaystyle=$ $\displaystyle\det\left(\begin{array}[]{lll}\delta^{i_{1}}_{j_{1}}&\cdots&% \delta^{i_{l}}_{j_{1}}\\ \vdots&\ddots&\vdots\\ \delta^{i_{1}}_{j_{l}}&\cdots&\delta^{i_{l}}_{j_{l}}\end{array}\right).$

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.

Title generalized Kronecker delta symbol GeneralizedKroneckerDeltaSymbol 2013-03-22 13:31:38 2013-03-22 13:31:38 matte (1858) matte (1858) 5 matte (1858) Definition msc 15A99 LeviCivitaPermutationSymbol3