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

Robert G. Donnelly, Molly W. Dunkum, Sasha V. Lišková and Alexandra Nance

Vol. 16 (2023), No. 2, 201–226
Abstract

We present a family of rank-symmetric diamond-colored distributive lattices that are naturally related to the Fibonacci sequence and some of its generalizations. These lattices reinterpret and unify descriptions of some uncolored 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 and open problems. Throughout, we make connections with integer sequences from the OEIS.

Keywords
diamond-colored distributive lattice, rank-generating function, skew-shaped semistandard tableau, skew Schur function, skew-tabular lattice, special linear Lie algebra representation, weight basis supporting graph/representation diagram
Mathematical Subject Classification
Primary: 05E16
Secondary: 20F55, 17B10
Milestones
Received: 1 January 2021
Revised: 7 May 2022
Accepted: 13 May 2022
Published: 26 May 2023

Communicated by Jim Haglund
Authors
Robert G. Donnelly
Department of Mathematics and Statistics
Murray State University
Murray, KY
United States
Molly W. Dunkum
Department of Mathematics
Western Kentucky University
Bowling Green, KY
United States
Sasha V. Lišková
Department of Mathematics
Western Kentucky University
Bowling Green, KY
United States
Alexandra Nance
Department of Mathematics and Statistics
Murray State University
Murray, KY
United States