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Cohomology of coloured partition algebras

James Cranch, Daniel Graves

Abstract

Coloured partition algebras were introduced by Bloss and exhibit a Schur-Weyl duality with certain complex reflection groups. In this paper we show that these algebras exhibit homological stability by demonstrating that their homology groups are stably isomorphic to the homology groups of a wreath product, generalizing work of Boyd--Hepworth--Patzt and Boyde for the usual partition algebras.

Cohomology of coloured partition algebras

Abstract

Coloured partition algebras were introduced by Bloss and exhibit a Schur-Weyl duality with certain complex reflection groups. In this paper we show that these algebras exhibit homological stability by demonstrating that their homology groups are stably isomorphic to the homology groups of a wreath product, generalizing work of Boyd--Hepworth--Patzt and Boyde for the usual partition algebras.

Paper Structure

This paper contains 14 sections, 17 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Conventions
  3. Coloured partition algebras
  4. Partitions and cospans
  5. Conventions for partition diagrams
  6. Partition category
  7. Coloured partitions
  8. Functoriality
  9. Conventions for coloured partition diagrams
  10. Defining the augmentation
  11. Proving Theorem \ref{['thm-A']}
  12. Idempotent covers and the Mayer-Vietoris complex
  13. A free idempotent left cover
  14. Extending the range

Key Result

Theorem 1.1

For $n\geqslant 2$, there exist natural isomorphisms of $k$-modules for $q\leqslant n$. These isomorphisms also hold for $n=1$ with $q=0$. Furthermore, if $\delta$ is invertible, these isomorphisms hold for all $n\geqslant 1$ and $q\geqslant 0$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 42 more