Volume Polynomials and Duality Algebras of Multi-Fans

Abstract

Abstract. We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan ∆ determines a volume polynomial V ∆ whose values are the volumes of multi-polytopes based on ∆. This homogeneous polynomial is further used to construct a Poincare duality algebra A˚p∆q. We study the structure and properties of V ∆ and A˚p∆q and give applications and connections to other subjects, such as Macaulay duality, Novik-Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the g-theorem does not hold for multi-polytopes.

The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand—Zetlin Polytope

“…For the Hessenberg function h = (3,4,4,4), an argument similar to Example 2.4 yields that Vol λ (Hess(S, h)) is the sum of the areas of the facets {x Figure 16. Face diagrams corresponding to Vol λ (Hess(S, h)) when h = (3,4,4,4) We take a moment to note that the results of Section 3 additionally allows us to explicitly and directly interpret the results of the partial derivative operations in the RHS of (4.6) in terms of the faces of GZ(λ), in a manner independent of Theorem 4.7.…”

“…2! Also note that T [2] and T [3] have the same diagonal entries, as do T [4] and T [5]; hence they contribute to the same monomial term in the formula. Putting this together, we obtain that the volume of the face F is equal to Figure 11.…”

Quasitoric Totally Normally Split Manifolds

“…It was proved by Timorin [20]. In case of a quasitoric manifold M 2n it follows from the results of Ayzenberg and Matsuda [4] about the dual algebra of a multifan, in the particular case of a simplicial complex on a sphere. Define the homogeneous form Q α : A 1 (V F ) → R of degree k by the formula Q α (x) := αx k V F , where k = 1, .…”

“…The conditions for odd k are always satisfied so not included in the above Theorem. In the case of even n, among these homogeneous forms one has the n-form corresponding to the volume polynomial of the multifan of M 2n [4]. There is a caveat: for any elements a, b ∈ H 2(n−2k) (M 2n ; Z) with admissible 2k-forms Q a , Q b the sum Q a+b is not admissible, generally speaking.…”

“…[4, Theorem 8.2] Let M 2n be a quasitoric manifold with the multifan F ⊂ R n having m rays. Then the isomorphism of algebrasH * (M 2n ; R) A * (V F ), a → D a ,holds.…”