generalized Kronecker delta symbol GeneralizedKroneckerDeltaSymbol 2003-03-22 15:57:54 2003-03-23 02:48:08 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 $\{1,\cdots, n\}$ for all $k$ in $\{1,\cdots, l\}$. Then the \emph{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 \begin{eqnarray*} \delta_{j_1\cdots j_n}\!\!\!\!\!\!\!\!\!\!\!\!^{i_1\cdots \,i_n}&=&\varepsilon^{i_1\cdots i_n}\varepsilon_{j_1\cdots j_n},\\ \delta_{j_1\cdots j_n}\!\!\!\!\!\!\!\!\!\!\!\!^{1\cdots \,n}&=&\varepsilon_{j_1\cdots j_n}, \end{eqnarray*} 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 \begin{eqnarray*} \delta_{j_1\cdots j_l}\!\!\!\!\!\!\!\!\!\!\!^{i_1\cdots i_l} &=& \sum_{\tau\in S(l)} \mbox{sign} \, \tau\, \delta^{i_{\tau(1)}}_{j_1}\cdots \delta^{i_{\tau(l)}}_{j_l} \\ &=& \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). \end{eqnarray*} 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.