  Matrix models with continuous symmetry are powerful tools for studying
quantum gravity and holography. Tensor models have also found applications in
holographic quantum gravity. Matrix models with discrete permutation symmetry
have been shown to satisfy large $N$ factorisation properties relevant to
holography, while also having applications to the statistical analysis of
ensembles of real-world matrices. Here we develop 3-index tensor models in
dimension $D$ with a discrete symmetry of permutations in the symmetric group
$S_D$. We construct the most general permutation invariant Gaussian tensor
model using the representation theory of symmetric groups and associated
partition algebras. We define a representation basis for the 3-index tensors,
where the two-point function is diagonalised. Inverting the change of basis
gives an explicit formula for the two-point function in the tensor basis for
general $D$.
