Vector invariant ideals of Abelian group algebras under the action of the symplectic groups. (Q2847691)

Let \(\mathrm{Sp}_{2\nu}(F)\) be the symplectic group over a finite field \(F\) and let \(K\) be a field. The authors prove the following results. Theorem 2.1. \dots Let \(G=\left\{(A)=\left(\begin{smallmatrix} A&0\\ 0&A\end{smallmatrix}\right)\mid A\in\mathrm{Sp}_{2\nu}(F)\right\}\) act on the vector space \(F^{2\nu}\oplus F^{2\nu}\); then \(G\) acts on the group algebra \(K[F^{2\nu}\oplus F^{2\nu}]\). Suppose that \(\mathrm{char\,}K\neq\mathrm{char\,}F\) and \(K\) contains a primitive \(p\)-th root of unity \(\varepsilon\). Then every \(G\)-stable ideal \(I\) of \(K[F^{2\nu}\oplus F^{2\nu}]\) is the intersection of finitely many ideals in set \(S\) (the same notation in the above remark), i.e. \(I=\bigcap_{I'\in S'}I'\), where \(S'\leq S\) is a subset of \(S\).NEWLINENEWLINE Theorem 3.1. \dots Let \(\mathrm{Sp}_{2\nu}(F)\) act on the \(F\)-vector space \(F^{2\nu}\); then \(\mathrm{Sp}_{2\nu}(F)\) acts on the group algebra \(K[F^{2\nu}]\). Hence it induces an action of \(G=\left\{(A)=\left(\begin{smallmatrix} A&0\\ 0&A\end{smallmatrix}\right)\mid A\in\mathrm{Sp}_{2\nu}(F)\right\}\) on the \(F\)-vector space \(F^{2\nu}\oplus F^{2\nu}\) and on the group algebra \(K[F^{2\nu}\oplus F^{2\nu}]\). Suppose that \(\mathrm{char\,}K\neq\mathrm{char\,}F\) and \(K\) contains a primitive \(p\)-th root of unity \(\varepsilon\). Then all the invariant ideals of \(K[F^{2\nu}]\) can be obtained by the natural projection of some vector invariant ideals of \(K[F^{2\nu}\oplus F^{2\nu}]\).