Ind-cluster algebras and infinite Grassmannians

TL;DR

This work establishes that the coordinate ring $\mathbb{C}[\mathrm{Gr}]$ of the Sato–Segal–Wilson Grassmannian carries a natural ind-cluster algebra structure, arising as the colimit of the finite Grassmannian rings $\mathbb{C}[\mathrm{Gr}_{m,n}]$ endowed with Scott’s cluster algebras. It introduces the concept of ind-cluster algebras as ind-objects in a category $\mathsf{mCl}$ of inducible melting cluster morphisms, proving that cluster algebras of infinite rank are precisely these ind-objects and that compact objects are the finite-rank seeds. The paper then connects this framework to KP-hierarchy via Plücker coordinates $\Delta_{\lambda}(W)$ and $\tau$-functions, showing that $ au$-functions form Laurent polynomials with positive coefficients in rectangular Plücker coordinates and that totally positive points $W\in\mathrm{Gr}^+$ correspond to positive Plücker data. Finally, it provides a combinatorial description through Postnikov diagrams, extending finite-grid mutations to an infinite setting and exposing how the KP equation emerges as a mutation relation in the ind-cluster algebra. This construction links infinite-dimensional geometry, integrable systems, and cluster combinatorics, offering a robust algebraic framework for KP-solutions and positivity phenomena on the Sato Grassmannian.

Abstract

A prototypical examples of a cluster algebra is the coordinate ring of a finite Grassmannian: using the Plücker embedding the cluster algebra structure allows one to move between `maximal sets' of algebraically independent Plücker coordinates via mutations. Fioresi and Hacon studied a specific colimit of the coordinate rings of finite Grassmannians and its link with the infinite Grassmannian introduced by Sato and independently by Segal and Wilson in connection with the Kadomtsev-Petiashvili (KP) hierarchy, an infinite set of nonlinear partial differential equations which possess soliton solutions. In this article we prove that this ring is a cluster algebra of infinite rank with the structure induced by the colimit construction. More generally, we prove that cluster algebras of infinite rank are precisely the ind-objects of a natural category of cluster algebras.

Figures (12)

• Figure 3.1: A graphical depiction of the bijection between Maya sequences and Young diagrams: each black go-stone in the left half of the Figure corresponds to a $45^\circ$ downward step and a white go-stone to a $45^\circ$ degree upward step when drawing the outline of a Young diagram. In the example shown the partition is $\lambda=(6,4,3,3,2,1)$ and the corresponding Maya sequence of charge $c$ is $a_\bullet=(c+5,c+2,c,c-1,c-3,c-5,c-6,c-8,c-9,\ldots)$, which are just the positions of the black go-stones on the left.
• Figure 3.2: The quiver $Q_{m,n}$ for the coordinate ring $\mathbb{C}[\mathrm{Gr}_{m,n}]$ of the (finite) Grassmannian $\mathrm{Gr}_{m,n}$. Here $n=4$ and $m=5$. The displayed sets of integers are the corresponding labels for the Plücker coordinates and the blue Young diagrams correspond to the frozen vertices.
• Figure 3.3: The rectangle-quiver $Q_{m,n}$ for the finite Grassmannian from Figure \ref{['fig:finite_quiver']} corresponds to the Postnikov diagram shown here.
• Figure 3.4: The (infinite) quiver $Q_\infty$ for the ind-cluster algebra $\mathbb{C}[\mathrm{Gr}]$. The vertex in the $i$th row and $j$th column is labelled by the rectangular partition of width $i$ and height $j$.
• Figure 3.5: The displayed double crossing of two paths, here labelled $i$ and $j$, is not allowed in a Postnikov diagram.

Theorems & Definitions (92)

• Definition 1.1: Sato sato1981solitonsato1983soliton, Segal-Wilson segal1985loop, Pressley-Segal pressley1985loop
• Remark 1.2
• Theorem 1.3: Theorem \ref{['T:filteredclosed']}
• Corollary 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Definition 1.9
• Corollary 1.10