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reference request – Lie powers of a graded vector house and Klyachko’s theorem

Let $V$ breathe a $mathbb{Z}_2$-graded vector house (aka tremendous vector house) and $L(V)$ breathe the free $mathbb{Z}_2$-graded Lie algebra (aka tremendous Lie algebra). The free tremendous Lie algebra is too graded by the variety of mills (the mills $V$ sit in diploma 1, and the Lie bracket itself has diploma zero on this sense). So it has a decomposition by levels, $L(V) = sum_{n=1}^infty L^n(V)$. The homogeneous parts $L^n(V)$ could breathe known as *(tremendous) Lie powers* of $V$ (by affinity with symmetric or alternating powers) and have an embedding $L^n(V) subset V^{otimes n}$, the place the Lie (tremendous)commutators are despatched to tensor (tremendous)commutators, $[a,b] = aotimes b – (-)^a b otimes a$.

**Q**: What is the $GL(V)$-representation theoretic description of $L^n(V)$?

When $V$ is solely plane (has no queer part), the query is answered by Klyachko’s theorem (as defined within the solutions to MO187545). So principally, I’m asking: What is the tremendous Lie algebra analog of Klyachko’s theorem? And what’s a reference that clearly states it?

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