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. Contents 1. Introduction 2.1. Partition algebras 2.2. Cellular algebras 2.3. Cellular partition theory 3. Cellular subalgebras 3.1. Half integer partition algebra 3.2. Quasi-partition algebra 3.3. Complex reflection group centralizer 3.4. Brauer algebra 3.5. Rook Brauer algebra 3.6. Rook algebra 4. Planar algebras 4.1. Temperley-Lieb and planar partition algebras 4.2. Planar uniform block algebra 4.3. Planar rook algebra 4.4. Motzkin algebra 4.5. Partial Temperley-Lieb algebra 4.6. Planar even algebra 4.7. Planar r-color algebra 4.8. Planar quasi-partition algebra 5. Alternative perspectives and generalizations 5.1. Blob algebra 5.2. Other Schur-Weyl duality algebras 5.3. Diagram algebras as categories 6. Wreath products References