Electrical networks and the Grove algebra

TL;DR

The paper constructs the grove algebra $G_n$, the coordinate ring of the planar electrical networks space ${\mathcal X}_n$, as an electrical analogue of the Plücker algebra. It develops the combinatorics of double groves, introduces the Bush basis for degree two indexed by $3$-noncrossing matchings, and proves a quadratic Gröbner basis for the grove ideal alongside a positive expansion of grove products in this basis. It further connects grove coordinates to Temperley–Lieb immanants via concordance and establishes electrical Plücker relations, while embedding ${\mathcal X}_n$ into Grassmannians and linking to the Lagrangian Grassmannian through symplectic representation theory. The work also outlines an electrical canonical basis with positivity conjectures and a tableaux-based perspective, providing a rich algebraic and combinatorial framework for planar electrical networks with deep ties to invariant theory and geometric representation theory.

Abstract

We study the ring of regular functions on the space of planar electrical networks, which we coin the grove algebra. This algebra is an electrical analogue of the Plücker ring studied classically in invariant theory. We develop the combinatorics of double groves to study the grove algebra, and find a quadratic Gröbner basis for the grove ideal.

Figures (20)

• Figure 1: Parallels between $\mathcal{X}_n$ and $\mathop{\mathrm{Gr}}\nolimits(k,n)$
• Figure 2: An example of a cactus network for $\zeta=(\bar{1}|\bar{2}|\bar{3},\bar{5}|\bar{4}|\bar{6}|\bar{7})$
• Figure 3: A cactus network $\Gamma$ with its medial graph $G(\Gamma)$ (left) and medial pairing $\tau(\Gamma)=\{(1,2),(3,11),(4,13),(5,12),(6,8),(7,9),(10,14)\}$ (right)
• Figure 4: A network $\Gamma$ and its dual $\Gamma^{\vee}$
• Figure 5: Example of a Dyck path of semilength 5 and its corresponding noncrossing matching under the bijection described above

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2: \ref{['thm:tableaux-basis']} and \ref{['prop:dim_G_nd']}
• Theorem 1.3
• Conjecture 1.4
• Conjecture 1.5
• Proposition 2.1: Proposition 2.12 of lam2004electroid
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Theorem 3.1: Corollary 5.4 of chen2007crossings