Half-Diagrams in Partition Algebras: A Geometric Perspective on Multiplicities

Abstract

This paper studies the restriction multiplicities of half-diagram modules for the partition algebra and their geometric interpretations. By specializing the Bowman-De Visscher-Orellana formula [BVC, Theorem 4.3] for restriction multiplicities of standard modules in the partition algebra, we compute these multiplicities and provide interpretations in terms of planar triangles and conic sections. Additionally, through the decomposition of half-diagrams, we explain the intrinsic reasons underlying this connection between geometry and algebra.

Comments are welcome. The authors wish to express their sincere gratitude to colleagues for their constructive comments during the development of this manuscript. Their valuable suggestions have significantly enhanced the quality of this work