---
tag: 76DF4G7J9S
kind: concept
name: Cellular algebra
msc: [16G10, 20C08]
specializes: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Cellular_algebra
same_as:
  wikidata: Q5058347
  wikipedia: Cellular algebra
---

## Summary

A cellular algebra is an [[associative algebra|3CTW1P4HBP]] with a special basis, the cellular basis.
This basis is compatible with an anti-involution and with a partial order on its labels.
The notion was introduced by Graham and Lehrer.
The basis produces cell modules, which play the role of the Specht modules of the symmetric group.
The simple modules come out of the cell modules in a uniform way.
[[Hecke algebras|5J2P2AGY4N]] of type $A$, [[Temperley–Lieb algebras|TZNR514JQN]] and Brauer algebras are cellular.

## Explanations

### L1 human

Many algebras that come from combinatorics have a basis indexed by pairs of objects of the same shape, such as pairs of tableaux.
In a cellular algebra, multiplying a basis element on the left only changes the first object of the pair, up to terms of smaller shape.
A reversing map swaps the two objects of each pair.
The model is the matrix algebra: the unit matrix $E_{ij}$ is labelled by the pair $(i, j)$, left multiplication mixes the first index, and transposition swaps the indices.
This structure lets one build all the simple modules from a single recipe.