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**First degree cohomology of Specht modules and extensions of symmetric powers**770 KB, PDF document

Journal | Advances in Mathematics |
---|---|

Date | Accepted/In press - 18 Dec 2018 |

Date | E-pub ahead of print - 17 Jan 2019 |

Date | Published (current) - 17 Mar 2019 |

Volume | 345 |

Number of pages | 84 |

Pages (from-to) | 618-701 |

Early online date | 17/01/19 |

Original language | English |

Let Σ
_{d} denote the symmetric group of degree d and let K be a field of positive characteristic p. For p>2 we give an explicit description of the first cohomology group H
^{1}(Σ
_{d},Sp(λ)), of the Specht module Sp(λ) over K, labelled by a partition λ of d. We also give a sufficient condition for the cohomology to be non-zero for p=2 and we find a lower bound for the dimension. The cohomology of Specht modules has been considered in many papers including [10], [12], [15] and [21]. Our method is to proceed by comparison with the cohomology for the general linear group G=GL
_{n}(K) and then to reduce to the calculation of Ext
_{B}
^{1}(S
^{d}E,K
_{λ}), where B is a Borel subgroup of G, where S
^{d}E denotes the dth symmetric power of the natural module E for G and K
_{λ} denotes the one dimensional B-module with weight λ. The main new input is the description of module extensions by: extensions sequences, coherent triples of extension sequences and coherent multi-sequences of extension sequences, and the detailed calculation of the possibilities for such sequences. These sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of G. Our methods are valid also in the quantised context and we aim to treat this in a separate paper.

© 2019 Published by Elsevier Inc. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.

- Cohomology, Extensions, General linear groups, Induced modules, Specht modules, Symmetric groups

## New perspectives on Buildings, Geometric Invariant Theory and Algebraic Groups

Project: Research project (funded) › Research

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