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We shall study the following complex Stiefel fibring:.In particular we shall study the problem: for what values of n and k does the fibring(1.1) admit a cross-section? A necessary condition for the existence of a cross-section has been found by Atiyah and Todd (8). We shall show (Theorem 1.1 below) that the condition of Atiyah and Todd is sufficient (as well as necessary) for the existence of a cross-section. The problem stated above is therefore completely solved.

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The purpose of this paper is to forge a direct link between the hit problem for the action of the Steenrod algebra A on the polynomial algebra P(n) = F 2 [x 1 , . . . , x n ], over the field F 2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n, F 2 ). The cohitsand the hit problem is to analyze this module. In certain generic degrees d we show how the semistandard Young tableaux can be used to index a set of monomials which span Q d (n). The hook formula, which calculates the number of semistandard Young tableaux, then gives an upper bound for the dimension of Q d (n). In the particular degree d where the Steinberg module appears for the first time in P(n) the upper bound is exact and Q d (n) can then be identified with the Steinberg module. 55S10; 20C20