Gorenstein Fano Generic Torus Orbit closures in $G/P$

Pierre-Louis Montagard (1) , Alvaro Rittatore (2)

1 IMAG - Institut Montpelliérain Alexander Grothendieck
2 CMAT - Centro de Matemática [Montevideo]

Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximal torus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial terms when the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\u{\i} and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures --- this list includes the reflexive root polytopes.

Domaines: Géométrie algébrique [math.AG]; Combinatoire [math.CO]; Théorie des représentations [math.RT]

Pierre-Louis Montagard, Alvaro Rittatore. Gorenstein Fano Generic Torus Orbit closures in $G/P$. Journal of Algebraic Combinatorics, 2023, 57, pp.439-460. ⟨10.1007/s10801-022-01198-z⟩. ⟨hal-01717935v3⟩