Vector invariant fields of finite classical groups (Q2312854)

Let \(W\) be an \(n\)-dimensional vector space over a finite field \(\mathbb{F}\). The general linear group \(\mathrm{GL}(W)\) acts naturally on \(mW\), the direct sum of \(m\) copies of \(W\). The authors first give an elementary proof of Steinberg's theorem providing \(mn\) explicit generators of the field \(\mathbb{F}(mW)^{\mathrm{GL}(W)}\) of rational invariants on \(mW\). As an application, for \(G\in \{\mathrm{GL}(W),\mathrm{SL}(W)\}\) they present \(mn\) homogeneous polynomials in the ring \(\mathbb{F}[mW]^G\) of polynomial invariants on \(mW\) that generate the field \(\mathbb{F}(mW)^G\) of rational invariants. Moreover, in odd characteristic they find analogous statements for the orthogonal group, the unitary group, and the symplectic group