Vector invariants of \(\mathsf{Syl}_p(\mathsf{GL}(n, \mathbb{F}_q))\) and their Hilbert ideals
Let \(\mathbb F=\mathbb F_q\) be a finite field with \(q=p^s\) elements for any prime \(p\). The upper unitriangular matrices in the general linear group \(GL(d,\mathbb F)\) constitute a \(p\)-Sylow subgroup \(P\) of \(GL(d,\mathbb F)\). In the present article are considered the \(n\)-fold invariants of \(P\), i.e., the embeddings of \(P\) in \(\mathrm{GL}(dn,\mathbb F)\). The Hilbert ideal of the ring of invariants \(\mathbb F[V]^P\) is defined as the ideal in the ambient ring of polynomials generated by all invariants of positive degree. The main result in the paper under review is that the Hilbert ideal is generated by the top orbit Chern classes of the standard dual basis of \(V^*\). This result actually follows from the second main theorem proved by the authors, which gives the generators of the ring of invariants \(\mathbb F[V]^P\).
scientific article; zbMATH DE number 6499704