De Rerum Natura

My previous post on precision error tensors was misleading. We tend to think of tensors as complicated mathematical structures. Vectors are rank 1 tensors. Driving home from work today, I realized that I had already shown that precision error vectors can be calculated in our horizontal decorrelation estimation paper. So mathematically speaking, I have already shown that precision error should be treated as tensors. The precision error vector is the rank 1 tensor example. The precision error covariance matrix is the rank-2 tensor. Two examples in the usual tensor progression. At some future time I should calculate the rank-3 tensor. How would one induce representations of the Symmetric group in rank-3 tensors?