On the saturation conjecture for $\mathrm{Spin}(2n)$

Abstract

In this paper we examine the saturation conjecture on decompositions of tensor products of irreducible representations for complex semisimple algebraic groups of type $D$ (the even \emph{spin} groups: Spin$(2n)$ for $n\ge 4$ an integer), extending work done by Kumar-Kapovich-Millson on Spin(8). Our main theorem asserts that the saturation conjecture holds for Spin(10) and Spin(12): for all triples of dominants weights $\lambda ,\mu ,\nu$ such that $\lambda +\mu +\nu$ is in the root lattice, and for any $N>0$, $${\left(V(\lambda )\otimes V(\mu )\otimes V(\nu )\right)}^G\ne 0$$ if and only if $${\left(V(N\lambda )\otimes V(N\mu )\otimes V(N\nu )\right)}^G\ne 0,$$ for $G=$ Spin(10) or Spin(12). Some related results for groups of other types are listed as well.

11 pages; v2 added a reference; v3 added update on saturation for D6