Toric orbifolds associated with partitioned weight polytopes in classical types

期刊论文
发表日期：

2024年9月27日

查看全文

Tatsuya Horiguchi,
Mikiya Masuda,
John Shareshian,
Jongbaek Song

摘要

Given a root system $\Phi $ $Φ$ of type $A_n$ $A_{n}$ , $B_n$ $B_{n}$ , $C_n$ $C_{n}$ , or $D_n$ $D_{n}$ in Euclidean space E, let W be the associated Weyl group. For a point $p \in E$ $p \in E$ not orthogonal to any of the roots in $\Phi $ $Φ$ , we consider the W-permutohedron $P_W$ $P_{W}$ , which is the convex hull of the W-orbit of p. The representation of W on the rational cohomology ring $H^*(X_\Phi )$ $H^{*} (X_{Φ})$ of the toric variety $X_\Phi $ $X_{Φ}$ associated to (the normal fan to) $P_W$ $P_{W}$ has been studied by various authors. Let $\{s_1,\ldots ,s_n\}$ ${s_{1}, \dots, s_{n}}$ be a complete set of simple reflections in W. For $K \subseteq [n]$ $K \subseteq [n]$ , let $W_K$ $W_{K}$ be the standard parabolic subgroup of W generated by $\{s_k:k \in K\}$ ${s_{k} : k \in K}$ . We show that the fixed subring $H^*(X_\Phi )^{W_K}$ $H^{*} {(X_{Φ})}^{W_{K}}$ is isomorphic to the cohomology ring of the toric variety $X_\Phi (K)$ $X_{Φ} (K)$ associated to a polytope obtained by intersecting $P_W$ $P_{W}$ with half-spaces bounded by reflecting hyperplanes for the given generators of $W_K$ $W_{K}$ . We also obtain explicit formulas for h-vectors of these polytopes. By a result of Balibanu–Crooks, the cohomology rings $H^*(X_\Phi (K))$ $H^{*} (X_{Φ} (K))$ are isomorphic with cohomology rings of certain regular Hessenberg varieties.