Kronecker symbol

Kronecker delta

The number $\delta^i_j$ defined by $$ \delta^i_j = \begin{cases} 1 & \text{if}\, i = j \\ 0 & \text{if}\, i \ne j \end{cases}\ \ , $$ $i,j = 1,2,\ldots$. When $1 \le i,j \le n$, the Kronecker symbol $\delta^i_j$ has $n^2$ components, and the matrix $(\delta^i_j)$ is the unit matrix. The Kronecker symbol was first used by L. Kronecker (1866).

The Kronecker symbol may be generalized, considering instead a set of quantities $\delta^{i_1\ldots i_p}_{j_1\ldots j_p}$ with $2p$ integer (upper and lower) indices, $i_\alpha\,,j_\beta = 1,\ldots,n$, equal to $+1$ (or $-1$) if the sequence $(i_1\ldots i_p)$ is an even (odd) permutation of the distinct indices $(j_1\ldots j_p)$ and zero otherwise. The numbers $\delta^{i_1\ldots i_p}_{j_1\ldots j_p}$ (when $p \ge 2$ often denoted by $\epsilon^{i_1\ldots i_p}_{j_1\ldots j_p}$) are called the components of the Kronecker symbol. An affine tensor of type $(p,p)$ whose components relative to some basis are equal to the components of the Kronecker symbol is isotropic: has the same components relative to any other basis.

The Kronecker symbol is convenient in various problems of tensor calculus. For example, the determinant $$ \left|{ \begin{array}{ccc} a^1_1 & \ldots & a^1_n \\ \vdots & \ddots & \vdots \\ a^n_1 & \ldots & a^n_n \end{array} }\right| $$ is equal to the sum $$ \sum \delta^{i_1\ldots i_n}_{1\ldots n} a^1_{i_1} \cdots a^n_{i_n} $$ where the summation is performed over all $n!$ permutations $( i_1\ldots i_n )$of the numbers $\{1,\ldots, n \}$. The alternant of the tensor $\{ a^{\alpha_1\ldots\alpha_n} : 1 \le \alpha_i \le n \}$ is given by $$ a^{[\alpha_1,\ldots,\alpha_p] } = \frac{1}{p!} \sum \delta^{\alpha_1\ldots \alpha_p}_{i_1\ldots i_p} a^{i_1\ldots i_p} \ . $$

References

[1]	L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903)

Comment

For the Kronecker symbol in number theory, see Legendre–Jacobi–Kronecker symbol‎