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On the decomposition numbers of the finite general linear groups


Author: Richard Dipper
Journal: Trans. Amer. Math. Soc. 290 (1985), 315-344
MSC: Primary 20G40; Secondary 20C20
DOI: https://doi.org/10.1090/S0002-9947-1985-0787968-5
MathSciNet review: 787968
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Abstract: Let $ G = {\text{GL}_n}(q)$, $ q$ a prime power, and let $ r$ be an odd prime not dividing $ q$. Let $ s$ be a semisimple element of $ G$ of order prime to $ r$ and assume that $ r$ divides. $ {q^{\deg (\Lambda )}} - 1$ for all elementary divisors $ \Lambda $ of $ s$. Relating representations of certain Hecke algebras over symmetric groups with those of $ G$, we derive a full classification of all modular irreducible modules in the $ r$-block $ {B_s}$ of $ G$ with semisimple part $ s$. The decomposition matrix $ D$ of $ {B_s}$ may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover $ D$ is lower unitriangular. This applies in particular to all $ r$-blocks of $ G$ if $ r$ divides $ q - 1$. Thus, in this case, the $ r$-decomposition matrix of $ G$ is lower unitriangular.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0787968-5
Article copyright: © Copyright 1985 American Mathematical Society

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