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

TL;DR

This paper explores the restriction multiplicities of half-diagram modules in partition algebras, offering geometric interpretations involving planar triangles and conic sections, and providing new insights into their algebraic structure.

Contribution

It specializes a known formula to compute restriction multiplicities and links these algebraic quantities to geometric figures, revealing intrinsic geometric-algebraic connections.

Findings

Computed restriction multiplicities using specialized formulas

Provided geometric interpretations involving triangles and conic sections

Explained the intrinsic geometric reasons behind algebraic 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.