Solution of the constraints for tensors generated by iterated integrals (Q1121354)

This paper gives the solutions of the reduced problems in the following problems. Let \({\mathcal T}\) be some tensor of rank N with tensor components \({\mathcal T}_{\mu_ I...\mu_ N}\in {\mathbb{C}}\), \(\mu_ i=0,1,...,d-1\), \(i=1,...,N\), with respect to a given fixed basis in \({\mathbb{C}}^ d\). Let \(\pi\) \({\mathcal T}\), \(\pi \in S_ N\) (the symmetric group), denote the tensor with components. \((\pi {\mathcal T})_{\mu_ 1...\mu_ N}={\mathcal T}_{\mu \pi (1)...\pi (N)}\) and X\({\mathcal T}\), \(X\in D_ N\) (the group algebra of the symmetric group), \(X=\sum_{\pi \in S_ N}\lambda_{\pi}\Pi\), \(\lambda_{\pi}\in {\mathbb{C}}\), denote the tensor \(X{\mathcal T}=\sum_{\pi \in S_ N}\lambda_{\pi}\Pi {\mathcal T}.\) Let the tensor \({\mathcal T}\) satisfy the following conditions \(X_{I,I}c{\mathcal T}=0\) for all shuffle elements \(X_{I,I}c\in D_ N\). Otherwise, let the tensor \({\mathcal T}\) be arbitrary. Problem 1. Find a basis of independently variable components of the tensor \({\mathcal T}\) along with a Z-module algorithm for the calculation of the remaining tensor components in terms of the basis components. Problem 2. Find a basis of independent variables for the linear span of all spacelike tensor components \({\mathcal T}_{\mu_ 1...\mu_ n}\), \(\mu_ i=0,1,...,d-1\), \(i=2,...,N-1\), \(\mu_ J=1,2,...,d-1\), \(j=1,N\), along with a \({\mathbb{Z}}\)-module algorithm for the calculation of an arbitrary spacelike tensor component in terms of these basis variables.