On automorphisms of matrix invariants
HTML articles powered by AMS MathViewer
- by Zinovy Reichstein PDF
- Trans. Amer. Math. Soc. 340 (1993), 353-371 Request permission
Abstract:
Let ${Q_{m,n}}$ be the space of $m$-tuples of $n \times n$-matrices modulo the simultaneous conjugation action of $PG{L_n}$. Let ${Q_{m,n}}(\tau )$ be the set of points of ${Q_{m,n}}$ of representation type $\tau$. We show that for $m \geq n + 1$ the group $\operatorname {Aut}({Q_{m,n}})$ of representation type preserving algebraic automorphisms of ${Q_{m,n}}$ acts transitively on each ${Q_{m,n}}(\tau )$. Moreover, the action of $\operatorname {Aut}({Q_{m,n}})$ on the Zariski open subset ${Q_{m,n}}(1,n)$ of ${Q_{m,n}}$ is $s$-transitive for every positive integer $s$. We also prove slightly weaker analogues of these results for all $m \geq 3$.References
- M. Artin, On Azumaya algebras and finite dimensional representations of rings, J. Algebra 11 (1969), 532–563. MR 242890, DOI 10.1016/0021-8693(69)90091-X
- Christine Bessenrodt and Lieven Le Bruyn, Stable rationality of certain $\textrm {PGL}_n$-quotients, Invent. Math. 104 (1991), no. 1, 179–199. MR 1094051, DOI 10.1007/BF01245071
- P. M. Cohn, Progress in free associative algebras, Israel J. Math. 19 (1974), 109–151. MR 379555, DOI 10.1007/BF02756628
- Edward Formanek, The center of the ring of $3\times 3$ generic matrices, Linear and Multilinear Algebra 7 (1979), no. 3, 203–212. MR 540954, DOI 10.1080/03081087908817278
- Edward Formanek, The center of the ring of $4\times 4$ generic matrices, J. Algebra 62 (1980), no. 2, 304–319. MR 563230, DOI 10.1016/0021-8693(80)90184-2
- Edward Formanek, Noncommutative invariant theory, Group actions on rings (Brunswick, Maine, 1984) Contemp. Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 87–119. MR 810646, DOI 10.1090/conm/043/810646
- P. I. Katsylo, Stable rationality of fields of invariants of linear representations of the groups $\textrm {PSL}_6$ and $\textrm {PSL}_{12}$, Mat. Zametki 48 (1990), no. 2, 49–52, 159 (Russian); English transl., Math. Notes 48 (1990), no. 1-2, 751–753 (1991). MR 1076933, DOI 10.1007/BF01262607
- Lieven Le Bruyn, Simultaneous equivalence of square matrices, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris, 1987/1988) Lecture Notes in Math., vol. 1404, Springer, Berlin, 1989, pp. 127–136. MR 1035223, DOI 10.1007/BFb0084074
- Lieven Le Bruyn and Claudio Procesi, Étale local structure of matrix invariants and concomitants, Algebraic groups Utrecht 1986, Lecture Notes in Math., vol. 1271, Springer, Berlin, 1987, pp. 143–175. MR 911138, DOI 10.1007/BFb0079236
- David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. MR 719371, DOI 10.1007/978-3-642-96676-7
- Claudio Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 8 (1967), 237–255 (English, with Italian summary). MR 224657
- C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
- Zinovy Reichstein, A functional interpretation of the ring of matrix invariants, J. Algebra 136 (1991), no. 2, 439–462. MR 1089308, DOI 10.1016/0021-8693(91)90055-D
- Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
- Aidan Schofield, Matrix invariants of composite size, J. Algebra 147 (1992), no. 2, 345–349. MR 1161297, DOI 10.1016/0021-8693(92)90209-5
- Gerald W. Schwarz, Lifting smooth homotopies of orbit spaces, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 37–135. MR 573821
- K. S. Sibirskiĭ, Algebraic invariants of a system of matrices, Sibirsk. Mat. . 9 (1968), 152–164 (Russian). MR 0223379
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 340 (1993), 353-371
- MSC: Primary 16R30
- DOI: https://doi.org/10.1090/S0002-9947-1993-1124173-1
- MathSciNet review: 1124173