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Cluster algebras III: Upper bounds and double Bruhat cells

Arkady Berenstein, Sergey Fomin, Andrei Zelevinsky

TL;DR

This work develops an upper-bound framework for cluster algebras via upper cluster algebras defined as intersections of Laurent rings, proves that acyclic, coprime seeds force equality of lower, upper, and cluster algebras, and establishes finite generation in these cases. It then applies this framework to double Bruhat cells, showing their coordinate rings are naturally isomorphic to explicitly defined upper cluster algebras built from combinatorial data tied to Weyl group elements. The paper provides detailed proofs of mutation-invariance, rank-2 reductions, and valuation-based separations, and classifies certain finite-type scenarios in rank 3. Overall, it connects cluster algebra structures to the geometry of double Bruhat cells, giving concrete algebro-geometric realizations and structural criteria for finiteness and equality with upper bounds.

Abstract

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.

Cluster algebras III: Upper bounds and double Bruhat cells

TL;DR

This work develops an upper-bound framework for cluster algebras via upper cluster algebras defined as intersections of Laurent rings, proves that acyclic, coprime seeds force equality of lower, upper, and cluster algebras, and establishes finite generation in these cases. It then applies this framework to double Bruhat cells, showing their coordinate rings are naturally isomorphic to explicitly defined upper cluster algebras built from combinatorial data tied to Weyl group elements. The paper provides detailed proofs of mutation-invariance, rank-2 reductions, and valuation-based separations, and classifies certain finite-type scenarios in rank 3. Overall, it connects cluster algebra structures to the geometry of double Bruhat cells, giving concrete algebro-geometric realizations and structural criteria for finiteness and equality with upper bounds.

Abstract

We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.

Paper Structure

This paper contains 20 sections, 41 theorems, 134 equations, 7 figures, 1 table.

Table of Contents

  1. Upper and lower bounds
  2. The upper bounds
  3. Seed mutations and invariance of upper bounds
  4. Lower bounds and cluster algebras
  5. Acyclicity and closing of the gaps
  6. Open problems
  7. Cluster algebras associated with double Bruhat cells
  8. Double Bruhat cells
  9. Combinatorial data
  10. Generalized minors
  11. Cluster algebra structures in $\mathbb{C}[G^{u,v}]$
  12. Double Bruhat cells associated with Coxeter elements
  13. Double Bruhat cells associated with base affine spaces
  14. Open double Bruhat cells
  15. Proof of Proposition \ref{['pr:coprime']}
  16. ...and 5 more sections

Key Result

Theorem 1.5

Assume that two seeds $\Sigma$ and $\Sigma'$ are related by a seed mutation, and are both coprime. Then the corresponding upper bounds coincide: $\mathcal{U}(\Sigma) = \mathcal{U}(\Sigma')$.

Figures (7)

  • Figure 1: Graph $\tilde{\Gamma}(\mathbf{i})$ for $\mathbf{i}=(1,2,1,-1,-2,-1)$
  • Figure 2: Matrix $\tilde{B}(\mathbf{i})$ and its $\mathbf{e}(\mathbf{i})^-\!\times\mathbf{e}(\mathbf{i})$ submatrix for $\mathbf{i}=(1,2,1,-1,-2,-1)$
  • Figure 3: Minors $\Delta(k;\mathbf{i})$ for $\mathbf{i}=(1,2,1,-1,-2,-1)$
  • Figure 4: Transformation of the graph $\Gamma(B)$ for $\mathbb{C}[G^{w_\circ,w_\circ}]$ in type $A_2$.
  • Figure 5: Diagrams for $\mathbb{C}[G^{e,w_\circ}]$ in type $A_4$.
  • ...and 2 more figures

Theorems & Definitions (94)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Corollary 1.7
  • Proposition 1.8
  • Corollary 1.9
  • Definition 1.10
  • ...and 84 more