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

TL;DR

This paper proves the saturation conjecture for tensor product decompositions of irreducible representations in Spin(10) and Spin(12), extending previous work on Spin(8) and contributing to understanding representation theory of even spin groups.

Contribution

The paper establishes the saturation conjecture for Spin(10) and Spin(12), advancing the knowledge of tensor product decompositions for these specific spin groups.

Findings

Saturation conjecture holds for Spin(10) and Spin(12).

Tensor product invariants scale consistently with tensor powers.

Related results for other group types are also discussed.

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.