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

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.

Figures (2)

• Figure 1: Degenerate polytope $P_{\lambda}$ of type $I_2(5)$ (left) and non-degenerate polytope $P_{\lambda}$ of type $A_2$ (right). The blue arrows indicate roots, the orange area indicates $P_{\lambda}\cap C_S$. Each $E_i$ denotes the edge.
• Figure 2: Three possible shapes of $P_{\Lambda}$. The orange area indicates $P_{\Lambda}\cap C_K$. When $|K|=1$, the blue polyline is actually a straight line.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 2.1: humphreysReflectionGroupsCoxeter1992
• Lemma 2.2
• proof
• Remark 2.3
• Proposition 2.4