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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 |
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Vol. 16 (2023), No. 2, 201–226
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Abstract |
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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. |
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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
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Mathematical Subject Classification
Primary: 05E16
Secondary: 20F55, 17B10
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Milestones
Received: 1 January 2021
Revised: 7 May 2022
Accepted: 13 May 2022
Published: 26 May 2023
Communicated by Jim Haglund
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Authors |
| Robert G. Donnelly | |
| Department of Mathematics and
Statistics Murray State University Murray, KY United States |
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| Molly W. Dunkum | |
| Department of Mathematics Western Kentucky University Bowling Green, KY United States |
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| Sasha V. Lišková | |
| Department of Mathematics Western Kentucky University Bowling Green, KY United States |
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| Alexandra Nance | |
| Department of Mathematics and
Statistics Murray State University Murray, KY United States |
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| © 2023 MSP (Mathematical Sciences Publishers). |
