Abstract
Let be a field of prime characteristic and let be a nonprincipal block of the group algebra of the symmetric group . The block component of the Lie module is projective, by a result of Erdmann and Tan, although itself is projective only when . Write , where , and let be the diagonal of a Young subgroup of isomorphic to . We show that . Hence we obtain a formula for the multiplicities of the projective indecomposable modules in a direct sum decomposition of . Corresponding results are obtained, when is infinite, for the -th Lie power of the natural module for the general linear group .
Citation
Roger Bryant. Karin Erdmann. "Block components of the Lie module for the symmetric group." Algebra Number Theory 6 (4) 781 - 795, 2012. https://doi.org/10.2140/ant.2012.6.781
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