Geometry of positive configurations in affine buildings.

*(English)*Zbl 1383.51009Summary: Positive configurations of points in the affine building were introduced in [the author, Geom. Topol. 20, No. 3, 1673–1735 (2016; Zbl 1348.30023)] as the basic object needed to define higher laminations. We start by giving a self-contained, elementary definition of positive configurations of points in the affine building and their basic properties. Then we study the geometry of these configurations. The canonical functions on triples of flags that were defined by V. Fock and A. Goncharov [Publ. Math., Inst. Hautes Étud. Sci. 103, 1–211 (2006; Zbl 1099.14025)] have a tropicalization that gives functions on triples of points in the affine Grassmannian. One expects that these functions, though of algebro-geometric origin, have a simple description in terms of the metric structure on the corresponding affine building.

We give a several conjectures describing the tropicalized canonical functions in terms of the geometry of affine buildings, and give proofs of some of them. The statements involve minimal networks and have some resemblance to the max-flow/min-cut theorem, which also plays a role in the proofs in unexpected ways. The conjectures can be reduced to purely algebraic statements about valuations of lattices that we argue are interesting in their own right.

One can view these conjectures as the first examples of intersection pairings between higher laminations. They fit within the framework of the Duality Conjectures of [Fock and Goncharov, loc. cit.].

We give a several conjectures describing the tropicalized canonical functions in terms of the geometry of affine buildings, and give proofs of some of them. The statements involve minimal networks and have some resemblance to the max-flow/min-cut theorem, which also plays a role in the proofs in unexpected ways. The conjectures can be reduced to purely algebraic statements about valuations of lattices that we argue are interesting in their own right.

One can view these conjectures as the first examples of intersection pairings between higher laminations. They fit within the framework of the Duality Conjectures of [Fock and Goncharov, loc. cit.].

##### MSC:

51E24 | Buildings and the geometry of diagrams |

14T05 | Tropical geometry (MSC2010) |

05E10 | Combinatorial aspects of representation theory |

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\textit{I. Le} and \textit{E. O'Dorney}, Doc. Math. 22, 1519--1538 (2017; Zbl 1383.51009)

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##### References:

[2] | V.V. Fock, A.B. Goncharov. Moduli spaces of local systems and higher Teichmuller theory. Publ. Math. IHES, n. 103 (2006) 1-212. math.AG/0311149 · Zbl 1099.14025 |

[3] | A.B. Goncharov, L. Shen. Geometry of canonical bases and mirror symmetry. arXiv:1309.5922 · Zbl 1355.14030 |

[4] | J. Kamnitzer. Hives and the fibres of the convolution morphism, Selecta Math. N.S. 13 no. 3 (2007), 483-496 · Zbl 1189.20040 |

[5] | I. Le. Higher Laminations and Affine Buildings. arXiv:1209.0812 · Zbl 1348.30023 |

[6] | A.G. Moshonkin. Concerning Hall’s Theorem, from Mathematics in St. Petersburg, eds. A. A. Bolibruch, A.S. Merkur’ev, N. Yu. Netsvetaev. Amererican Mathematical Society Translations, Series 2, Volume 174, 1996 · Zbl 0852.05012 |

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