# The mapping to alternating tensors

by yifli
Tags: alternating, mapping, tensors
 P: 70 I'm wondering why $1/k!$ is needed in Alt(T), which is defined as: $$\frac{1}{k!}\sum_{\sigma \in S_k} \mbox{sgn}\sigma T(v_{\sigma(1)},\cdots,v_{\sigma(k)})$$ After removing $1/k!$, the new $\mbox{Alt}$, $\overline{\mbox{Alt}}$, still satisfies $\overline{\mbox{Alt}}(T)(v_1,\cdots,v_i,\cdots,v_j,\cdots,v_k)=-\overline{\mbox{Alt}}(T)(v_1,\cdots,v_j,\cdots,v_i,\cdots,v_k)$, which means $\overline{\mbox{Alt}}$ is an alternating tensor