Invariant operators, orthogonal bases and correlators in general tensor models

Abstract We study invariant operators in general tensor models. We show that representation theory provides an efficient framework to count and classify invariants in tensor models of (gauge) symmetry G d = U ( N 1 ) ⊗ ⋯ ⊗ U ( N d ) . As a continuation and completion of our earlier work, we present two natural ways of counting invariants, one for arbitrary G d and another valid for large rank of G d . We construct bases of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank of G d diagonalizes the two-point function of the free theory. It is analogous to the restricted Schur basis used in matrix models. We show that the constructions get almost identical as we swap the Littlewood–Richardson numbers in multi-matrix models with Kronecker coefficients in general tensor models. We explore the parallelism between matrix model and tensor model in depth from the perspective of representation theory and comment on several ideas for future investigation.