On the volume functions and the cohomology rings of special weight varieties of type A

TL;DR

The paper studies the topology and volumes of special weight varieties ${\mathcal{M}}_{T}$ arising as torus quotients of products of coadjoint orbits for type $A$ (specifically $G=SU(l+1)$). It shows that, under suitable assumptions, the symplectic volume ${\rm vol}({\mathcal{M}}_{T})$ equals the flow-polytope volume ${v_{l,n}(\lambda-\mu)}$ when $\mu$ is near $\lambda$, and develops a differential-equation framework for the flow-polytope volumes that yields an explicit presentation of the cohomology ring ${H^{*}({\mathcal{M}}_{T};\mathbb{R})}$. The cohomology is generated by two-forms $z_i$ and $v_j$, with a Kirwan-map-based construction, and, in the nice chamber, one obtains the concrete ring presentation ${H^{*}({\mathcal{M}}_{T};\mathbb{R})} \cong \mathbb{R}[z_{1},\dots,z_{l}]/(z_{l}^{n}, z_{l-1}^{n}(z_{l-1}+z_{l})^{n},\dots,(z_{1}+\cdots+z_{l})^{n})$. This provides a tractable route to compute intersection pairings, Betti numbers, and the overall cohomology structure for these weight varieties.

Abstract

In this paper, we consider the cohomology rings of some multiple weight varieties of type A, that is, symplectic torus quotients for a direct product of several coadjoint orbits of the special unitary group. Under some specific assumptions, we prove the symplectic volumes of multiple weight varieties are equal to the volumes of flow polytopes. Using differential equations satisfied by the volume functions of flow polytopes, we give an explicit presentation of the cohomology ring of the multiple weight variety of special type.

Theorems & Definitions (29)

• Theorem 1.1: see Theorem 2.1
• Corollary 1.2: see Corollary 2.2
• Theorem 1.3: see Theorem 3.3
• Theorem 1.4: see Theorem 3.9
• Theorem 2.1
• proof
• Corollary 2.2
• proof
• Corollary 2.3
• Definition 2.4