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...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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2012
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| _version_ | 1797095237057773568 |
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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 |
| record_format | dspace |
| 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 |