Rank-One Juggling Varieties

• Rank-One Juggling Varieties are algebraic and geometric objects defined by imposing rank-one constraints on tensor and matrix spaces, with combinatorial interpretations via juggling patterns.
• They are closely linked to positroid stratifications in Grassmannians and quiver Grassmannians, exhibiting desirable properties such as normality, Cohen–Macaulayness, and rational singularities.
• Their explicit constructions, including computable cohomology and GKM formulations, provide practical tools for analyzing integrability in exterior differential systems and PDEs.

Rank-one juggling varieties are algebraic and geometric objects that arise in several contexts, notably in the theory of involutive tableaux and exterior differential systems, the stratifications of Grassmannians and flag manifolds, and the study of quiver Grassmannians with torus actions. These varieties are characterized by the imposition of rank-one constraints on certain tensor or matrix spaces and admit combinatorial descriptions via juggling patterns and cyclic quiver representations. Their geometry, cohomology, and stratification structure are explicitly computable in the rank-one case, exhibiting normality, Cohen–Macaulayness, and rational singularities.

1. Definitions and Core Constructions

The rank-one juggling variety admits several formulations depending on context, but central to all is the representation of rank-one conditions as algebraic equations on vector bundles or coordinate spaces:

• Grassmann Bundle Formulation: Given a smooth manifold $M$ of dimension $n + r$, the Grassmann bundle $\mathrm{Gr}_n(TM)\rightarrow M$ comprises $n$-planes in each tangent space. The vertical tangent bundle $A=\ker(\pi_*: TM^{(1)}\to TM)$ is realized as a subbundle of $W\otimes V^*$, with $W \simeq T_pM/e$ and $V^* \simeq e^*$. The rank-one cone $\mathrm{Cone}_e$ in each fiber consists of those $\pi \in A_e$ such that $\operatorname{rank}\pi = 1$, i.e., $$\mathrm{Cone}_e = A_e \cap \{\operatorname{rank}\pi=1\} = A_e\cap \operatorname{Var}(\mathcal{R})$$, with $\mathcal{R}$ the ideal of $2\times 2$ minors.
• Quiver Grassmannians: For the cyclic quiver $\Delta_n$, a rank-$(n-1)$ juggling variety is $$X(k,n,\omega)=\mathrm{Gr}_{(k\omega,\ldots,k\omega)}(U_{\omega n})$$. The rank-one case is $X_m = X(1,2,m) = \mathrm{Gr}_{(m,m)}(U_{2m})$.
• Positroid/Juggling Patterns in Grassmannians: Rank-one positroid (juggling) varieties in $Gr(1,n)\cong \mathbb{P}^{n-1}$ correspond to patterns $t=(t_1,\ldots,t_n)\in\{0,1,\ldots,n\}^n$ with $\sum t_i = n$, interpreted as the unique open stratum $$\Pi_t^\circ = \{ [p_1:\cdots:p_n]\;|\; p_i\neq 0 \iff t_i > 0\}$$ and its closure $$\Pi_t = \{[p]:p_i=0 \text{ for }t_i=0\}\cong \mathbb{P}^{d-1}$$, where $d = |\{i\mid t_i>0\}|$.

2. Combinatorial Characterizations

Rank-one juggling varieties are closely tied to combinatorial data:

• Juggling Patterns: A juggling pattern of length $n$ and ball number $k=1$ is a periodic sequence $(t_1,\ldots,t_n)$ with $\sum t_i = n$ and $0 \le t_i \le n$. Every such sequence determines a bounded affine permutation $f(i)=i+t_{\bar i}$ with $\bar i \equiv i \bmod n$. At $k=1$, these are in bijection with weak compositions of $n$ into $n$ nonnegative parts; their count is ${2n-1 \choose n}$. Each such pattern directly encodes the support of Plücker coordinates in $\mathbb{P}^{n-1}$.
• Poset of Closures: The closure relations among strata are given by coordinate inclusion: $\Pi_{t'} \subset \Pi_t$ if and only if $\{ i : t'_i > 0 \} \subseteq \{ i : t_i > 0 \}$.

3. Algebraic and Topological Properties

Rank-one juggling varieties possess several favorable geometric and algebraic properties:

• Normality and Cohen–Macaulayness: Each closed positroid variety $\Pi_t\subset \mathbb{P}^{n-1}$ is a coordinate linear subspace, hence normal and Cohen–Macaulay with rational singularities. The defining ideal is simply $I(\Pi_t) = (p_i \,:\; t_i=0)$; all such varieties are reduced and all equations are explicit.
• Cohomology Classes: The cohomology class $[\Pi_t]$ in $H^*(\mathbb{P}^{n-1})\cong \mathbb{Z}[x]/(x^n)$ is $x^{n-d}$, with $d$ the number of non-vanishing entries in $t$.
• Affine Stanley Functions: For $k=1$, the affine Stanley symmetric function $\widetilde{F}_f(x)$ attached to the permutation $f$ is $x^d$, and its projection to $H^*(Gr(1,n))$ gives the relevant cohomology class.

4. Equivariant Cohomology and GKM Theory

Rank-one juggling varieties $X_m$ exhibit a torus action compatible with GKM theory, allowing explicit computation of their equivariant cohomology:

• $(\mathbb{C}^*)^2$-action and Fixed Points: The action of $T \cong (\mathbb{C}^*)^2$ on $X_m$ produces finitely many fixed points and one-dimensional orbits, yielding a moment graph suitable for GKM analysis.
• Generators and Relations: $H_T^*(X_m;\mathbb{Q})$ is generated as a $\mathbb{Q}[\tau_1,\tau_2]$-algebra by two degree-2 GKM classes $\xi^{\pm}$, subject to three explicit polynomial relations.

| Generator | Degree | Description | |-------------------|--------|-----------------------------------------------| | $\xi^+$ | 2 | $\xi(m+1,m-1)$ | | $\xi^-$ | 2 | $\xi(m-1,m+1)$ | | $\tau_1,\tau_2$ | 2 | Equivariant parameters of $T$ |

• Knutson–Tao-type Basis: There exists a basis $$\{\xi(a,b)\;|\; (a,b)=(m\pm q, m\mp q), 0\le q\le m\}$$ of $2m+1$ elements, forming a free $\mathbb{Q}[\tau_1,\tau_2]$-module. Each basis vector is uniquely specified by local vanishing conditions and edge weights in the moment graph.
• Multiplication Rules and Structure Constants: Multiplication of basis elements yields explicit closed-form expansion in the same basis. All structure constants $c_{ij}^k(\tau_1,\tau_2)$ are integral polynomials in $\mathbb{Z}[\tau_1,\tau_2]$.

5. Incidence Correspondence and Application to PDEs

The geometric construction of rank-one juggling varieties interfaces with the study of exterior differential systems and the characteristic variety of PDEs:

• Rank-One Incidence and Characteristic Variety: The rank-one cone in each fiber projects to a characteristic variety $\Xi \subset P V^*$, characterizing covectors $[\xi]$ for which there exists $0\neq \pi\in A_e$ of rank one and proportional to $w\otimes \xi$.
• Guillemin Normal Form: For involutive tableaux $A\subset W \otimes V^*$, the symbol family $B(\varphi):W\to W$ with $v \mapsto \sum \varphi_i B^{(i)}(v)$ preserves the “mutual eigenvector” line $W^1(\xi)$ where $B(\xi)(v)w=\xi(v)w$ for all $v$. The locus $\operatorname{rank}\sigma(\xi)=1$ picks out such covectors.
• Dimension and Integrability: The dimension of the rank-one fiber $V^1(\xi) = \ker \sigma_\xi$ is generically $\sum s_i$, the sum of Cartan characters. If $A$ is involutive, standard theorems imply the characteristic variety is a finite branched cover and that associated eikonal systems are involutive.
• Worked Example: For $r=n=3$ and Cartan characters $(2,1,0)$, imposing the rank-one condition on matrix entries gives an explicit quadratic equation whose solution invokes elementary algebraic geometry.

6. Worked Examples and Explicit Calculations

The nature of rank-one juggling varieties allows for direct calculation in small cases:

• Positroid Varieties in $\mathbb{P}^3$ ($n=4$):

| Pattern $t$ | Nonzero $t_i$ | Variety $\Pi_t$ | Ideal $I(\Pi_t)$ | Class $[\Pi_t]$ | |--------------|--------------|---------------------------------|-------------------------|-----------------| | $(1,1,1,1)$ | 4 | $\mathbb{P}^3$ | $(0)$ | $1$ | | $(2,1,1,0)$ | 3 | Hyperplane $p_4=0$ ($\mathbb{P}^2$) | $(p_4)$ | $x$ | | $(0,2,2,0)$ | 2 | Plane $p_1=p_4=0$ ($\mathbb{P}^1$) | $(p_1,p_4)$ | $x^2$ | | $(4,0,0,0)$ | 1 | Point $p_1 \neq 0$ | $(p_2,p_3,p_4)$ | $x^3$ |

• Equivariant GKM calculation for $m=2$: Six fixed points correspond to $(4,0), (3,1), (2,2), (1,3), (0,4)$. The multiplication of basis elements (e.g., $\xi(3,1)\xi(1,3)=\xi(4,0)+\xi(0,4)$) demonstrates the integral structure of the cohomology ring.

7. Significance and Broader Connections

Rank-one juggling varieties serve as explicit test cases linking combinatorics, algebraic geometry, and the geometric theory of PDEs:

• Degeneration of the Positroid Framework: At $k=1$, positroid strata degenerate to coordinate linear subspaces, rendering all general results about normality, shellability, Gröbner degenerations, and cohomology classes elementary.
• Recovering Classical Geometric Structures: In the limit $m\to\infty$, the GKM model for $X_m$ recovers the affine flag variety of type $A_1$.
• Framework for PDE Integrability: Involutive tableaux and their characteristic/rank-one varieties allow a precise description of hyperbolicity criteria (Yang), integrability by the Cartan–Kähler theorem, and the eikonal construction.

A plausible implication is that the rank-one case, due to its explicitness, is an ideal ground for the study of more general phenomena in the interplay of combinatorics, algebraic geometry, and the theory of exterior differential systems.