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arXiv · 2608.27870

On cluster deep loci in double Bruhat cells

Hanwen Quan
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Abstract

We study deep loci for the reduced-word Berenstein-Fomin-Zelevinsky cluster atlas on type-A double Bruhat cells. As defined in CGSS24, for a seed collection on a cluster variety X, the deep locus is the complement of the union of all corresponding cluster tori. We focus on double Bruhat cells G(u,v) for G = SL(n). Our main results are several theorems describing the structure of these deep loci, including their interaction with the Poisson structure, the relationships between the deep loci of different cells, and a tracking formula that allows us to compute the reduced-word deep locus of a cell from a lower-dimensional one. The paper also includes explicit low-rank calculations for double Bruhat cells in SL(3) and Borel double Bruhat cells in SL(4). Finally, we discuss the relationship between these results and some general properties of deep loci.

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Hanwen Quan. 2026-08-28. On cluster deep loci in double Bruhat cells. https://arxiv.org/abs/2608.27870

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