ABSTRACT
Abstract We prove a formula for the multiplicity of the irreducible representation V (n) of sl(2,C) as a direct summand of its own exterior cube Λ3V (n). From this we determine that V (n) occurs exactly once as a summand of Λ3V (n) if and only if n = 3, 5, 6, 7, 8, 10. These representations admit a unique sl(2)-invariant alternating ternary structure obtained from the projection Λ3V (n)→ V (n). We calculate the structure constants for each of these alternating triple systems and use computer algebra to determine their polynomial identities of degree ≤ 7. We discover a remarkable 14-term identity in degree 7. The variety defined by this identity contains V (3), V (5), and V (7).
