Automorphism group schemes of basic matrix invariants
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- by William C. Waterhouse PDF
- Trans. Amer. Math. Soc. 347 (1995), 3859-3872 Request permission
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.References
- LeRoy B. Beasley, Linear transformations on matrices: The invariance of the third elementary symmetric function, Canadian J. Math. 22 (1970), 746–752. MR 268201, DOI 10.4153/CJM-1970-084-x
- Hans-Jürgen Hoehnke, Über komponierbare Formen und konkordante hyperkomplexe Grössen, Math. Z. 70 (1958), 1–12 (German). MR 99353, DOI 10.1007/BF01558573
- N. Jacobson, Structure groups and Lie algebras of Jordan algebras of symmetric elements of associative algebras with involution, Advances in Math. 20 (1976), no. 2, 106–150. MR 407103, DOI 10.1016/0001-8708(76)90183-3
- D. G. James, Linear transformations of the second elementary function, Linear and Multilinear Algebra 10 (1981), no. 4, 347–349. MR 638129, DOI 10.1080/03081088108817425
- Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
- Amos Kovacs, Trace preserving linear transformations on matrix algebras, Linear and Multilinear Algebra 4 (1976/77), no. 4, 243–250. MR 429951, DOI 10.1080/03081087708817158
- Marvin Marcus and Roger Purves, Linear transformations on algebras of matrices: the invariance of the elementary symmetric functions, Canadian J. Math. 11 (1959), 383–396. MR 105425, DOI 10.4153/CJM-1959-039-4
- Bernard R. McDonald, $R$-linear endomorphisms of $(R)_{n}$ preserving invariants, Mem. Amer. Math. Soc. 46 (1983), no. 287, iv+67. MR 719156, DOI 10.1090/memo/0287
- William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117
- William C. Waterhouse, Linear maps preserving reduced norms, Linear Algebra Appl. 43 (1982), 197–200. MR 656445, DOI 10.1016/0024-3795(82)90253-1
- William C. Waterhouse, Invertibility of linear maps preserving matrix invariants, Linear and Multilinear Algebra 13 (1983), no. 2, 105–113. MR 697321, DOI 10.1080/03081088308817510
- William C. Waterhouse, Automorphisms of $\textrm {det}(X_{ij})$: the group scheme approach, Adv. in Math. 65 (1987), no. 2, 171–203. MR 900267, DOI 10.1016/0001-8708(87)90021-1
Additional Information
- © Copyright 1995 American Mathematical Society
- 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