Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The homology of the partition algebras

Rachael Boyd, Richard Hepworth, Peter Patzt

Abstract

We show that the homology of the partition algebras, interpreted as appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range of degrees that increases with the number of nodes. Furthermore, we show that when the defining parameter $δ$ of the partition algebra is invertible, the homology of the partition algebra is in fact isomorphic to the homology of the symmetric group in all degrees. These results parallel those obtained for the Brauer algebras in the authors' earlier work, but with significant differences and difficulties in the inductive resolution and high acyclicity arguments required to prove them. Our results join the growing literature on homological stability for algebras, which now encompasses the Temperley-Lieb, Brauer and partition algebras, as well as the Iwahori-Hecke algebras of types A and B.

The homology of the partition algebras

Abstract

We show that the homology of the partition algebras, interpreted as appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range of degrees that increases with the number of nodes. Furthermore, we show that when the defining parameter of the partition algebra is invertible, the homology of the partition algebra is in fact isomorphic to the homology of the symmetric group in all degrees. These results parallel those obtained for the Brauer algebras in the authors' earlier work, but with significant differences and difficulties in the inductive resolution and high acyclicity arguments required to prove them. Our results join the growing literature on homological stability for algebras, which now encompasses the Temperley-Lieb, Brauer and partition algebras, as well as the Iwahori-Hecke algebras of types A and B.
Paper Structure (13 sections, 19 theorems, 47 equations, 9 figures)

This paper contains 13 sections, 19 theorems, 47 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Outline, and comparison to previous work
  3. Acknowledgements
  4. Partition algebras
  5. The principle of inductive resolutions
  6. Inductive resolutions
  7. Reducing to $\mathcal{A}_{X,x}$ and $\mathcal{B}_{X,x}$
  8. Resolving $\mathcal{A}_{X,x}$
  9. Resolving $\mathcal{B}_{X,x}$
  10. Proof of Theorem \ref{['theorem-inductive']}
  11. Replacing Shapiro's Lemma
  12. High connectivity
  13. Proof of \ref{['thmb']}

Key Result

Theorem A

Suppose that $\delta$ is invertible in $R$. Then the homology of the partition algebra is isomorphic to the homology of the symmetric group: Indeed, the inclusion and projection maps induce inverse isomorphisms

Figures (9)

  • Figure 1: Visualization of the partition $\{\{-5,-3\},\{-4,-2,-1,3,4\},\{1,5\},\{2\}\}$
  • Figure 2: Multiplication in the partition algebra
  • Figure 3: The elements $S_2,V_{13},T_3\in\operatorname{P}_4$
  • Figure 4: The module structure of $\operatorname{P}_5\otimes_{\operatorname{P}_3}\mathbbm{1}$
  • Figure 5: The resolution $C(X,x,y)\to\mathcal{A}_{X,x}$
  • ...and 4 more figures

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Corollary C
  • Definition 1: The partition algebra Jones_PottsMartin_JKTR
  • Definition 2: The trivial module $\mathbbm{1}$
  • Definition 3
  • Proposition 1: PatztRepstab
  • Example 1
  • Theorem 1
  • proof
  • ...and 37 more