Projects per year
Abstract
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 nonzero 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 Bmodule with weight λ. The main new input is the description of module extensions by: extensions sequences, coherent triples of extension sequences and coherent multisequences 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.
Original language  English 

Pages (fromto)  618701 
Number of pages  84 
Journal  Advances in Mathematics 
Volume  345 
Early online date  17 Jan 2019 
DOIs  
Publication status  Published  17 Mar 2019 
Bibliographical note
© 2019 Published by Elsevier Inc. This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy.Keywords
 Cohomology
 Extensions
 General linear groups
 Induced modules
 Specht modules
 Symmetric groups
Profiles
Projects
 1 Finished

New perspectives on Buildings, Geometric Invariant Theory and Algebraic Groups
1/02/14 → 31/01/17
Project: Research project (funded) › Research