generalized Kronecker delta symbol

Let $l$ and $n$ be natural numbers such that $1\le l\le n$.Further, let $i_k$ and $j_k$ be natural numbers in $\{1,\mathrm{\cdots },n\}$for all $k$ in $\{1,\mathrm{\cdots },l\}$.Then thegeneralized Kronecker delta symbol, denoted by${\delta }_{j_1\mathrm{\cdots }{j}_l}^{i_1\mathrm{\cdots }{i}_l}$,is zero if $i_r=i_s$or $j_r=j_s$ for some $r\ne s$, or if$\{i_1,\mathrm{\cdots },i_l\}\ne \{j_1,\mathrm{\cdots },j_l\}$ as sets.If none of the above conditions are met, then${\delta }_{j_1\mathrm{\cdots }{j}_l}^{i_1\mathrm{\cdots }{i}_l}$is defined as the sign of the permutation that maps$i_1\mathrm{\cdots }{i}_l$ to $j_1\mathrm{\cdots }{j}_l$.

From the definition, it follows that when $l=1$,the generalized Kronecker delta symbol reduces tothe traditional delta symbol ${\delta }_j^i$.Also, for $l=n$, we obtain

${\delta }_{j_1\mathrm{\cdots }{j}_n}^{i_1\mathrm{\cdots }{i}_n}$$=$${\epsilon }^{i_1\mathrm{\cdots }{i}_n}{\epsilon }_{j_1\mathrm{\cdots }{j}_n},$

${\delta }_{j_1\mathrm{\cdots }{j}_n}^{1\mathrm{\cdots }n}$$=$${\epsilon }_{j_1\mathrm{\cdots }{j}_n},$

where ${\epsilon }_{j_1\mathrm{\cdots }{j}_n}$ is the Levi-Civita permutation symbol.

For any $l$ we can write the generalized delta functionas a determinant of traditional delta symbols. Indeed,if $S(l)$ is the permutation group of $l$ elements, then

${\delta }_{j_1\mathrm{\cdots }{j}_l}^{i_1\mathrm{\cdots }{i}_l}$$=$${\displaystyle \sum _{\tau \in S(l)}}\text{sign}\tau {\delta }_{j_1}^{i_{\tau (1)}}\mathrm{\cdots }{\delta }_{j_l}^{i_{\tau (l)}}$

$=$

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