Alternating quaternary algebra structures on irreducible representations of sl(2,C)

TL;DR

This paper investigates the structure of certain algebraic representations of sl(2,C), identifying multiplicities and polynomial identities for specific cases using computational methods.

Contribution

It determines multiplicities of irreducible representations within exterior powers and classifies polynomial identities for invariant algebra structures.

Findings

Multiplicity is 1 for n=4,6 and 2 for n=8,10.

Polynomial identities of degree ≤7 are classified for these cases.

Computational linear algebra is used to analyze algebraic identities.

Abstract

We determine the multiplicity of the irreducible representation V(n) of the simple Lie algebra sl(2,C) as a direct summand of its fourth exterior power $\Lambda^4 V(n)$. The multiplicity is 1 (resp. 2) if and only if n = 4, 6 (resp. n = 8, 10). For these n we determine the multilinear polynomial identities of degree $\le 7$ satisfied by the sl(2,C)-invariant alternating quaternary algebra structures obtained from the projections $\Lambda^4 V(n) \to V(n)$. We represent the polynomial identities as the nullspace of a large integer matrix and use computational linear algebra to find the canonical basis of the nullspace.