# Connecting Permutation Equivariant Neural Networks and Partition Diagrams

arXiv: https://arxiv.org/abs/2212.08648 · HTML: https://mlsanity.com/paper/2212.08648/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 44 references that are in the map.
- First posted on arXiv 16 Dec 2022.

## Builds on (most cited first, 6 of 6)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [The general theory of permutation equivarant neural networks and higher order graph variational encoders](https://mlsanity.com/paper/2004.03990/) (2020)
- [What is an equivariant neural network?](https://mlsanity.com/paper/2205.07362/) (2022)

## Built on by (most cited first, 9 of 9)

- [Fast computation of permutation equivariant layers with the partition algebra](https://mlsanity.com/paper/2303.06208/) (2023)
- [How Jellyfish Characterise Alternating Group Equivariant Neural Networks](https://mlsanity.com/paper/2301.10152/) (2023)
- [Graph Automorphism Group Equivariant Neural Networks](https://mlsanity.com/paper/2307.07810/) (2023)
- [Categorification of Group Equivariant Neural Networks](https://mlsanity.com/paper/2304.14144/) (2023)
- [Separation Power of Equivariant Neural Networks](https://mlsanity.com/paper/2406.08966/) (2024)
- [Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations](https://mlsanity.com/paper/2410.06665/) (2024)
- [An Algorithm for Computing with Brauer's Group Equivariant Neural Network Layers](https://mlsanity.com/paper/2304.14165/) (2023)
- [Compact Matrix Quantum Group Equivariant Neural Networks](https://mlsanity.com/paper/2311.06358/) (2023)
- [Permutation Equivariant Neural Networks for Symmetric Tensors](https://mlsanity.com/paper/2503.11276/) (2025)

## Abstract (arXiv)

Permutation equivariant neural networks are often constructed using tensor powers of ℝn as their layer spaces. We show that all of the weight matrices that appear in these neural networks can be obtained from Schur-Weyl duality between the symmetric group and the partition algebra. In particular, we adapt Schur-Weyl duality to derive a simple, diagrammatic method for calculating the weight matrices themselves.