The unipotent modules of GL _n(F_q) via tableaux

We construct the irreducible unipotent modules of the finite general linear groups from actions on tableaux. Our approach is analogous to that of James (Bull Lond Math Soc 8:229–232, 1976) for the symmetric groups, answering an open question as to whether such a construction exists. We show that our modules are isomorphic to those previously constructed by James (Representations of general linear groups, London Mathematical Society Lecture Note Series, vol. 94. Cambridge University Press, Cambridge, 1984. doi:10.1017/CBO9780511661921) , although the two presentations are quite different. Key to our construction are the generalized Gelfand–Graev representations of Kawanaka (Generalized Gel’fand-Graev representations and Ennola duality. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), advanced studies in pure math., vol. 6, pp. 175–206. North-Holland, Amsterdam 1985)..