Cadabra manual: Reduce

reduce_delta

Simplify a self-contracted generalised delta.

Reduce a self-contracted generalised Kronecker delta symbol to a simpler expression without self-contractions, according to

$\begin{equation} n! \, \delta^{a_1\cdots a_n}_{b_1\cdots b_n}\, \delta^{b_1}_{a_1} \cdots \delta^{b_m}_{a_m} = \Big[\prod_{i=1}^m \big( d-(n-i) \big) \Big] \, (n-m)!\, \delta^{a_{m+1}\cdots a_n}_{b_{m+1}\cdots b_n}\, . \end{equation}$ Here is an example:

\delta{#}::KroneckerDelta; {m,n,q}::Integer(0..3); ex:=\delta_{m}^{n}_{n}^{q};

$\displaystyle{}\text{Attached property KroneckerDelta to }\delta\left(\#\right).$

$\displaystyle{}\text{Attached property Integer to }\left(m, n, q\right).$

$\displaystyle{}\delta_{m}\,^{n}\,_{n}\,^{q}$

reduce_delta(_);

$\displaystyle{} - \frac{3}{2}\delta_{m}\,^{q}$

Note that this requires that the indices on the Kronecker delta symbol also carry an Integer property to specify their range.