Automorphism group schemes of basic matrix invariants

Abstract:

For $3 \leqslant k < n,\quad {\text {let}}\quad {E_k}(X)$ be the polynomial in ${n^2}$ variables defined by ${\text {det}}(X + \lambda I) = \sum {{E_k}(X){\lambda ^{n - k}}}$. Let $R$ be a ring containing a field of characteristic $p \geqslant 0$. If $p$ does not divide $n - k + 1$, the invertible linear transformations on matrices preserving ${E_k}(X)$ up to scalars are (in essence) just the obvious ones arising from scaling, similarities, and transposition. If the power ${p^s}$ dividing $n - k + 1$ is greater than $k$, then we have these elements times maps of the form $X \mapsto X + f(X)I$. When smaller powers ${p^s}$ divide $n - k + 1$, the group scheme is like the first with an infinitesimal part of the second. One corollary is that every division algebra of finite dimension ${n^2} > 4$ over its center carries a canonical cubic form that determines it up to antiisomorphism.