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Transactions of the American Mathematical Society

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Automorphism group schemes of basic matrix invariants


Author: William C. Waterhouse
Journal: Trans. Amer. Math. Soc. 347 (1995), 3859-3872
MSC: Primary 14L15; Secondary 14L27, 16K20
DOI: https://doi.org/10.1090/S0002-9947-1995-1303128-X
MathSciNet review: 1303128
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1303128-X
Article copyright: © Copyright 1995 American Mathematical Society

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