Permutation Centralizer Algebras and Multi-Matrix Invariants

TL;DR

This paper introduces permutation centralizer algebras that underpin the combinatorics of multi-matrix gauge invariants, providing new tools for counting, diagonalizing, and understanding symmetries in multi-matrix models.

Contribution

It defines a new class of permutation centralizer algebras, analyzes their structure, and connects them to multi-matrix invariants, Schur operators, and gauge theory symmetries.

Findings

Decomposition explains counting of restricted Schur operators.

Algebra structure relates to Littlewood-Richardson numbers.

Provides a star product and efficient correlator computations.

Abstract

We introduce a class of permutation centralizer algebras which underly the combinatorics of multi-matrix gauge invariant observables. One family of such non-commutative algebras is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of 2-matrix models. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The centre of the algebra allows efficient computation of a sector of multi-matrix correlators. These generate the counting of a certain class of bi-coloured ribbon graphs with arbitrary genus.