Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras

TL;DR

This paper introduces symmetric Fibonaccian distributive lattices linked to Fibonacci sequences, unifying various lattice descriptions and realizing representations of special linear Lie algebras with potential extremal properties.

Contribution

It presents a new family of lattices related to Fibonacci numbers that realize Lie algebra representations and unify existing lattice structures in the literature.

Findings

Lattices naturally realize certain Lie algebra representations.

New formulas for lattice cardinalities and rank generating functions.

Connections established with OEIS integer sequences.

Abstract

We present a family of rank symmetric diamond-colored distributive lattices that are naturally related to the Fibonacci sequence and certain of its generalizations. These lattices re-interpret and unify descriptions of some un- or differently-colored lattices found variously in the literature. We demonstrate that our symmetric Fibonaccian lattices naturally realize certain (often reducible) representations of the special linear Lie algebras, with weight basis vectors realized as lattice elements and Lie algebra generators acting along the covering digraph edges of each lattice. We present evidence that each such weight basis possesses certain distinctive extremal properties. We provide new descriptions of the lattice cardinalities and rank generating functions and offer several conjectures/open problems. Throughout, we make connections with integer sequences from the OEIS.