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Cellular subalgebras of the partition algebra

Travis Scrimshaw

Abstract

We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram algebras surveyed here are all cellular algebras and parameterize their cell modules. We give a new construction to build new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra that we call the cellular wreath product.

Cellular subalgebras of the partition algebra

Abstract

We describe various diagram algebras and their representation theory using cellular algebras of Graham and Lehrer and the decomposition into half diagrams. In particular, we show the diagram algebras surveyed here are all cellular algebras and parameterize their cell modules. We give a new construction to build new cellular algebras from a general cellular algebra and subalgebras of the rook Brauer algebra that we call the cellular wreath product.
Paper Structure (30 sections, 29 theorems, 88 equations, 4 tables)

This paper contains 30 sections, 29 theorems, 88 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Partition algebras
  4. Cellular algebras
  5. Cellular partition theory
  6. Cellular subalgebras
  7. Half integer partition algebra
  8. Quasi-partition algebra
  9. Complex reflection group centralizer
  10. Uniform block algebra
  11. Parity matching algebra
  12. Colored permutation symmetrizer
  13. Brauer algebra
  14. Rook Brauer algebra
  15. Rook algebra
  16. ...and 15 more sections

Key Result

Theorem 2.4

There exists a surjection $\mathcal{P}_n(m) \to \mathop{\mathrm{End}}\nolimits_{\Sigma_m} \mathbf{V}^{\otimes n}$. Furthermore, this map is a bijection if and only if $m \geq 2n$.

Theorems & Definitions (62)

  • Example 2.1
  • Remark 2.2
  • Example 2.3
  • Theorem 2.4: Jones94
  • Remark 2.5
  • Definition 2.6: Cellular algebra GL96
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • Theorem 2.9: GL96
  • ...and 52 more