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  "body": "\n## Summary\n\nA cellular algebra is an [[associative algebra|3CTW1P4HBP]] with a special basis, the cellular basis.\nThis basis is compatible with an anti-involution and with a partial order on its labels.\nThe notion was introduced by Graham and Lehrer.\nThe basis produces cell modules, which play the role of the Specht modules of the symmetric group.\nThe simple modules come out of the cell modules in a uniform way.\n[[Hecke algebras|5J2P2AGY4N]] of type $A$, [[Temperley–Lieb algebras|TZNR514JQN]] and Brauer algebras are cellular.\n\n## Explanations\n\n### L1 human\n\nMany algebras that come from combinatorics have a basis indexed by pairs of objects of the same shape, such as pairs of tableaux.\nIn a cellular algebra, multiplying a basis element on the left only changes the first object of the pair, up to terms of smaller shape.\nA reversing map swaps the two objects of each pair.\nThe 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.\nThis structure lets one build all the simple modules from a single recipe.\n",
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