Symmetric Powers of the Half-Spin Representation: Howe Duality and a Conjecture of Manivel

This work proves a conjecture formulated by Laurent Manivel in 2009 concerning the symmetric powers of the half-spin representation. In his study of secant varieties of spinor varieties, Manivel conjectured that the multiplicities in the decomposition of the fourth symmetric power of the half-spin representation of the orthogonal Lie algebra are independent of the rank, and he proposed explicit generating functions for three families of these multiplicities.The article proves this conjecture and generalizes it to any symmetric power. To this end, we use the skew Howe duality for the pair consisting of the orthogonal Lie algebra and the orthogonal group acting on an exterior algebra. This tool allows us to identify the symmetric power of the half-spin representation with the space of invariants of the hyperoctahedral group (the Weyl group of type B) inside said exterior algebra.Consequently, the multiplicities are expressed as dimensions of spaces of invariants in irreducible representations of the orthogonal group, which proves their independence from the rank.For the particular case of the fourth symmetric power, we recover and prove the three generating functions conjectured by Manivel, using classical invariant theory of reflection groups and the rotation group of the cube. The work also includes an independent computational verification for small rank cases, whose scripts are attached as supplementary material.