Higher identities for the ternary commutator

TL;DR

This paper investigates polynomial identities of the ternary commutator in free associative algebras, discovering new identities at degree 11 using computer algebra and representation theory.

Contribution

It identifies and explicitly constructs new polynomial identities for the ternary commutator at degree 11, expanding known algebraic structure insights.

Findings

New identities found in degree 11 for the ternary commutator.

Explicit multilinear identity constructed for partition 2^5 1.

Demonstration of a non-multilinear identity involving permutations of variables.

Abstract

We use computer algebra to study polynomial identities for the trilinear operation [a,b,c] = abc - acb - bac + bca + cab - cba in the free associative algebra. It is known that [a,b,c] satisfies the alternating property in degree 3, no new identities in degree 5, a multilinear identity in degree 7 which alternates in 6 arguments, and no new identities in degree 9. We use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11. The only irreducible representations of dimension < 400 with new identities correspond to partitions 2^5 1 and 2^4 1^3 and have dimensions 132 and 165. We construct an explicit new multilinear identity for partition 2^5 1 and we demonstrate the existence of a new non-multilinear identity in which the underlying variables are permutations of a^2 b^2 c^2 d^2 e^2 f.