Extremal rays of the embedded subgroup saturation cone

Kiers, Joshua

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 511-585.

Nous examinons les rayons extrémaux dans le cône des poids dominants des groupes $G\subseteq \widehat{G}$ pour lesquels il existe $N\gg 0$ tels que

$${\left(V\left(N\mu \right)\otimes V\left(N\widehat{\mu }\right)\right)}^G\ne \left(0\right).$$

Nous donnons des formules pour une classe de rayons («  type I  ») sur une face régulière arbitraire. Ces rayons sont identifiés grace à une généralisation d’une conjecture de Fulton que nous démontrons. Nous vérifions que tous les autres rayons («  type 2  ») sur la face viennent d’un cône plus petit grâce à une application dont la formule est donnée. On décrit un processus pour trouver les rayons qui ne sont pas sur une face régulière. Ceci generalize les résultats de Belkale et Kiers sur les rayons extrémaux du cône tensoriel saturé ; les résultats de ce papier sont démontrés ici quand on prend l’application diagonale de $G$ dans $G\times G$. Plusieurs exemples sont fournis.

We examine the extremal rays of the cone of dominant weights $(\mu ,\widehat{\mu })$ for groups $G\subseteq \widehat{G}$ for which there exists $N\gg 0$ such that

$${\left(V\left(N\mu \right)\otimes V\left(N\widehat{\mu }\right)\right)}^G\ne \left(0\right).$$

We exhibit formulas for a class of rays (“type I”) on any regular face. These rays are identified thanks to a generalization of Fulton’s conjecture, which we prove. We verify that the remaining rays (“type II”) on the face come from a smaller cone under a map whose formula is given. A procedure is given for finding the rays not on any regular face. This generalizes work of Belkale and Kiers on extremal rays for the saturated tensor cone; the specialization is given by $\widehat{G}=G\times G$ with the diagonal embedding of $G$. We include several examples to illustrate the formulas.

Classification : 14C17, 14M15, 15A42, 22E46
Keywords: Representation theory, algebraic geometry, branching, invariant theory.
Mot clés : Théorie des représentations, géométrie algébrique, branchement, théorie des invariants.

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