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arXiv · 2504.14012

Cluster structures on schemes of bands

Luca Francone·Bernard Leclerc
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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.

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Luca Francone, Bernard Leclerc. 2025-04-18. Cluster structures on schemes of bands. https://arxiv.org/abs/2504.14012

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