A Large Dihedral Symmetry of the Set of Alternating Sign Matrices

2000/06/30 by Benjamin Wieland, Wieland, Benjamin
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CO #math.MP

paper · pdf · doi:10.48550/arxiv.math/0006234

13 pages, 6 figures

openalex publication_date 2000/06/30 · arxiv created 2000/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs). We examine data arising from the representation of an ASM as a collection of paths connecting 2n vertices and show it to be invariant under the dihedral group D_2n rearranging those vertices, which is much bigger than the group of symmetries of the square. We also generalize conjectures of Propp and Wilson relating some of this data for different values of n.

Related

• Symmetry Classes of Alternating Sign Matrices
• A poset Φ_n whose maximal chains are in bijection with the n × n alternating sign matrices
• Refined Enumeration of Symmetry Classes of Alternating Sign Matrices
• 1/N phenomenon for some symmetry classes of the odd alternating sign matrices
• The many faces of alternating-sign matrices