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

Pei Wang, Changjing Zhuge

TL;DR

This work analyzes restriction multiplicities for half-diagram modules in the partition algebra by specializing the Bowman–De Visscher–Orellana formula to one-part partitions, reducing the problem to a Diophantine system whose solutions count the multiplicities. It provides a diagrammatic inflation and walled-half-diagram decomposition that explains why each solution contributes exactly 1, and translates the counting problem into geometric terms involving triangles, lattice points, and conic sections. The results yield explicit, geometry-laden expressions for the multiplicities E_{p,q}^r, and extend to Temperley–Lieb algebras, where the Grothendieck ring structure constants are governed by triangle inequalities. The approach bridges representation theory with classical geometry, offering a concrete, computable framework for understanding partition-algebra restriction phenomena and their TL-subalgebra counterparts.

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.

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

TL;DR

This work analyzes restriction multiplicities for half-diagram modules in the partition algebra by specializing the Bowman–De Visscher–Orellana formula to one-part partitions, reducing the problem to a Diophantine system whose solutions count the multiplicities. It provides a diagrammatic inflation and walled-half-diagram decomposition that explains why each solution contributes exactly 1, and translates the counting problem into geometric terms involving triangles, lattice points, and conic sections. The results yield explicit, geometry-laden expressions for the multiplicities E_{p,q}^r, and extend to Temperley–Lieb algebras, where the Grothendieck ring structure constants are governed by triangle inequalities. The approach bridges representation theory with classical geometry, offering a concrete, computable framework for understanding partition-algebra restriction phenomena and their TL-subalgebra counterparts.

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.

Paper Structure

This paper contains 18 sections, 14 theorems, 63 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Partition algebra
  3. Definitions
  4. Half-Diagrams and Inflation
  5. Standard Modules
  6. Multiplicities
  7. The Restriction of a standard module
  8. L-R and Kronecker Coefficients for One-Part Partitions
  9. Diophantine equation
  10. Walled Diagrams and Parameterization
  11. Diagram Decomposition via Inflation
  12. Multiplicity Reduction
  13. Geometric interpretation
  14. Geometric problem
  15. Lattice Point Enumeration
  16. ...and 3 more sections

Key Result

Proposition 2.1

Keep the notations as above. Let $M$ and $N$ be $\mathbb{C}\mathfrak{S}_{r}$-modules, and let $V(n, r)$ be the module of all $(n, r)$-diagrams having precisely $r$ propagating blocks of $P_{n}$. Then

Figures (21)

  • Figure 1: Partition diagrams
  • Figure 2: Multiplication of diagrams
  • Figure 3: Generators of $P_n$
  • Figure 4: Diagram decomposition
  • Figure 5: Action on half-diagram
  • ...and 16 more figures

Theorems & Definitions (20)

  • Proposition 2.1
  • proof
  • Theorem 3.1: Littlewood–Richardson Rule
  • Theorem 3.2: BVC, Theorem 4.3
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 10 more