Finite diamond-colored modular and distributive lattices with\n applications to combinatorial Lie representation theory

2018/12/03 by Robert G. Donnelly, Donnelly, Robert G. · 1 citation
Mathematics · #06C99 #06D99 (Secondary) #17B10 (Primary) 06A07 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1812.04434

openalex publication_date 2018/12/03 · openalex created_date 2022/08/01 · openalex updated_date 2026/07/28

Abstract

A modular or distributive lattice is `diamond-colored' if its order diagram\nedges are colored in such a way that, within any diamond of edges, parallel\nedges have the same color. Such lattices arise naturally in combinatorial\nrepresentation theory, particularly in the study of poset models for semisimple\nLie algebra representations and their companion Weyl group symmetric functions.\nOne of our goals is to gather in one place some elementary but foundational\nresults concerning these lattice structures; this includes some new results as\nwell as some new interpretations of classical results. We then develop many\npoints of contact between diamond-colored modular/distributive lattices and\ncombinatorial Lie representation theory, leading to some new Dynkin diagram\nclassification results and some new results concerning minuscule and\nquasi-minuscule representations.\n

Citations

• The Partial Order of Dominant Weights
• The numbers game and Dynkin diagram classification results
• Poset models for Weyl group analogs of symmetric functions and Schur functions

Cited by

• Analogs of Schur functions for rank two Weyl groups obtained from\n grid-like posets

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