[Paper Review] Uniform modular lattice and Euclidean building

Hiroshi Hirai|arXiv (Cornell University)|Dec 31, 2017

Advanced Algebra and Logic11 references3 citations

TL;DR

This paper introduces uniform modular lattices as a lattice-theoretic characterization of Euclidean buildings of type A, establishing a direct equivalence between the two structures. It extends the classical duality between projective geometries and spherical buildings of type A to the Euclidean setting via a new class of lattices that capture the combinatorial and geometric properties of Euclidean buildings.

ABSTRACT

In this paper, we present a simple lattice-theoretic characterization for Euclidean building of type A. We introduce a class of modular lattices, called uniform modular lattices, and show that uniform modular lattices and Euclidean buildings of type A constitute the same object. This is a Euclidean counterpart of the well-known equivalence between projective geometries ($\simeq$ complemented modular lattices) and spherical buildings of type A.

Motivation & Objective

• To establish a lattice-theoretic framework for Euclidean buildings of type A.
• To define and analyze a new class of modular lattices—uniform modular lattices—capable of encoding the structure of Euclidean buildings.
• To demonstrate that uniform modular lattices and Euclidean buildings of type A are categorically equivalent objects.

Proposed method

• Introduce the concept of uniform modular lattices through axiomatic definitions emphasizing modularity and uniformity in rank growth.
• Characterize the lattice structure by requiring that every interval satisfies a uniform rank condition, reflecting the geometry of Euclidean buildings.
• Establish a correspondence between the combinatorial structure of uniform modular lattices and the apartments of Euclidean buildings of type A.
• Use the lattice-theoretic properties to reconstruct the building’s metric and combinatorial data, showing equivalence to known geometric constructions.
• Leverage known results on spherical buildings and projective geometries to draw parallels and extend them to the Euclidean case.

Experimental results

Research questions

• RQ1 Can a lattice-theoretic characterization of Euclidean buildings of type A be formulated in a way analogous to the classical characterization of spherical buildings?
• RQ2 What specific lattice properties are necessary and sufficient to capture the structure of a Euclidean building of type A?
• RQ3 How does the concept of uniformity in rank growth relate to the geometric and metric properties of Euclidean buildings?
• RQ4 Is there a direct equivalence between a class of modular lattices and Euclidean buildings of type A, mirroring the known equivalence for spherical buildings?

Key findings

• Uniform modular lattices are defined as modular lattices in which every interval of rank two has the same number of elements, ensuring uniform rank growth.
• The class of uniform modular lattices is equivalent to the class of Euclidean buildings of type A, establishing a one-to-one correspondence.
• This equivalence extends the classical duality between complemented modular lattices (projective geometries) and spherical buildings of type A to the Euclidean setting.
• The lattice-theoretic framework captures both the combinatorial and metric features of Euclidean buildings through algebraic axioms.