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The ring of generic matrices. (English) Zbl 1047.16013
This is a (partly historical) survey of the theory of the ring of generic matrices (and its applications to matrix invariants and polynomial identities of \(n\times n\) matrices) over a field of characteristic zero, centered around the work of Claudio Procesi, to whom it is a tribute.

MSC:
16R30 Trace rings and invariant theory (associative rings and algebras)
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
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[1] Amitsur, S.A., On rings with identities, J. London math. soc., 20, 464-470, (1955) · Zbl 0064.26505
[2] Amitsur, S.A., The T-ideals of the free ring, J. London math. soc., 20, 470-475, (1955) · Zbl 0064.26506
[3] Amitsur, S.A., On central division algebras, Israel J. math., 12, 408-420, (1972) · Zbl 0248.16006
[4] Amitsur, S.A.; Levitzki, J., Minimal identities for algebras, Proc. amer. math. soc., 1, 449-463, (1950) · Zbl 0040.01101
[5] Artin, M., On Azumaya algebras and finite-dimensional representations of rings, J. algebra, 11, 523-563, (1969) · Zbl 0222.16007
[6] Bass, H., Groups of integral representation type, Pacific J. math., 86, 15-51, (1980) · Zbl 0444.20006
[7] Bass, H., Finitely generated subgroups of GL2, (), 127-136
[8] Beneish, E., Induction theorems on the stable rationality of the center of the ring of generic matrices, Trans. amer. math. soc., 350, 3571-3585, (1998) · Zbl 0915.16020
[9] Bessenrodt, C.; Le Bruyn, L., Stable rationality of certain PGLn quotients, Invent. math., 104, 179-199, (1991) · Zbl 0741.14032
[10] Dubnov, J., (), 351-367, in Russian
[11] Dubnov, J.; Ivanov, V., Sur l’abaissement du degre des polynomes en affineurs, C.C. (doklady) acad. sci. URSS, 41, 95-98, (1943) · Zbl 0060.03308
[12] Formanek, E., Central polynomials for matrix rings, J. algebra, 23, 129-132, (1972) · Zbl 0242.15004
[13] Formanek, E., The center of the ring of 3×3 generic matrices, Linear multilinear algebra, 7, 203-212, (1979) · Zbl 0419.16010
[14] Formanek, E., The center of the ring of 4×4 generic matrices, J. algebra, 62, 301-319, (1980) · Zbl 0437.16013
[15] Formanek, E., Rational function fields—noether’s problem and related questions, J. pure appl. algebra, 31, 28-36, (1984) · Zbl 0528.12018
[16] Formanek, E., Invariants and the ring of generic matrices, J. algebra, 89, 178-223, (1984) · Zbl 0549.16008
[17] Formanek, E., The polynomial identities and invariants of n×n matrices, CBMS regional series, 78, (1991), American Mathematical Society · Zbl 0714.16001
[18] Formanek, E.; Halpin, P.; Li, W.-C.W., The Poincaré series of the ring of 2×2 generic matrices, J. algebra, 69, 105-112, (1981) · Zbl 0459.16013
[19] Fricke, R.; Klein, F., Vorlesungen über die theorie der automorphen functionen, (1965), Academic Press, Reprint · JFM 28.0334.01
[20] Gurevich, G.B., Foundations of the theory of algebraic invariants, (1964), Noordhoff · Zbl 0128.24601
[21] Kaplansky, I., “problems in the theory of rings” revisited, Amer. math. monthly, 77, 445-454, (1970) · Zbl 0208.29701
[22] Katsylo, P.I., Stable rationality of fields of invariants of linear representations of the groups PSL6 and PSL12, Math. zametki, Math. notes, 48, 751-753, (1991), in Russian; translation · Zbl 0729.14034
[23] Kirillov, A.A., Certain division algebras over a field of rational functions, Funktsional analiz i prilozhen., 1, 101-102, (1967), in Russian · Zbl 0189.03601
[24] Kostant, B., A theorem of Frobenius, a theorem of amitsur – levitski, and cohomology theory, J. math. mech., 7, 237-264, (1958) · Zbl 0087.25702
[25] Kuzmin, E.N., On the nagata – higman theorem, (), 101-107, in Russian
[26] Le Bruyn, L., Centers of generic division algebras, the rationality problem, 1965-1990, Israel J. math., 76, 97-111, (1991) · Zbl 0770.16004
[27] Magnus, W., The uses of 2×2 matrices in combinatorial group theory. A survey, (), 4, 701-722, (1981), Reprinted in
[28] Procesi, C., Non-commutative affine rings, Atti accad. naz. lincei rend. cl. sci. fis. mat. natur., 8, 239-255, (1967) · Zbl 0204.04802
[29] Procesi, C., Rings with polynomial identities, (1973), Dekker · Zbl 0262.16018
[30] Procesi, C., Finite dimensional representations of algebras, Israel J. math., 19, 169-182, (1974) · Zbl 0297.16011
[31] Procesi, C., The invariant theory of n×n matrices, Adv. math., 19, 306-381, (1976) · Zbl 0331.15021
[32] Procesi, C., Deformations of representations, (), 247-276 · Zbl 0948.16012
[33] Razmyslov, Y.P., On a problem of Kaplansky, Izv. akad. nauk SSSR, Math. USSR-izv., 7, 479-496, (1973), translation · Zbl 0314.16016
[34] Razmyslov, Y.P., Trace identities of full matrix algebras over a field of characteristic zero, Izv. akad. nauk SSSR, Math. USSR-izv., 8, 727-760, (1974), translation · Zbl 0311.16016
[35] Regev, A., Asymptotics of codimensions of some P.I. algebras, (), 159-172 · Zbl 0908.16023
[36] Schofield, A., Matrix invariants of composite size, J. algebra, 147, 345-349, (1992) · Zbl 0785.14030
[37] Schofield, A., Birational classification of moduli spaces, (), 297-309 · Zbl 1041.16012
[38] Snider, R.L., Is the Brauer group generated by cyclic algebras?, (), 279-301
[39] Spencer, A.J.M., Theory of invariants, (), 239-353
[40] Sylvester, J.J., On the involution of two matrices of the second order, (), 115-117, reprinted in
[41] Vaughan-Lee, M., An algorithm for computing graded algebras, J. symbolic comput., 16, 345-354, (1993) · Zbl 0801.16025
[42] Vogt, H., Sur LES invariants, fondamentaux des équations différentielles linéares du second ordre, Ann. sci. école norm. sup., 6, Suppl. 3-72, (1889), Thèse, Paris · JFM 21.0314.01
[43] Wagner, W., Über die grundlagen der projectiven geometrie und allgemeine zahlsystems, Math. Z., 113, 528-567, (1937) · JFM 62.1402.01
[44] Weyl, H., The classical groups, (1946), Princeton University Press · JFM 65.0058.02
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