On the homology of split extensions with \(p\)-elementary kernel (Q2731901)

Let \(p\) be a prime number and \(0\to A\to H\to G\to 1\) be a split group extension with \(A\cong(\mathbb{Z}/p)^n\). Thus, there is a Hochschild-Serre spectral sequence with abutment \(H_n(H;\mathbb{F}_q)\) and \(E^2\)-term (for \(p\) odd) NEWLINE\[NEWLINEE^2_{i,j}=H_i(G;H_j(A;\mathbb{F}_p))=\bigoplus_{k+2\ell=j}H_i(G;\Lambda^k(A)\otimes D^\ell(A)).NEWLINE\]NEWLINE (Here \(D^j(A)\) is the \(j\)-th divided power and \(\otimes=\otimes_{\mathbb{F}_p}\).) Let \(\Delta\colon\Lambda^*(A)\to A^{\otimes*}\) be the natural embedding, and \(\Delta_*=H_i(G;\Delta)\) the induced map on homology. Let \(H_i(G;\Lambda^*(A))_{\text{reg}}\) denote \(H_i(G;\Lambda^*(A))/\ker\Delta_*\). The main theorem of this paper states that the spaces \(H_i(G;\Lambda^*(A))_{\text{reg}}\) `survive to infinity' in the spectral sequence. For \(p=2\) an analogous result is true with \(\Lambda^*(A)\) replaced by \(D^*(A)\).NEWLINENEWLINENEWLINEIn the last section of the paper, the author examines in more detail the following example: Consider the split extension of \(\mathbb{F}_p\)-algebras \(\mathbb{F}_p\to\mathbb{F}_p[x]/x^2\to\mathbb{F}_p\). This gives a split extension of groups of the type considered above: NEWLINE\[NEWLINE0\to M(\mathbb{F}_p)\to\text{GL}(\mathbb{F}_p[x]/x^2)\to\text{GL}(\mathbb{F}_p)\to 1.NEWLINE\]NEWLINE (Here \(M(R)\) denotes the union of the matrix groups \(M_n(R)\) and \(\text{GL}(S)\) denotes the union of the general linear groups \(\text{GL}_n(S)\).) For \(p\) odd, the author shows using invariant theory that the dimension of the \(\mathbb{F}_p\)-vector space \(H_0(\text{GL}(\mathbb{F}_p);\Lambda^j(M(\mathbb{F}_p)))_{\text{reg}}\) is equal to the number of partitions of \(j\) into different odd numbers prime to \(p\).