有限伪反射群的相对不变式的Poincare级数
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摘要
设G是作用在特征数为0的域F上的向量空间V的有限伪反射群，χ∶G→F*是G的1-维表示，本文证明对(?)g∈G，χ=(g)=(detg)~α(0≤α≤r-1)，其中r为g的阶。另外指出了该群的相对不变式和一般不变式的关系，并根据一般不变式的Poincare级数的Molien公式，计算出相对不变式的Poincare级数。
Let F be a field with characteristic 0, V = F~n be the n-dimensional vector space over F and G be a finite pseudo-reflection group acts on V. Letχ: G→F~* be a 1-dimensional representation of G. In this article we show thatχ(g) = (detg)~α(0≤α≤r-1), where g∈G and r is the order of g. In addition, we characterize the relation between relative invariants and invariants of the group G, and then we use Molien's Theorem of invariants to compute the Poincare series of relative invariants.
Let F be a field with characteristic 0, V = F~n be the n-dimensional vector space over F and G be a finite pseudo-reflection group acts on V. Letχ: G→F~* be a 1-dimensional representation of G. In this article we show thatχ(g) = (detg)~α(0≤α≤r-1), where g∈G and r is the order of g. In addition, we characterize the relation between relative invariants and invariants of the group G, and then we use Molien's Theorem of invariants to compute the Poincare series of relative invariants.
引文
[1] L.C.Grove, C.T.Benson.Finite reflection groups.Spinger,1984.
[2] 万哲先．有限反射群的不变式论．上海交通大学出版社，1997．
[3] D.J.Benson. Polynomial invariants of finite groups. Cambridge University Press, 1993.
[4] L.Smith. On the invariants of finite pseudoreflection groups,Arch.Math.44,1985,225-228.
[5] L.Smith. On the invariants of finite pseudoreflection groups:Orbit polynomials and splitting principles,J.Algebra 110,1987,134-157.
[6] L.Smith. Polynomial invariants of finite groups. A survey of rccent developments,Bull.Amer.Math.Soc.34,1997,211-250.
[7] L.Smith. Polynomial invariants of finite groups. A.K.Peters,LTD,Wellesey, MA,1995.
[8] L.Smith. Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay. Proceedings of the American Mathematical Socity, Volume 134,Number 8,2006,2205-2212.
[9] Ian Hughes, Gregor Kemper. Symmetric powers of modular representations for groups with a Sylow subgroup of prime order.J.Algebra 241, 2001,759-788.
[10] Richard Kane. Reflection groups and invariant theory. Springer, 2001.
[11] Chevalley. Invariants of finite groups generated by reflections, Amer.J.Math.1955,778-782.
[12] Flatto. Basic sets of invariants for finite reflection groups,Bull.Amer.Math.Soc.74,1968,730-734.
[13] Flatto. Invariants of finite reflection groups, Enseign.Math.1978,237-292.
[14] H.L.Hiller.Geometry of Coxeter groups, Research Notes in Mathematics,No.54,Pitman,Boston,1982.
[15] J.E.Humphreys.Reflection groups and Coxeter groups,Cambridge univeristy Press,Great Britian,1990.
[16] I.G.Macdanald.The poincare series of a Coxeter group,Math.Ann.199,1972,161-174.
[17] H.Nakajima. Invariants of finite groups generated by pseudoreflections in positive charac- teristic,Tsukuba J.Math.3,1979,109-122.
[18] G.C.Shephard. Some problems on finite reflection groups,Enseign.Math.2,1956,42-48.
[19] L.Solomon.Invariants of Euclidean reflection groups,Traus.Amer.Math.Soc. 113,1964,287- 286.
[20] L.Solomon,Orlik. Unitary reflection groups and cohomology,Invent.Math.59,1980,77-94.
[21] R.P.Stanley. Invariants of finite groups and their applications to combinatorics,Bul.Amer.Math.Soc.1,1979.475-511.
[22] R..P. Stanley. Relative invariants of finite groups generated by pseudoreflections, J.of Algebra 49,1977,134-148.
[23] R.Steinberg. Invarints of finite reflection groups, Canad.J.Math. 12,1960,616-618.
[24] J.Hartmann,A.Shepler,Jacobians of reflection groups over finite fields,Preprint,2004.
[25] T.A.Springer.Regular elements of finite reflection groups,Invent.Math.25,1974,159-198.
[26] T.A.Springer. Invariant theory,Lecture Notes in Math.585,Spinger-Verlag,Berlin,1977.
[27] A.M.Cohen. Finite complex reflection groups,Ann.Sci.9,1976,379-436.
[28] A.Dress. On finite groups generated by pseudorefiections,J.Algebra 11,1969,1-5.
[2] 万哲先．有限反射群的不变式论．上海交通大学出版社，1997．
[3] D.J.Benson. Polynomial invariants of finite groups. Cambridge University Press, 1993.
[4] L.Smith. On the invariants of finite pseudoreflection groups,Arch.Math.44,1985,225-228.
[5] L.Smith. On the invariants of finite pseudoreflection groups:Orbit polynomials and splitting principles,J.Algebra 110,1987,134-157.
[6] L.Smith. Polynomial invariants of finite groups. A survey of rccent developments,Bull.Amer.Math.Soc.34,1997,211-250.
[7] L.Smith. Polynomial invariants of finite groups. A.K.Peters,LTD,Wellesey, MA,1995.
[8] L.Smith. Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay. Proceedings of the American Mathematical Socity, Volume 134,Number 8,2006,2205-2212.
[9] Ian Hughes, Gregor Kemper. Symmetric powers of modular representations for groups with a Sylow subgroup of prime order.J.Algebra 241, 2001,759-788.
[10] Richard Kane. Reflection groups and invariant theory. Springer, 2001.
[11] Chevalley. Invariants of finite groups generated by reflections, Amer.J.Math.1955,778-782.
[12] Flatto. Basic sets of invariants for finite reflection groups,Bull.Amer.Math.Soc.74,1968,730-734.
[13] Flatto. Invariants of finite reflection groups, Enseign.Math.1978,237-292.
[14] H.L.Hiller.Geometry of Coxeter groups, Research Notes in Mathematics,No.54,Pitman,Boston,1982.
[15] J.E.Humphreys.Reflection groups and Coxeter groups,Cambridge univeristy Press,Great Britian,1990.
[16] I.G.Macdanald.The poincare series of a Coxeter group,Math.Ann.199,1972,161-174.
[17] H.Nakajima. Invariants of finite groups generated by pseudoreflections in positive charac- teristic,Tsukuba J.Math.3,1979,109-122.
[18] G.C.Shephard. Some problems on finite reflection groups,Enseign.Math.2,1956,42-48.
[19] L.Solomon.Invariants of Euclidean reflection groups,Traus.Amer.Math.Soc. 113,1964,287- 286.
[20] L.Solomon,Orlik. Unitary reflection groups and cohomology,Invent.Math.59,1980,77-94.
[21] R.P.Stanley. Invariants of finite groups and their applications to combinatorics,Bul.Amer.Math.Soc.1,1979.475-511.
[22] R..P. Stanley. Relative invariants of finite groups generated by pseudoreflections, J.of Algebra 49,1977,134-148.
[23] R.Steinberg. Invarints of finite reflection groups, Canad.J.Math. 12,1960,616-618.
[24] J.Hartmann,A.Shepler,Jacobians of reflection groups over finite fields,Preprint,2004.
[25] T.A.Springer.Regular elements of finite reflection groups,Invent.Math.25,1974,159-198.
[26] T.A.Springer. Invariant theory,Lecture Notes in Math.585,Spinger-Verlag,Berlin,1977.
[27] A.M.Cohen. Finite complex reflection groups,Ann.Sci.9,1976,379-436.
[28] A.Dress. On finite groups generated by pseudorefiections,J.Algebra 11,1969,1-5.