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Cluster structures on schemes of bands

Luca Francone, Bernard Leclerc

TL;DR

This work constructs spaces of $(G,c)$-bands $B(G,c)$ as infinite-dimensional affine pro-varieties whose coordinate rings $R(G,c)$ carry natural cluster algebra structures. It proves that the cluster algebra of $R(G,c)$ is governed by an isomorphism with the infinite-rank cluster algebra $\,\\mathcal{A}$ from representation theory, via initial seeds mapped to families of generalized minors translated by the Coxeter element, and it identifies meaningful cluster subalgebras given by $R(G,c)^U$ and $R(G,c)^G$. The authors develop a detailed finite-band reduction, glueing formulas, and a Starfish-type argument to control mutations, and they connect the geometry to the representation theory of quantum affine algebras, including $O^{+}_{\mathbb{Z}}$, ${\mathcal{C}}^{\mathrm{shift}}_{\mathbb{Z}}$, and ${\mathcal{C}}_{\mathbb{Z}}$, via a precise dictionary between cluster variables and (renormalized) $Q$-characters, prefundamental modules, and Kirillov–Reshetikhin modules. This yields both a geometric realization of known infinite-rank cluster structures and a unified framework linking cluster algebras with the representation theory of quantum affine algebras and shifted variants. The results offer a robust toolkit for translating between geometric objects $B(G,c)$ and categorical Grothendieck rings, enabling new perspectives on cluster dynamics in infinite-rank settings.

Abstract

We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared in different contexts related to the representation theories of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. In this paper we show that they all belong to a common geometric setting.

Cluster structures on schemes of bands

TL;DR

This work constructs spaces of -bands as infinite-dimensional affine pro-varieties whose coordinate rings carry natural cluster algebra structures. It proves that the cluster algebra of is governed by an isomorphism with the infinite-rank cluster algebra from representation theory, via initial seeds mapped to families of generalized minors translated by the Coxeter element, and it identifies meaningful cluster subalgebras given by and . The authors develop a detailed finite-band reduction, glueing formulas, and a Starfish-type argument to control mutations, and they connect the geometry to the representation theory of quantum affine algebras, including , , and , via a precise dictionary between cluster variables and (renormalized) -characters, prefundamental modules, and Kirillov–Reshetikhin modules. This yields both a geometric realization of known infinite-rank cluster structures and a unified framework linking cluster algebras with the representation theory of quantum affine algebras and shifted variants. The results offer a robust toolkit for translating between geometric objects and categorical Grothendieck rings, enabling new perspectives on cluster dynamics in infinite-rank settings.

Abstract

We introduce new objects, called -bands, associated with a simple simply-connected algebraic group , and a Coxeter element in its Weyl group. We show that bands of a given type are the -points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared in different contexts related to the representation theories of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. In this paper we show that they all belong to a common geometric setting.

Paper Structure

This paper contains 37 sections, 48 theorems, 241 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setup
  3. Algebraic groups
  4. Schemes
  5. The case of $SL(n)$ and $c_{st}$
  6. $(SL(n), c_{st})$-bands and their cluster structure
  7. Finite bands
  8. $U$-invariants
  9. $G$-invariants
  10. Bands
  11. Limits and pro-varieties
  12. The affine case and infinite products
  13. Finite $(G,c)$-bands
  14. $(G,c)$-bands
  15. The case of $SL(n)$ and $c_{st}$ revisited
  16. ...and 22 more sections

Key Result

Theorem 1.2

The $(G,c)$-bands over $\mathbb{C}$ are the $\mathbb{C}$-rational points of an infinite-dimensional affine integral scheme $B(G,c)$. The ring $R(G,c)$ of regular functions on $B(G,c)$ is a unique factorization domain.

Figures (11)

  • Figure 1: A finite segment of an initial seed of the cluster algebra structure on $\mathcal{R}_3$
  • Figure 2: The iced-quivers $\Gamma_{c_{st}, -2,1}$ and $\Gamma_{c_{st}, -1,1}$ for $n=3$
  • Figure 3: A finite segment of an initial seed of the cluster algebra structure on $\mathcal{R}_3^U$
  • Figure 4: A finite segment of an initial seed of the cluster algebra structure on $\mathcal{R}_3^G$
  • Figure 5: The quiver $\Gamma_{\widetilde{c}}$ for $\widetilde{c} = s_2s_1s_3$ in type $A_3$, and the corresponding initial seed of $R(G, \widetilde{c})$.
  • ...and 6 more figures

Theorems & Definitions (110)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 3.1
  • Example 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • ...and 100 more