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On Specht modules of general linear groups. (English) Zbl 1037.20046
Let $$F$$ be a field of characteristic $$p$$ where $$p\nmid q$$ and let $$S^\lambda$$ be a Specht module for $$\text{GL}(\mathbb{F}_q)$$ over $$F$$ as in [G. D. James, Representations of general linear groups, Lond. Math. Soc. Lect. Note Ser. 94 (1984; Zbl 0541.20025)]. A nonzero vector in $$S^\lambda$$ is a linear combination of $$\lambda$$-flags, but not every linear combination occurs. The author shows in particular that if $$T$$ is the row standard tableaux defined by a term in the linear combination that is maximal according to a certain partial order, then $$T$$ must be a standard tableaux. This is analogous to a result for Specht modules of the symmetric group, but in contrast to that case one does not have a simple connection between standard tableaux and elements of some basis of $$S^\lambda$$.

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields 05E10 Combinatorial aspects of representation theory
##### Keywords:
Specht modules; standard tableaux; general linear groups
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##### References:
 [1] R. Dipper, G.D. James, On Specht modules for general linear groups, submitted for publication · Zbl 1071.20041 [2] James, G.D., The representation theory of the symmetric groups, Lecture notes in math., 682, (1978), Springer-Verlag New York · Zbl 0393.20009 [3] James, G.D., Representations of general linear groups, London math. soc. lecture notes, 94, (1984), Cambridge University Press Cambridge · Zbl 0541.20025
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