典型群的特殊子群的BN-对及其不变式
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摘要
典型群及其特殊子群是代数学的重要研究对象。本文主要研究局部环上典型群的BN-对,二面体群的BN-对和不变式环,有限域上典型群的极大子群和根子群的有理不变式域的结构。
第一章考虑任意局部环上典型群的BN-对问题,构造了局部环上一般线性群、辛群、正交群的BN-对,并且证明了局部环上一般线性群与对应的BN-对之间满足群交换图关系。
第二章研究群环R上酉群的结构及其应用。首先确定群环R上Hermitian矩阵的合同标准型。其次,讨论R上酉群的BN-对和计数问题。最后,作为应用构造了一个Cartesian认证码,并计算了相应码的参数。
第三章讨论一类重要的有限反射群-二面体群D_n的BN-对和Building。并且,在不变式环方面证明一个有趣的结论:当n=3m,m∈Z~+时,D_n的BN-对中的子群B的不变式环是D_n的不变式环的一个整扩张,并且R_2作为R_1-自由模的基底的个数等于[D_n:B],从而刻划了群和对应的不变式环之间的数值关系。
第四章在有限典型群的极大子群和根子群的有理不变式域方面做了一些讨论。通过构造有理不变式域的超越基,从而回答了有限典型群的G_1和G_2类极大子群及其根子群Noether问题。另外,证明有限域上一般线性群的C_1类极大子群和特殊线性群的根子群的不变式环是多项式的。
It is well known that the classical groups and their special subgroups are important research objects.In this thesis,we study BN-pairs of classical groups over local rings, BN-pairs and rings of polynomial invariants of dihedral groups,the rational invariant fields of the maximal subgroups and root subgroups of classical groups over finite fields.
In Chapter 1,we investigate BN-pairs of classical groups over local rings.We construct the BN-pairs arising from classical groups of linear,orthogonal and symplectic type over local rings.Furthermore,we consider the relations between the general linear groups and the corresponding BN-pairs and show that they satisfy a commutative diagram of groups.
In Chapter 2,we study the structure of unitary groups over finite group rings R and its applications.First,we determine all the normal forms of Hermitian matrices over R.Secondly,we obtain BN-pairs and some Anzahl theorems of unitary groups over R. Finally,as applications,we construct a.Cartesian authentication code and compute its parameters.
In Chapter 3,we discuss the BN-pairs and buildings of dihedral groups.And we also obtain an interesting conclusion for the invariant rings of dihedral groups D_n with n=3m,m∈Z~+.Denote the invariant rings of D_n by R_2 and that of the subgroup B by R_1.Then R_1 is an integral extension over R_2.Moreover,the ring R_1,as a free R_2-module, has a basis with[D_n:B]elements.Consequently,we determine the relations between groups and invariant rings.
In Chapter 4,we focus on the structure of the rational invariant fields of the maximal subgroups and root subgroups of finite classical groups.We answer Noether's problem for those maximal subgroups of classes C_1 and C_2 and the root subgroups by constructing the explicit,transcendental bases of their invariant subfields.In addition,we prove that the invariant rings of the maximal subgroups of class C_1 of the general linear group are polvnomial,and so do the invariant rings of root subgroups of the special linear groups.
第一章考虑任意局部环上典型群的BN-对问题,构造了局部环上一般线性群、辛群、正交群的BN-对,并且证明了局部环上一般线性群与对应的BN-对之间满足群交换图关系。
第二章研究群环R上酉群的结构及其应用。首先确定群环R上Hermitian矩阵的合同标准型。其次,讨论R上酉群的BN-对和计数问题。最后,作为应用构造了一个Cartesian认证码,并计算了相应码的参数。
第三章讨论一类重要的有限反射群-二面体群D_n的BN-对和Building。并且,在不变式环方面证明一个有趣的结论:当n=3m,m∈Z~+时,D_n的BN-对中的子群B的不变式环是D_n的不变式环的一个整扩张,并且R_2作为R_1-自由模的基底的个数等于[D_n:B],从而刻划了群和对应的不变式环之间的数值关系。
第四章在有限典型群的极大子群和根子群的有理不变式域方面做了一些讨论。通过构造有理不变式域的超越基,从而回答了有限典型群的G_1和G_2类极大子群及其根子群Noether问题。另外,证明有限域上一般线性群的C_1类极大子群和特殊线性群的根子群的不变式环是多项式的。
It is well known that the classical groups and their special subgroups are important research objects.In this thesis,we study BN-pairs of classical groups over local rings, BN-pairs and rings of polynomial invariants of dihedral groups,the rational invariant fields of the maximal subgroups and root subgroups of classical groups over finite fields.
In Chapter 1,we investigate BN-pairs of classical groups over local rings.We construct the BN-pairs arising from classical groups of linear,orthogonal and symplectic type over local rings.Furthermore,we consider the relations between the general linear groups and the corresponding BN-pairs and show that they satisfy a commutative diagram of groups.
In Chapter 2,we study the structure of unitary groups over finite group rings R and its applications.First,we determine all the normal forms of Hermitian matrices over R.Secondly,we obtain BN-pairs and some Anzahl theorems of unitary groups over R. Finally,as applications,we construct a.Cartesian authentication code and compute its parameters.
In Chapter 3,we discuss the BN-pairs and buildings of dihedral groups.And we also obtain an interesting conclusion for the invariant rings of dihedral groups D_n with n=3m,m∈Z~+.Denote the invariant rings of D_n by R_2 and that of the subgroup B by R_1.Then R_1 is an integral extension over R_2.Moreover,the ring R_1,as a free R_2-module, has a basis with[D_n:B]elements.Consequently,we determine the relations between groups and invariant rings.
In Chapter 4,we focus on the structure of the rational invariant fields of the maximal subgroups and root subgroups of finite classical groups.We answer Noether's problem for those maximal subgroups of classes C_1 and C_2 and the root subgroups by constructing the explicit,transcendental bases of their invariant subfields.In addition,we prove that the invariant rings of the maximal subgroups of class C_1 of the general linear group are polvnomial,and so do the invariant rings of root subgroups of the special linear groups.
引文
[1]华罗庚.万哲先.典型群[M].上海:上海科技出版社.1963.
[2]万哲先.有限几何和不完全区组设计的一些研究[M].北京:科学出版社,1966.
[3]万哲先,霍元极.有限典型群子空间轨道生成的格[M].北京:科学出版社,1999.
[4]VAN Z X.Further construction of Cartesian authentication codes from symplectic geometry [J].Northeastern Mathematical Journal,1992,8:4-20.
[5]YOU H,NAN J Z.Using normal form of matrices over finite fields to construct Cartesian authentication codes[J].Journal of Mathematical Research and Exposition,1998,18(3):341-346.
[6]游宏,高有.有限交换环上典型群阶的计算[J].科学通报,1994,39(14):289-292.
[7]YOU H,NAN J Z.Some anzahl theorems in vector space over Z/p~kZ[J].Acta Math.Scientia,1996,16(1):81-88.
[8]TITS J.Buildings of spherical type and finite BN-pairs[M].Lecture Notes in Mathematics,Vol.386,Berlin-New York:Springer-Verlag,1974.
[9]FONG P,SEITZ G.Groups with a BN-pair of rank 2.Ⅰ[J].Invent.Math.,1973,21:1-57.
[10]FONG P,SEITZ G.Groups with a BN-pair of rank 2.Ⅱ[J].Invent.Math.,1974,24:191-239.
[11]TENT K.Split BN-pairs of rank 2:the octagons[J].Advances in Mathematics,2004,181:308-320.
[12]CHERLIN G.Good tori in groups of finite Morley rank[J].J.Group Theory,2005,8(5):613-622.
[13]BOROVIK A,NESIN A.Groups of finite Morley rank[M].New York:Oxford University Press.1994.
[14]CARTER R W.Finite groups of Lie type:Conjugacy Class and Complex Characters[M].London:Wiley-Interscience,1985.
[15]GARRETT P.Buildings and classical groups[M].London:Chapman and Hall,1997.
[16]BROWN K S.Buildings[M].New York:Springer-Verlag,1989.
[17]IWAHORI N,MATSUMOTO H.On some bruhat decomposition and the structure of the hecke ring of p-adic chevalley groups[J].Inst.Hautes Etudes Sci.Publ.Math.,1965.25:5-48.
[18]ASCHBACHER M.On the maximal subgoups of the finite classical groups[J].Invent.Math.,1984,76:469-514.
[19]KLEIDMAN P,LIEBECK M.A survey of the maximal subgroups of the finite simple groups[M].Geom.Dedicata.1985.
[20]KLEIDMAN P,LIEBECK M.The subgroup structure of the finite classical groups[M].Cambridge:Cambridge Univ.Press,1989.
[21]SEITZ G.The maximal subgroups of the classical algebraic groups[M].Mere.AMS 67,No.365,1993.
[22]NOETHER E.Gleichungen mit vorgeschriebener gruppe[J].Mat.Ann.,1918,78(216):221-229.
[23]SWAN R G.Invariant rational functions and a problem of steenrod[J].Invent.Math.,1969,7(2):148-158.
[24]LENSTRA H W.Jr.:Rational functions invariant under a cyclic group[J].Proc.Number Theory Conf.,Kingston,Queen's Papers Pure Appl.Math.,1980,54:91-99.
[25]Saltman D J.Noether's problem over an algebraically closed field[J].Invent.Math.1984,77:71-84.
[26]SONN J.Nonabelien counterexamples to the Noether problem[J].Proc.Amer.Math.Soc.,1985,93:225-226.
[27]HAJJA M.The alternating functions of three and of four variables[J].Algebras Groups Geom.,1989,6:49-55.
[28]MASUDA K.On a problem of Chevalley[J].Nagoya Math.J.,1955,8:59-63.
[29]MASUDA K.Application of theory of the group of classes of projective modules to existence problem of independent parameters of invariant[J].J.Math.Soc.Japan,1968,20:223-232.
[30]VOSKRESENSKII V E.Byrational properties of linear algebraic groups(Russian)[J].Engl.transl.:Math.USSR Izv.,1970,4:1-7.
[31]VOSKRESENSKII V E.On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field Q(x_1,…,x_n)[J].Engl.transl.:Math.USSR Izv.,1970,4:371-380.
[32]VOSKRESENSKII V E.Rationality of certain algebraic tori[J].Engl.transl.:Math.USSR.Izv.,1971,5:1049-1056.
[33]SHEPHARD G C,TODD J A.Finite unitary reflection groups[J].Canad.J.Math.,1954(6):274-304.
[34]CHAVELLEY C.Invariants of finite groups generated by reflections[J].Amer.J.Math.,1955,67:778-782.
[35]BENSON D J.Polynomial invariants of finite groups[M].LMS.Lecture Set.190,Cambridge:Cambridge University Press,1993.
[36]KEMPER G,MALLE G.Invariant fields of finite irreducible reflection groups[J].Math.Ann.,1999,315(4):569-586.
[37]FISHER E.Die isomorphie invariantenkorper der endiichen abelschen Gruppen linearer transformationen[J].Nachr.d.Ges.d.Wiss.zu Gottingen,1915,77-80.
[38]ENDO S,MIYATA T.Invariants of finite abelian groups[J].J.Math.Soc.Japan,1973,25:7-26.
[39]ENDO S,MIYATA T.Quasi-permutation modules over finite groups Ⅰ[J].J.Math.Soc.Japan,1973,25:397-421.
[40]ENDO S;MIYATA T.Quasi-permutation modules over finite groups Ⅱ[J].J.Math.Soc.Japan,1974,26:698-713.
[41]LENSTRA H W.Jr.:Rational functions invariant under a finite abelian group[J].Invent.Math.,1974,25:299-325.
[42]DICKSON L E.A fundamental system of invariants of the general modular linear group with a solution of the form problem[J].Trans.Amer.Math.Soc.,1911,12:75-98.
[43]CARLISLE D.KROPHOLLER P H.Rational invariants of certain orthogonal and unitary groups[J].Bull.London Math.Soc..1992,24:57-60.
[44]RAJAEI S M.Rational invariants of certain orthogonal groups over finite fields of characteristic two[J].Comm.Algebra,2000,28:2367-2393.
[45]TANG Z M,WAN Z X.A matrix approach to the rational invariants of certain classical groups over finite fields of characteristic two[J].Finite Fields and Their Applications,2006,12:186-210.
[46]COHEN S D.Eational function invariant under an orthogonal group[J].Bull.London Math.Soc.,1990,22:217-221.
[47]CHU H.Orthogonal group actions on rational function fields[J].Bull.Inst.Math.Acad.Sinica,1988,16:115-122.
[48]CHU H.Supplementary note on 'rational invariants of certain orthogonal and unitary groups'[J].Bull.London Math.Soc.,1997,29:37-42.
[49]Miyata T.Invariants of certain groups Ⅰ[J].Nagoya Math.J..1971,41:69-73.
[50]ATIYAH M F,MACDONALD I C.Introduction to commutative algebra[M].Addison-Wesley publishing Company.1969.
[51]ROTMAM J J.An introduction to homological algebra[M].London:Academic Press,1979.
[52]OSBORNE M.Basic homological algebra[M].Springer-Verlag,2000.
[53]WAN Z X.Geometry of classical groups over finite fields[M].Beijing/New York:Science Press,2002.
[54]SIMMONS G J.Authentication theory/Coding theory[M].Berlin:Springer-Verlag,1984:411-431.
[55]KARPILOVSKY G.Commutative group algebra[M].New York:Marcel Dekker,1983.
[56]高有.有限局部环上酉群阶的计算[J].数学物理学报,2005,25A(4):564-568.
[57]FRALEIGH J B.A first course in abstract algebra[M].Zaned Adchison Wealey Pub.Co.,1978.
[58]H.S.M.Coxeter.Discrete groups generated by reflections[J].Annal.Math.,1934,(35):588-621.
[59]E.Witt.Spiegelungsgruppen und Aufz(a|¨)hlung halbeinfachen Lieschen ringe[J].Abhandl.Math.Sem.Univ.Hamburg,1941,(14):289-337.
[60]P.Cartier.Groupes finis engendr(?)s par des sym(?)tries,(?)eminaire[M].Paris:Cheballey,Expose 14.1958.
[61]N.Bourbaki.Groupes et Alg(?)bres de Lie.chaps.4,5 and 6[M].Fascicule ⅩⅩⅩⅥ,Paris:El(?)ments de Math(?)matique,Hermann,1968.
[62]CHEVALLEY C.Invariant of finite groups generated by reflections[J].Amer.J.Math.1955.77:778-782.
[63]SMITH L.On the invariant theory of finite pseudoreflection groups[J].Arch.Math.,1957,9:273-276.
[64]SOLOMO L.Invariant of finite reflection groups[J].Nagoya J.Math.,1963,22:57-64.
[65]SPRINGER T A.Regular elements of finite reflection groups[J].Invent.Math.1974,25:159-198.
[66]STANLEY R P.Relative invariants of finite groups generated by pseudoreflections[J].J.Algebra,1977,49:134-148.
[67]KANE R.Reflection groups and invariant theory[M].New York:Springer-Verlag Inc.,2001.
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[71]GROVE L C,BENSON C T.Finite reflection groups[M].Springer-Verlag,1985.
[72]万哲先.有限反射群的不变式论[M].上海:上海交通大学出版社,1997.
[73]WILF H S.Generating functionology[M].Second edition.San Diego:Academic Press.1994.
[74]MAEDA T.Noether's problem for A_5[J].J.Algebra,1989,125:418-430.
[75]李尚志.典型群的子群结构[M].上海:上海科技出版社,1998.
[76]KLEIDMAN P.LIEBECK M W.The subgroup structure of the finite classical groups[M].Cambridge:Cambridge Univ.Press,1990.
[77]HUNZIKER M.Classical invariant theory for finite reflection groups[J].Transformation Groups,1997.2(2):147-163.
[78]WILKERSON C.A primer on the dickson invariants[J].Amer.Math.Soc.Contemp.Math.,1983,19:421-434.
[79]NAKAJIMA H.Invariants of finite groups generated by pseudo-reflections in positive characteristic [J].Tsukuba J.Math.,1979,3:109-122.
[80]KEMPER G.Calculating invariant rings of finite groups over arbitrary fields[J].J.Symbolic Computation,1996,21:351-366.
[2]万哲先.有限几何和不完全区组设计的一些研究[M].北京:科学出版社,1966.
[3]万哲先,霍元极.有限典型群子空间轨道生成的格[M].北京:科学出版社,1999.
[4]VAN Z X.Further construction of Cartesian authentication codes from symplectic geometry [J].Northeastern Mathematical Journal,1992,8:4-20.
[5]YOU H,NAN J Z.Using normal form of matrices over finite fields to construct Cartesian authentication codes[J].Journal of Mathematical Research and Exposition,1998,18(3):341-346.
[6]游宏,高有.有限交换环上典型群阶的计算[J].科学通报,1994,39(14):289-292.
[7]YOU H,NAN J Z.Some anzahl theorems in vector space over Z/p~kZ[J].Acta Math.Scientia,1996,16(1):81-88.
[8]TITS J.Buildings of spherical type and finite BN-pairs[M].Lecture Notes in Mathematics,Vol.386,Berlin-New York:Springer-Verlag,1974.
[9]FONG P,SEITZ G.Groups with a BN-pair of rank 2.Ⅰ[J].Invent.Math.,1973,21:1-57.
[10]FONG P,SEITZ G.Groups with a BN-pair of rank 2.Ⅱ[J].Invent.Math.,1974,24:191-239.
[11]TENT K.Split BN-pairs of rank 2:the octagons[J].Advances in Mathematics,2004,181:308-320.
[12]CHERLIN G.Good tori in groups of finite Morley rank[J].J.Group Theory,2005,8(5):613-622.
[13]BOROVIK A,NESIN A.Groups of finite Morley rank[M].New York:Oxford University Press.1994.
[14]CARTER R W.Finite groups of Lie type:Conjugacy Class and Complex Characters[M].London:Wiley-Interscience,1985.
[15]GARRETT P.Buildings and classical groups[M].London:Chapman and Hall,1997.
[16]BROWN K S.Buildings[M].New York:Springer-Verlag,1989.
[17]IWAHORI N,MATSUMOTO H.On some bruhat decomposition and the structure of the hecke ring of p-adic chevalley groups[J].Inst.Hautes Etudes Sci.Publ.Math.,1965.25:5-48.
[18]ASCHBACHER M.On the maximal subgoups of the finite classical groups[J].Invent.Math.,1984,76:469-514.
[19]KLEIDMAN P,LIEBECK M.A survey of the maximal subgroups of the finite simple groups[M].Geom.Dedicata.1985.
[20]KLEIDMAN P,LIEBECK M.The subgroup structure of the finite classical groups[M].Cambridge:Cambridge Univ.Press,1989.
[21]SEITZ G.The maximal subgroups of the classical algebraic groups[M].Mere.AMS 67,No.365,1993.
[22]NOETHER E.Gleichungen mit vorgeschriebener gruppe[J].Mat.Ann.,1918,78(216):221-229.
[23]SWAN R G.Invariant rational functions and a problem of steenrod[J].Invent.Math.,1969,7(2):148-158.
[24]LENSTRA H W.Jr.:Rational functions invariant under a cyclic group[J].Proc.Number Theory Conf.,Kingston,Queen's Papers Pure Appl.Math.,1980,54:91-99.
[25]Saltman D J.Noether's problem over an algebraically closed field[J].Invent.Math.1984,77:71-84.
[26]SONN J.Nonabelien counterexamples to the Noether problem[J].Proc.Amer.Math.Soc.,1985,93:225-226.
[27]HAJJA M.The alternating functions of three and of four variables[J].Algebras Groups Geom.,1989,6:49-55.
[28]MASUDA K.On a problem of Chevalley[J].Nagoya Math.J.,1955,8:59-63.
[29]MASUDA K.Application of theory of the group of classes of projective modules to existence problem of independent parameters of invariant[J].J.Math.Soc.Japan,1968,20:223-232.
[30]VOSKRESENSKII V E.Byrational properties of linear algebraic groups(Russian)[J].Engl.transl.:Math.USSR Izv.,1970,4:1-7.
[31]VOSKRESENSKII V E.On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field Q(x_1,…,x_n)[J].Engl.transl.:Math.USSR Izv.,1970,4:371-380.
[32]VOSKRESENSKII V E.Rationality of certain algebraic tori[J].Engl.transl.:Math.USSR.Izv.,1971,5:1049-1056.
[33]SHEPHARD G C,TODD J A.Finite unitary reflection groups[J].Canad.J.Math.,1954(6):274-304.
[34]CHAVELLEY C.Invariants of finite groups generated by reflections[J].Amer.J.Math.,1955,67:778-782.
[35]BENSON D J.Polynomial invariants of finite groups[M].LMS.Lecture Set.190,Cambridge:Cambridge University Press,1993.
[36]KEMPER G,MALLE G.Invariant fields of finite irreducible reflection groups[J].Math.Ann.,1999,315(4):569-586.
[37]FISHER E.Die isomorphie invariantenkorper der endiichen abelschen Gruppen linearer transformationen[J].Nachr.d.Ges.d.Wiss.zu Gottingen,1915,77-80.
[38]ENDO S,MIYATA T.Invariants of finite abelian groups[J].J.Math.Soc.Japan,1973,25:7-26.
[39]ENDO S,MIYATA T.Quasi-permutation modules over finite groups Ⅰ[J].J.Math.Soc.Japan,1973,25:397-421.
[40]ENDO S;MIYATA T.Quasi-permutation modules over finite groups Ⅱ[J].J.Math.Soc.Japan,1974,26:698-713.
[41]LENSTRA H W.Jr.:Rational functions invariant under a finite abelian group[J].Invent.Math.,1974,25:299-325.
[42]DICKSON L E.A fundamental system of invariants of the general modular linear group with a solution of the form problem[J].Trans.Amer.Math.Soc.,1911,12:75-98.
[43]CARLISLE D.KROPHOLLER P H.Rational invariants of certain orthogonal and unitary groups[J].Bull.London Math.Soc..1992,24:57-60.
[44]RAJAEI S M.Rational invariants of certain orthogonal groups over finite fields of characteristic two[J].Comm.Algebra,2000,28:2367-2393.
[45]TANG Z M,WAN Z X.A matrix approach to the rational invariants of certain classical groups over finite fields of characteristic two[J].Finite Fields and Their Applications,2006,12:186-210.
[46]COHEN S D.Eational function invariant under an orthogonal group[J].Bull.London Math.Soc.,1990,22:217-221.
[47]CHU H.Orthogonal group actions on rational function fields[J].Bull.Inst.Math.Acad.Sinica,1988,16:115-122.
[48]CHU H.Supplementary note on 'rational invariants of certain orthogonal and unitary groups'[J].Bull.London Math.Soc.,1997,29:37-42.
[49]Miyata T.Invariants of certain groups Ⅰ[J].Nagoya Math.J..1971,41:69-73.
[50]ATIYAH M F,MACDONALD I C.Introduction to commutative algebra[M].Addison-Wesley publishing Company.1969.
[51]ROTMAM J J.An introduction to homological algebra[M].London:Academic Press,1979.
[52]OSBORNE M.Basic homological algebra[M].Springer-Verlag,2000.
[53]WAN Z X.Geometry of classical groups over finite fields[M].Beijing/New York:Science Press,2002.
[54]SIMMONS G J.Authentication theory/Coding theory[M].Berlin:Springer-Verlag,1984:411-431.
[55]KARPILOVSKY G.Commutative group algebra[M].New York:Marcel Dekker,1983.
[56]高有.有限局部环上酉群阶的计算[J].数学物理学报,2005,25A(4):564-568.
[57]FRALEIGH J B.A first course in abstract algebra[M].Zaned Adchison Wealey Pub.Co.,1978.
[58]H.S.M.Coxeter.Discrete groups generated by reflections[J].Annal.Math.,1934,(35):588-621.
[59]E.Witt.Spiegelungsgruppen und Aufz(a|¨)hlung halbeinfachen Lieschen ringe[J].Abhandl.Math.Sem.Univ.Hamburg,1941,(14):289-337.
[60]P.Cartier.Groupes finis engendr(?)s par des sym(?)tries,(?)eminaire[M].Paris:Cheballey,Expose 14.1958.
[61]N.Bourbaki.Groupes et Alg(?)bres de Lie.chaps.4,5 and 6[M].Fascicule ⅩⅩⅩⅥ,Paris:El(?)ments de Math(?)matique,Hermann,1968.
[62]CHEVALLEY C.Invariant of finite groups generated by reflections[J].Amer.J.Math.1955.77:778-782.
[63]SMITH L.On the invariant theory of finite pseudoreflection groups[J].Arch.Math.,1957,9:273-276.
[64]SOLOMO L.Invariant of finite reflection groups[J].Nagoya J.Math.,1963,22:57-64.
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