On invariants of representations of Weyl groups associated with the cohomology of toric varieties

TL;DR

This paper establishes explicit algebra isomorphisms between the cohomology rings of certain toric varieties associated with Weyl groups and their quotients, extending to broader classes and answering open questions.

Contribution

It constructs explicit isomorphisms between cohomology rings of toric varieties related to Weyl groups and their quotients, generalizing previous results.

Findings

Explicit algebra isomorphisms between cohomology rings established

Generalization to intermediate lattices and non-degenerate W-symmetric polytopes achieved

Answers to open questions by Horiguchi--Masuda--Shareshain--Song provided

Abstract

For a Weyl group $W$ and a $W$-permutohedron $P$, there are associated toric varieties $X_P$ and $X_{P/W_K}$ for any parabolic subgroup $W_K$ of $W$, since the quotient $P/W_K$ can be identified with a polytope inside $P$. We construct an explicit algebra isomorphism between $H^*(X_{P/W_K};\mathbb{Q})$ and $H^*(X_P;\mathbb{Q})^{W_K}$. We further generalize this isomorphism to intermediate lattices, to finite Coxeter groups, and to non-degenerate $W$-symmetric polytopes. Our results give affirmative answers to two open questions of Horiguchi--Masuda--Shareshain--Song.