On invariant ideals associated to classical groups
Assume that \(G_1\) is a classical group from the list: \(GL_n(k)\), \(Sp_n(k)\) (\(n\) even), and \(SO_n(k)\) (\(n\) odd), where \(k\) stands for an infinite field of characteristic zero. Let \(G_2\) be a group from the similar list of \(m\times m\) matrices. The variety \(X(G_1,G_2)\) of nullforms of \(G_1\times G_2\) is an affine variety of \(n\times m\) matrices \(M\) over \(k\) satisfying equalities \(M^T J_{G_1} M=0\) and \(M J_{G_2} M^T=0\), where \(J_{G_i}\) is the defining matrix of \(G_i\). (If \(G_i\) is the general linear group, then \(J_{G_i}=0\).) The group \(G_1\times G_2\) acts on \(X(G_1,G_2)\) via \((g,h)\cdot M=gMh^{-1}\). Thus the coordinate ring \(A(G_1,G_2)\) of \(X(G_1,G_2)\) is also a \(G_1\times G_2\)-module. For a partition \(\lambda=(\lambda_1,\ldots,\lambda_r)\) denote by \(\nabla_{G_i}(\lambda)\) the Schur module of \(G_i\) that corresponds to \(\lambda\). The ring \(A(G_1,G_2)\) decomposes as \(A(G_1,G_2)=\sum M_{\lambda}\), where \(M_{\lambda}\simeq \nabla_{G_1}(\lambda^{\ast})\otimes \nabla_{G_2}(\lambda)\) is an irreducible \(G_1\times G_2\)-module. Denote by \(I_{\lambda}=\langle M_{\lambda}\rangle\) the \(G_1\times G_2\)-invariant ideal of \(A(G_1,G_2)\) generated by \(M_{\lambda}\). In the paper under review it is shown that \(I_{\lambda}=\sum_{\lambda_i\leq\mu_i} M_{\mu}\), where \(\lambda_1\leq \mu_1\leq \text{min}\{\text{rk}\, G_1,\text{rk}\, G_2\}\). Earlier, this result for \(GL_n\times GL_m\) and \(Sp_n\times GL_m\) was obtained by \textit{C. DeConcini, D. Eisenbud} and \textit{C. Procesi} [Invent.~Math. 56, 129--165 (1980; Zbl 0435.14015)] and by \textit{E. Strickland} [J.~Algebra 66, 511--533 (1980; Zbl 0448.20040)], respectively. A decomposition of \(M_{\lambda} M_{\mu}\) into irreducible \(G_1\times G_2\)-modules is established in the paper under review. Earlier, this result for \(GL_n\times GL_m\) was obtained by \textit{K. Whitehead} [Thesis, University of Minnesota, (1982)]. Finally, in the paper under review the prime and primary \(G_1\times G_2\)-ideals are described and a primary decomposition of \(G_1\times G_2\)-invariant ideals \(I\subset A(G_1,G_2)\) is given.
scientific article; zbMATH DE number 5768598