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

1. Problem Statement and Motivation

The paper addresses a specific problem in algebraic combinatorics and toric geometry concerning the relationship between the cohomology rings of toric varieties associated with Weyl group orbits and their quotients.

• Context: Let $W$ be a Weyl group acting on a Euclidean space $V$. For a point $\lambda$ in the fundamental chamber, the convex hull $P_\lambda = \text{conv}(W(\lambda))$ is a permutohedron, which defines a toric variety $X_{P_\lambda}$.
• The Quotient: For a parabolic subgroup $W_K \subset W$ (determined by a subset of simple roots $K$), the quotient polytope $P_\lambda / W_K$ can be identified with the intersection $P_\lambda \cap C_K$ (where $C_K$ is the closed polyhedral cone). This quotient defines another toric variety $X_{P_\lambda / W_K}$.
• The Open Question: Horiguchi, Masuda, Shareshian, and Song [Hor+24] explicitly constructed an algebra isomorphism between the rational cohomology ring of the quotient variety and the $W_K$-invariant subring of the original variety, i.e., $H^*(X_{P_\lambda / W_K}; \mathbb{Q}) \cong H^*(X_{P_\lambda}; \mathbb{Q})^{W_K}$, but only for classical Lie types. They asked whether this isomorphism holds for all root systems (including exceptional types) and for more general settings (intermediate lattices, non-crystallographic systems).
• Goal: The author aims to provide a type-uniform proof of this isomorphism, generalize it to intermediate lattices and non-degenerate symmetric polytopes, and extend the framework to non-crystallographic root systems using "polytopal algebras."

2. Methodology

The paper employs a combination of combinatorial geometry, invariant theory, and the algebraic topology of toric varieties.

A. Combinatorial Geometry of Polytopes

• Face Structure Analysis: The author analyzes the face structure of the $W$-symmetric polytope $P$ and its quotient $P/W_K$.
• Stanley-Reisner Rings: The cohomology of toric varieties is described via the Stanley-Reisner ring $SR(P)$ modulo an ideal generated by linear forms (the Danilov-Jurkiewicz presentation).
• Non-Degeneracy: The paper distinguishes between non-degenerate polytopes (vertices do not lie on the boundary of the fundamental chamber $C_S$) and degenerate ones. The main theorems focus on the non-degenerate case where the polytope is simple (each vertex lies in exactly $n$ facets).
• Bijection of Facets: A key combinatorial step (Proposition 2.4) establishes a bijection between the facets of the quotient $P/W_K$ and a specific subset of facets of $P$ combined with the walls of the cone $C_K$.

B. Construction of the Isomorphism

The core of the methodology is the explicit construction of a ring homomorphism $\Phi$ (or $\phi$) from the cohomology of the quotient to the invariants of the original cohomology.

1. Generator Mapping:
   • Generators corresponding to facets $F$ in the fundamental chamber are mapped to the sum of their $W_K$-orbits: $\Phi(X_{F,e}) = \sum_{s \in W_K/W_F} X_{F,s}$.
   • Generators corresponding to the walls of the cone ($Y_k$) are mapped to linear combinations of orbit sums, weighted by coefficients $C_{F,s,k}$ derived from the relation $s(\ell_F) - \ell_F = \sum C_{F,s,k} \ell_{H_k}$.
2. Verification: The author proves that this map:
   • Preserves the ideal relations (maps the Stanley-Reisner ideal and the linear ideal of the quotient to zero in the target).
   • Is surjective onto the invariant subring.
   • Is an isomorphism by utilizing the Poincaré Duality Algebra (PDA) property. Since both rings are PDAs and the map is a surjection that preserves the top-degree component, it must be an isomorphism.

C. Generalization to Polytopal Algebras

To handle non-crystallographic root systems (where no lattice exists to define a toric variety), the author introduces the Polytopal Algebra $A(P)$.

• This is defined over $\mathbb{R}$ as a quotient of the Stanley-Reisner ring by an ideal generated by linear forms involving normal vectors on the unit sphere.
• This algebra serves as a topological analogue to the cohomology ring, allowing the results to be extended beyond crystallographic settings.

3. Key Contributions and Results

Theorem 1.1 (Main Result for Permutohedra)

For any parabolic subgroup $W_K$ of a Weyl group $W$, there is an explicit ring isomorphism:
$$H^*(X_{P_\lambda / W_K}; \mathbb{Q}) \cong H^*(X_{P_\lambda}; \mathbb{Q})^{W_K}$$
This resolves the open questions [Hor+24, Questions 8.1, 8.2] for all root systems, not just classical types.

Theorem 1.2 (Generalization to Intermediate Lattices)

The isomorphism holds for any simple non-degenerate $W$-symmetric polytope $P_\Lambda$ (where $\Lambda$ is a finite set of vertices) and any intermediate lattice $M$ between the root and weight lattices.
$$H^*(X_{P_\Lambda / W_K}; \mathbb{Q}) \cong H^*(X_{P_\Lambda}; \mathbb{Q})^{W_K}$$
This result is independent of the specific choice of the lattice $M$.

Theorem 1.4 (Non-Crystallographic Case)

For a simple non-degenerate $W$-symmetric polytope $P_\Lambda$ associated with a reduced root system (not necessarily crystallographic), there is an explicit ring isomorphism between the polytopal algebras:
$$A(P_\Lambda / W_K) \cong A(P_\Lambda)^{W_K}$$
This generalizes the result to cases where the Weyl group does not preserve a lattice.

Theorem 6.1 (Degenerate Case in Dimension 2)

The isomorphism is extended to degenerate $W$-symmetric polygons (dimension 2). This recovers and generalizes previous results by Song [Son22] for rank-2 systems.

4. Significance and Impact

1. Unification: The paper provides a type-uniform proof, eliminating the need for case-by-case analysis of classical Lie types ($A, B, C, D$) versus exceptional types ($E, F, G$).
2. Resolution of Open Problems: It definitively answers the questions posed by Horiguchi et al. regarding the existence of such isomorphisms for general root systems.
3. Topological vs. Combinatorial: By introducing the polytopal algebra $A(P)$, the author bridges the gap between the topological cohomology of toric varieties (which requires a lattice) and the purely combinatorial geometry of polytopes associated with non-crystallographic groups.
4. Representation Theory: The results deepen the understanding of the $W$-representation on the cohomology of toric varieties. The paper notes that for non-degenerate permutohedra, the cohomology ring is a permutation representation (a result previously known for crystallographic cases, now generalized).
5. Future Directions: The paper concludes by posing questions about the isomorphism for degenerate polytopes in dimensions $n \geq 3$ and the conditions under which the polytopal algebra remains a permutation representation.

In summary, Tao Gong's work establishes a robust, general framework linking the invariant theory of Weyl groups acting on polytopes to the cohomology of the associated toric varieties (or their algebraic analogues), proving that the cohomology of the quotient variety is precisely the invariant subring of the original variety.