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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
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[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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