Block components of the Lie module for the symmetric group
Let F be a field of prime characteristic p and let B be a nonprincipal block of the group algebra F S r of the symmetric group S r. The block component Lie(r) B of the Lie module Lie(r) is projective, by a result of Erdmann a...
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Format: | Journal article |
Language: | English |
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2012
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author | Bryant, R Erdmann, K |
author_facet | Bryant, R Erdmann, K |
author_sort | Bryant, R |
collection | OXFORD |
description | Let F be a field of prime characteristic p and let B be a nonprincipal block of the group algebra F S r of the symmetric group S r. The block component Lie(r) B of the Lie module Lie(r) is projective, by a result of Erdmann and Tan, although Lie(r) itself is projective only when p {does not divide} r. Write r = p m k, where p {does not divide} k, and let S z.ast;k be the diagonal of a Young subgroup of S r isomorphic to S k × · · · × S k. We show that p m Lie(r) B = (Lie(k)↑ SrS z.ast;k)B. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of Lie(r) B. Corresponding results are obtained, when F is infinite, for the r -th Lie power L r (E) of the natural module E for the general linear group GL n (F). © 2012 by Mathematical Sciences Publishers. |
first_indexed | 2024-03-07T04:24:58Z |
format | Journal article |
id | oxford-uuid:cc4d999c-165b-4c5f-a03b-5dc9c29a2c6d |
institution | University of Oxford |
language | English |
last_indexed | 2024-03-07T04:24:58Z |
publishDate | 2012 |
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spelling | oxford-uuid:cc4d999c-165b-4c5f-a03b-5dc9c29a2c6d2022-03-27T07:20:59ZBlock components of the Lie module for the symmetric groupJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:cc4d999c-165b-4c5f-a03b-5dc9c29a2c6dEnglishSymplectic Elements at Oxford2012Bryant, RErdmann, KLet F be a field of prime characteristic p and let B be a nonprincipal block of the group algebra F S r of the symmetric group S r. The block component Lie(r) B of the Lie module Lie(r) is projective, by a result of Erdmann and Tan, although Lie(r) itself is projective only when p {does not divide} r. Write r = p m k, where p {does not divide} k, and let S z.ast;k be the diagonal of a Young subgroup of S r isomorphic to S k × · · · × S k. We show that p m Lie(r) B = (Lie(k)↑ SrS z.ast;k)B. Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of Lie(r) B. Corresponding results are obtained, when F is infinite, for the r -th Lie power L r (E) of the natural module E for the general linear group GL n (F). © 2012 by Mathematical Sciences Publishers. |
spellingShingle | Bryant, R Erdmann, K Block components of the Lie module for the symmetric group |
title | Block components of the Lie module for the symmetric group |
title_full | Block components of the Lie module for the symmetric group |
title_fullStr | Block components of the Lie module for the symmetric group |
title_full_unstemmed | Block components of the Lie module for the symmetric group |
title_short | Block components of the Lie module for the symmetric group |
title_sort | block components of the lie module for the symmetric group |
work_keys_str_mv | AT bryantr blockcomponentsoftheliemoduleforthesymmetricgroup AT erdmannk blockcomponentsoftheliemoduleforthesymmetricgroup |