## Abstract

We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is nonzero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree one Hochschild cohomology as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn is a consequence of a general formula that makes more precise a theorem of our previous paper describing the generating functions for the dimensions of Hochschild cohomology of symmetric groups.

Original language | English |
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Article number | 20 |

Number of pages | 5 |

Journal | Research in Mathematical Sciences |

Volume | 10 |

Issue number | 2 |

DOIs | |

Publication status | Published - 5 May 2023 |

## Keywords

- Blocks
- Hochschild cohomology
- Partition identities
- Symmetric groups