Cellular subalgebras of the partition algebra
Travis Scrimshaw
TL;DR
This paper explores the cellular structure of various diagram algebras, providing new constructions and parameterizations of their modules, enhancing understanding of their representation theory.
Contribution
It introduces the cellular wreath product, a novel construction for building new cellular algebras from existing ones and subalgebras.
Findings
All surveyed diagram algebras are cellular.
Cell modules are explicitly parameterized.
New cellular algebra constructions are provided.
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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Algebra and Geometry · Algebraic structures and combinatorial models