Invariant Operators, Orthogonal Bases and Correlators in General Tensor Models

TL;DR

This paper develops a representation-theoretic framework to classify and compute invariant operators and correlators in general tensor models, extending previous work and drawing parallels with matrix models.

Contribution

It introduces two counting methods for invariants, constructs a basis of invariant operators, and explores the analogy between tensor and matrix models using representation theory.

Findings

Constructed a basis of invariant operators that diagonalizes the two-point function.

Established a parallel between Kronecker coefficients in tensor models and Littlewood-Richardson numbers in matrix models.

Provided methods for counting invariants for arbitrary and large group ranks.

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. In continuation and completion of our earlier work, we present two natural ways of counting invariants, one for arbitrary rank of the group and another valid for large rank. We construct basis of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank diagonalizes two-point function. 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 this parallelism between matrix model and tensor model in depth from the perspective of representation theory and comment on several ideas for future investigation.