On the Steinberg module, representations of the symmetric groups, and the Steenrod algebra (Q1076135)

Let \(A\) denote the mod \(p\) Steenrod algebra. Let \(B\) denote one of the subalgebras A(n) of \(A\) or \(P(n)\) of \(A/A\beta A\). \(B\) admits a left \(A\)-module structure extending its \(B\)-module structure. This result has independently been proved by the author [Topology 24, 227--246 (1985; Zbl 0568.55021)] and J. Smith. In the paper under review it is shown that for \(B=P(n)\) (or \(p=2\) and \(B=A(n))\) these two \(A\)-module structures are the same. The proof uses the fact that the Steinberg module is isomorphic to the Weyl module associated to the partition \(\Delta_ n=((p-1)(n-1),(p-1)(n-2),\ldots,p-1)\). Combining this Weyl module characterization of the Steinberg module with an unpublished result of S. Piddy, the author can deduce the following theorem of J. Smith: Let \(f\) denote the classical primitive idempotent corresponding to \(\Delta_ n\). Then \((\otimes^{(p-1)\binom{n}{2}}\mathbb Z_ n)f\) is a free \(P(n-2)\)-module on one generator.