Young tableaux and the Steenrod algebra
نویسندگان
چکیده
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) = F2[x1, . . . , xn], over the field F2 of two elements, and semistandard Young tableaux as they apply to the modular representation theory of the general linear group GL(n,F2). The cohits Qd(n) = Pd(n)/Pd(n)∩A+(P(n)) form a modular representation of GL(n,F2) and 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 Qd(n). The hook formula, which calculates the number of semistandard Young tableaux, then gives an upper bound for the dimension of Qd(n). In the particular degree d where the Steinberg module appears for the first time in P(n) the upper bound is exact and Qd(n) can then be identified with the Steinberg module.
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