Relatively free invariant algebras of finite reflection groups

Domokos, Mátyás

doi:10.1090/S0002-9947-96-01687-X

Relatively free invariant algebras of finite reflection groups
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by Mátyás Domokos

Trans. Amer. Math. Soc. 348 (1996), 2217-2234

DOI: https://doi.org/10.1090/S0002-9947-96-01687-X

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Abstract:

Let $G$ be a finite subgroup of $Gl_{n}(K)$ $(K$ is a field of characteristic $0$ and $n\geq 2)$ acting by linear substitution on a relatively free algebra $K\langle x_{1},\dots ,x_{n}\rangle /I$ of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if $G$ is a pseudo-reflection group and $I$ contains the polynomial $[[x_{2},x_{1}],x_{1}].$

References

N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379–436. MR 422448, DOI 10.24033/asens.1313
Warren Dicks and Edward Formanek, Poincaré series and a problem of S. Montgomery, Linear and Multilinear Algebra 12 (1982/83), no. 1, 21–30. MR 672913, DOI 10.1080/03081088208817467
Vesselin Drensky, Polynomial identities for $2\times 2$ matrices, Acta Appl. Math. 21 (1990), no. 1-2, 137–161. MR 1085776, DOI 10.1007/BF00053295
Vesselin Drensky, Computational techniques for PI-algebras, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 17–44. MR 1171223
V. Drensky, Private communication, 1993.
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
Robert M. Guralnick, Invariants of finite linear groups on relatively free algebras, Linear Algebra Appl. 72 (1985), 85–92. MR 815254, DOI 10.1016/0024-3795(85)90144-2
V. K. Kharchenko, Noncommutative invariants of finite groups and Noetherian varieties, J. Pure Appl. Algebra 31 (1984), no. 1-3, 83–90. MR 738208, DOI 10.1016/0022-4049(84)90079-3
D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438. MR 325658, DOI 10.1090/S0002-9947-1973-0325658-5
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
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Louis Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963), 57–64. MR 154929, DOI 10.1017/S0027763000011028
T. A. Springer, Regular elements of finite reflection groups, Invent. Math. 25 (1974), 159–198. MR 354894, DOI 10.1007/BF01390173
T. A. Springer, Invariant theory, Lecture Notes in Mathematics, Vol. 585, Springer-Verlag, Berlin-New York, 1977. MR 0447428, DOI 10.1007/BFb0095644