rep - 青山学院大学kyo/preprint/ohp1_karnataka2007.pdf1 intro duction 2 first f undamental theo...

Representations of redu tive groupsand invariant theoryKyo NishiyamaGraduate S hool of S ien eKyoto UniversityInternational onferen e on Re ent Advan es in Mathemati sand its appli ations.(September 24{30, 2007)Department of Mathemati s, KU Post Graduate CenterBelgaum, Karnataka INDIAKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 1 / 24

1 Introdu tion2 First Fundamental Theorem (FFT)3 Se ond Fundamental Theorem (SFT)4 Geometri invariant theory (a �rst step)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 2 / 24

Introdu tionWhat is invariants?

Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 3 / 24

Introdu tionWhat is invariants?Examples of invariants in broader sensenumbers : : : of �nite sets


Introdu tionWhat is invariants?Examples of invariants in broader sensenumbers : : : of �nite setsdim : : : of �nite ve tor spa es


Introdu tionWhat is invariants?Examples of invariants in broader sensenumbers : : : of �nite setsdim : : : of �nite ve tor spa esrank : : : of matrix (or linear map)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 3 / 24

Introdu tionWhat is invariants?Examples of invariants in broader sensenumbers : : : of �nite setsdim : : : of �nite ve tor spa esrank : : : of matrix (or linear map)genus : : : of ompa t 2-dimensional surfa esor we should say...Euler hara teristi : : : of manifoldsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 3 / 24

Introdu tionWhat is invariants?Examples of invariants in broader sensenumbers : : : of �nite setsdim : : : of �nite ve tor spa esrank : : : of matrix (or linear map)genus : : : of ompa t 2-dimensional surfa esor we should say...Euler hara teristi : : : of manifoldsClassi� ation problem=) study of equivalen e lasses=) invariantsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 3 / 24

Introdu tionThere are more sophisti ated invariants...


Introdu tionThere are more sophisti ated invariants... whi h I vaguely understand


Introdu tionThere are more sophisti ated invariants... whi h I vaguely understandAlexander/Jones/Hom y/Kau�man polynomials : : : of knots andlinksVasiliev invariantsChern-Simons invariantsNow there are so many invariants, quantum invariants, : : :Donaldson invariantsSeiberg-Witten invariantsGromov-Witten invariants : : : quantum ohomologyIwasawa invariants : : : for lass �eld theoryKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 4 / 24

Introdu tionThere are more sophisti ated invariants... whi h I vaguely understandAlexander/Jones/Hom y/Kau�man polynomials : : : of knots andlinksVasiliev invariantsChern-Simons invariantsNow there are so many invariants, quantum invariants, : : :Donaldson invariantsSeiberg-Witten invariantsGromov-Witten invariants : : : quantum ohomologyIwasawa invariants : : : for lass �eld theoryStudy of invariants ; Study of mathemati s !Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 4 / 24

Introdu tionThere are more sophisti ated invariants... whi h I vaguely understandAlexander/Jones/Hom y/Kau�man polynomials : : : of knots andlinksVasiliev invariantsChern-Simons invariantsNow there are so many invariants, quantum invariants, : : :Donaldson invariantsSeiberg-Witten invariantsGromov-Witten invariants : : : quantum ohomologyIwasawa invariants : : : for lass �eld theoryStudy of invariants ; Study of mathemati s !: : : Too big subje t for usKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 4 / 24

Introdu tionThere are more sophisti ated invariants... whi h I vaguely understandAlexander/Jones/Hom y/Kau�man polynomials : : : of knots andlinksVasiliev invariantsChern-Simons invariantsNow there are so many invariants, quantum invariants, : : :Donaldson invariantsSeiberg-Witten invariantsGromov-Witten invariants : : : quantum ohomologyIwasawa invariants : : : for lass �eld theoryStudy of invariants ; Study of mathemati s !: : : Too big subje t for us (at least for me)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 4 / 24

Introdu tionSo what is invariant theory?


Introdu tionSo what is invariant theory?Examples of lassi al invariants in invariant theory1 square of distan e r2 = x21 + � � � + x2n : orthogonal invariant


Introdu tionSo what is invariant theory?Examples of lassi al invariants in invariant theory1 square of distan e r2 = x21 + � � � + x2n : orthogonal invariantOr more generally,quadrati form of signature (p; q) (p + q = n)x21 + � � �+ x2p � x2p+1 � � � � � x2p+q=) Sylvester's law of inertiaKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 5 / 24

Introdu tionSo what is invariant theory?Examples of lassi al invariants in invariant theory1 square of distan e r2 = x21 + � � � + x2n : orthogonal invariantOr more generally,quadrati form of signature (p; q) (p + q = n)x21 + � � �+ x2p � x2p+1 � � � � � x2p+q=) Sylvester's law of inertia2 determinant : detXdet(gXg�1) = detX g : invertible matrixdet(gX ) = detX g : unimodular matrixKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 5 / 24

Introdu tionSo what is invariant theory?Examples of lassi al invariants in invariant theory1 square of distan e r2 = x21 + � � � + x2n : orthogonal invariantOr more generally,quadrati form of signature (p; q) (p + q = n)x21 + � � �+ x2p � x2p+1 � � � � � x2p+q=) Sylvester's law of inertia2 determinant : detXdet(gXg�1) = detX g : invertible matrixdet(gX ) = detX g : unimodular matrixtra e : tra eXtra e(gXg�1) = tra eX g : invertible matrixKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 5 / 24

Introdu tion3 dis riminant : �(f ) � � � SL2-invariantf (x) = a0xn + a1xn�1 + � � �+ an�1x + an= a0Qnj=1(x � �j)�(f ) := a2n�20 Qi<j(�i � �j)2 : : : polynomial in a = (a0; a1; : : : ; an)


Introdu tion3 dis riminant : �(f ) � � � SL2-invariantf (x) = a0xn + a1xn�1 + � � �+ an�1x + an= a0Qnj=1(x � �j)�(f ) := a2n�20 Qi<j(�i � �j)2 : : : polynomial in a = (a0; a1; : : : ; an)4 resultant : R(f ; g) � � � SL2-invariantf (x) = a0xn + a1xn�1 + � � �+ an�1x + an = a0Qni=1(x � �i )g(x) = b0xm + b1xm�1 + � � �+ bm�1x + bm = b0Qmj=1(x � �j)R(f ; g) := am0 bn0Qi ;j(�i � �j) � � � polynomial in a & bKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 6 / 24

Introdu tionThese are all polynomial invariants and


Introdu tionThese are all polynomial invariants andrelated to some group a tion


Introdu tionThese are all polynomial invariants andrelated to some group a tionHere is a summary:distan e On y Rn : orthogonal groupquadrati form Op;q y Rn : inde�nite orth groupdetX ; tra eX GLn y Mn : adjoint a tion�(f );R(f ; g) SL2 y C [x ; y ℄nHere GLn = fg : n � n-matrix j 9g�1 () det g 6= 0gOp;q = fg 2 GLn j kgxkp;q = kxkp;qg (n = p + q)where kxkp;q = x21 + � � � + x2p � x2p+1 � � � � � x2p+qSLn = fg 2 GLn j det g = 1g example of redu tive groupsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 7 / 24

Introdu tionAbstra t settingG : (linear) group y X � C N : algebrai a tion of G on X


Introdu tionAbstra t settingG : (linear) group y X � C N : algebrai a tion of G on XDe�nition (algebrai a tion)� : G � X ! X : polynomial fun tion s.t.1 �(e; x) = x2 �(gh; x) = �(g ; �(h; x))Notation: g � x = gx = �(g ; x)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 8 / 24

Introdu tionAbstra t settingG : (linear) group y X � C N : algebrai a tion of G on XDe�nition (algebrai a tion)� : G � X ! X : polynomial fun tion s.t.1 �(e; x) = x2 �(gh; x) = �(g ; �(h; x))Notation: g � x = gx = �(g ; x)Polynomial fun tions and invariants:C [X ℄ := ff : X ! C j f is polynomial fun tiong : ring of regular fun tionsC [X ℄G := ff 2 C [X ℄ j f (g�1 � x) = f (x) (8g 2 G )g : ring of invariantsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 8 / 24

Introdu tionAbstra t settingG : (linear) group y X � C N : algebrai a tion of G on XDe�nition (algebrai a tion)� : G � X ! X : polynomial fun tion s.t.1 �(e; x) = x2 �(gh; x) = �(g ; �(h; x))Notation: g � x = gx = �(g ; x)Polynomial fun tions and invariants:C [X ℄ := ff : X ! C j f is polynomial fun tiong : ring of regular fun tionsC [X ℄G := ff 2 C [X ℄ j f (g�1 � x) = f (x) (8g 2 G )g : ring of invariantsfun tions whi h are onstant along orbitsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 8 / 24

Introdu tionAbstra t settingG : (linear) group y X � C N : algebrai a tion of G on XDe�nition (algebrai a tion)� : G � X ! X : polynomial fun tion s.t.1 �(e; x) = x2 �(gh; x) = �(g ; �(h; x))Notation: g � x = gx = �(g ; x)Polynomial fun tions and invariants:C [X ℄ := ff : X ! C j f is polynomial fun tiong : ring of regular fun tionsC [X ℄G := ff 2 C [X ℄ j f (g�1 � x) = f (x) (8g 2 G )g : ring of invariantsfun tions whi h are onstant along orbits=) C [X ℄G is graded by degree of polynomials, i.e., graded algebraKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 8 / 24

Introdu tionWhy invariants?


Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati s


Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje ts


Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y X


Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,I homogeneous spa es, or prehomogeneous spa esKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,I homogeneous spa es, or prehomogeneous spa esI symmetri spa es, symmetri domains, et .Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,I homogeneous spa es, or prehomogeneous spa esI symmetri spa es, symmetri domains, et .Harmoni analysis relative to the a tion G y XKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,I homogeneous spa es, or prehomogeneous spa esI symmetri spa es, symmetri domains, et .Harmoni analysis relative to the a tion G y XI invariant integral, or Haar measureKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,I homogeneous spa es, or prehomogeneous spa esI symmetri spa es, symmetri domains, et .Harmoni analysis relative to the a tion G y XI invariant integral, or Haar measureI invariant di�erential operators (Lapla ian) and spheri al fun tionsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionWhy invariants?As we saw, invariants are ubiquitos in mathemati sIn parti ular...Classifying mathemati al obje tsUnderstanding the original a tion G y XI orbits,I homogeneous spa es, or prehomogeneous spa esI symmetri spa es, symmetri domains, et .Harmoni analysis relative to the a tion G y XI invariant integral, or Haar measureI invariant di�erential operators (Lapla ian) and spheri al fun tionsI Fourier transform, et .Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 9 / 24

Introdu tionFundamental problems of invariant theoryTwo lassi al problems ...


Introdu tionFundamental problems of invariant theoryTwo lassi al problems ...1 Find the ring generators f�igi2I � C [X ℄GQuestion : 9 �nite number of generators?Can hoose a good basis?FFT = First Fundamental Theorem


Introdu tionFundamental problems of invariant theoryTwo lassi al problems ...1 Find the ring generators f�igi2I � C [X ℄GQuestion : 9 �nite number of generators?Can hoose a good basis?FFT = First Fundamental Theorem2 Find all the relations among f�igi2IQuestion: trans ending degree?singularities?SFT = Se ond Fundamental TheoremKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 10 / 24

Introdu tionFundamental problems of invariant theoryTwo lassi al problems ...1 Find the ring generators f�igi2I � C [X ℄GQuestion : 9 �nite number of generators?Can hoose a good basis?FFT = First Fundamental Theorem2 Find all the relations among f�igi2IQuestion: trans ending degree?singularities?SFT = Se ond Fundamental TheoremThere are many kinds of answers � � �Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 10 / 24

Introdu tionFundamental problems of invariant theoryTwo lassi al problems ...1 Find the ring generators f�igi2I � C [X ℄GQuestion : 9 �nite number of generators?Can hoose a good basis?FFT = First Fundamental Theorem2 Find all the relations among f�igi2IQuestion: trans ending degree?singularities?SFT = Se ond Fundamental TheoremThere are many kinds of answers � � �Final GoalBetter understanding of C [X ℄G in geometri terms.Understanding of the original a tion G y X through it.Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 10 / 24

First Fundamental Theorem (FFT)Basi settingG : redu tive, linear algebrai group


First Fundamental Theorem (FFT)Basi settingG : redu tive, linear algebrai groupDe�nition (algebrai group)algebrai group = Zariski losed subgroup in GLN(C )Zariski losed = solutions of the system of polynomial equations


First Fundamental Theorem (FFT)Basi settingG : redu tive, linear algebrai groupDe�nition (algebrai group)algebrai group = Zariski losed subgroup in GLN(C )Zariski losed = solutions of the system of polynomial equationsDe�nition (redu tive)redu tive group = nilpotent radi al is trivial(if =k , k being algebrai ally losed, har k = 0)= 8 �nite dim representation is ompletely redu ible= 8 �nite dim repr is de omposed into the dire t sum of irredu iblesV : redu ible () V = U1 � U2 (9Ui : subrepresentation)irredu ibles = basi unit (atom) of representationKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 11 / 24

First Fundamental Theorem (FFT)Redu tive groupsExample (redu tive groups)T = (C �)m : torusGLn; SLn; On; SOn = On \ SLn; Sp2n : lassi al groupsG2; F4; E6; E7; E8 : ex eptional groups


First Fundamental Theorem (FFT)Redu tive groupsExample (redu tive groups)T = (C �)m : torusGLn; SLn; On; SOn = On \ SLn; Sp2n : lassi al groupsG2; F4; E6; E7; E8 : ex eptional groupsAlso we have a general ma hinery to produ e redu tive groupsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 12 / 24

First Fundamental Theorem (FFT)Redu tive groupsExample (redu tive groups)T = (C �)m : torusGLn; SLn; On; SOn = On \ SLn; Sp2n : lassi al groupsG2; F4; E6; E7; E8 : ex eptional groupsAlso we have a general ma hinery to produ e redu tive groupsTheorem1 Produ t of redu tive groups is redu tiveKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 12 / 24

First Fundamental Theorem (FFT)Redu tive groupsExample (redu tive groups)T = (C �)m : torusGLn; SLn; On; SOn = On \ SLn; Sp2n : lassi al groupsG2; F4; E6; E7; E8 : ex eptional groupsAlso we have a general ma hinery to produ e redu tive groupsTheorem1 Produ t of redu tive groups is redu tive2 G : redu tive, H � G : normal =) G=H redu tive (quotient)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 12 / 24

First Fundamental Theorem (FFT)Redu tive groupsExample (redu tive groups)T = (C �)m : torusGLn; SLn; On; SOn = On \ SLn; Sp2n : lassi al groupsG2; F4; E6; E7; E8 : ex eptional groupsAlso we have a general ma hinery to produ e redu tive groupsTheorem1 Produ t of redu tive groups is redu tive2 G : redu tive, H � G : normal =) G=H redu tive (quotient)3 G Æ : redu tive =) G : redu tive (G Æ : identity omponent)Extension by �nite group (#G=G Æ <1)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 12 / 24

First Fundamental Theorem (FFT)Finite generation of invariantsHere is one of the best answer to FFTTheorem (D. Hilbert 1990, 1993)G : redu tive y V = C n : ve tor spa e (linear repr)=) C [V ℄G : �nitely generated algebra =CKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 13 / 24

First Fundamental Theorem (FFT)Finite generation of invariantsHere is one of the best answer to FFTTheorem (D. Hilbert 1990, 1993)G : redu tive y V = C n : ve tor spa e (linear repr)=) C [V ℄G : �nitely generated algebra =CRemark9 ounter example for non-redu tive G: : : Nagata (1959) : Hilbert's 14th problemRe ent work by Mukai (2005) : : : Ri h examples of �nite generation evenwhen G is not redu tiveKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 13 / 24

First Fundamental Theorem (FFT)Example of a tions of �nite groups#G <1 =) G : redu tiveC [V ℄G = fR(f ) j R(f )(x) = 1#G Pg2G f (g�1x)gR(f ) : Reynolds operator (proje tion to invariants)


First Fundamental Theorem (FFT)Example of a tions of �nite groups#G <1 =) G : redu tiveC [V ℄G = fR(f ) j R(f )(x) = 1#G Pg2G f (g�1x)gR(f ) : Reynolds operator (proje tion to invariants)Theorem (E. Noether 1916)C [V ℄G : generated by polynomials of degree � #GKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 14 / 24

First Fundamental Theorem (FFT)Example of a tions of �nite groups#G <1 =) G : redu tiveC [V ℄G = fR(f ) j R(f )(x) = 1#G Pg2G f (g�1x)gR(f ) : Reynolds operator (proje tion to invariants)Theorem (E. Noether 1916)C [V ℄G : generated by polynomials of degree � #GMolien series : 1Pk=0 dim(C [V ℄Gk ) tk = 1#G Pg2G 1det(1� t�(g))Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 14 / 24

First Fundamental Theorem (FFT)Example of a tions of �nite groups#G <1 =) G : redu tiveC [V ℄G = fR(f ) j R(f )(x) = 1#G Pg2G f (g�1x)gR(f ) : Reynolds operator (proje tion to invariants)Theorem (E. Noether 1916)C [V ℄G : generated by polynomials of degree � #GMolien series : 1Pk=0 dim(C [V ℄Gk ) tk = 1#G Pg2G 1det(1� t�(g))TheoremG : a �nite re e tion groupf�1; : : : ;�lg � C [V ℄G : minimal homogeneous generators=) fdk = deg�k j 1 � k � lg : uniquely determined (exponents)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 14 / 24

First Fundamental Theorem (FFT)Rational invariants and Galois theoryTheoremG : �nite group =) C (V )G = Q(C [V ℄G ) : quotient �eld &[C (V ) : C (V )G ℄ = #GKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 15 / 24

First Fundamental Theorem (FFT)Rational invariants and Galois theoryTheoremG : �nite group =) C (V )G = Q(C [V ℄G ) : quotient �eld &[C (V ) : C (V )G ℄ = #GC (V ) : Galois extension of C (V )G with Galois group Gi.e., Study of C [V ℄G $ Galois theory for ringsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 15 / 24

First Fundamental Theorem (FFT)Example: symmetri group a tionG = Sn y V = C n : a tion by oordinate hangeC [V ℄G = fsymmetri polynomialsg : invariants


First Fundamental Theorem (FFT)Example: symmetri group a tionG = Sn y V = C n : a tion by oordinate hangeC [V ℄G = fsymmetri polynomialsg : invariantsGenerators of the ring of invariants C [V ℄G :felementary symm fun ek (1 � k � n)g nQi=1(1 + txi) = nPk=0 ek (x)tkKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 16 / 24

First Fundamental Theorem (FFT)Example: symmetri group a tionG = Sn y V = C n : a tion by oordinate hangeC [V ℄G = fsymmetri polynomialsg : invariantsGenerators of the ring of invariants C [V ℄G :felementary symm fun ek (1 � k � n)g nQi=1(1 + txi) = nPk=0 ek (x)tkfpower sum pk(1 � k � n)g pk(x) = nPi=1 xkiKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 16 / 24

First Fundamental Theorem (FFT)Example: symmetri group a tionG = Sn y V = C n : a tion by oordinate hangeC [V ℄G = fsymmetri polynomialsg : invariantsGenerators of the ring of invariants C [V ℄G :felementary symm fun ek (1 � k � n)g nQi=1(1 + txi) = nPk=0 ek (x)tkfpower sum pk(1 � k � n)g pk(x) = nPi=1 xkif omplete symm fun hk (1 � k � n)g nQi=1(1� txi)�1 = 1Pk=0 hk(x)tkKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 16 / 24

First Fundamental Theorem (FFT)Example: symmetri group a tionG = Sn y V = C n : a tion by oordinate hangeC [V ℄G = fsymmetri polynomialsg : invariantsGenerators of the ring of invariants C [V ℄G :felementary symm fun ek (1 � k � n)g nQi=1(1 + txi) = nPk=0 ek (x)tkfpower sum pk(1 � k � n)g pk(x) = nPi=1 xkif omplete symm fun hk (1 � k � n)g nQi=1(1� txi)�1 = 1Pk=0 hk(x)tkExponents f1; 2; : : : ; ngGneretors are alegbrai ally independentKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 16 / 24

First Fundamental Theorem (FFT)Example: symmetri group a tion ( ontinued)De�ne a qutient map � : V ! C n by �(v) = (e1(v); e2(v); : : : ; en(v))


First Fundamental Theorem (FFT)Example: symmetri group a tion ( ontinued)De�ne a qutient map � : V ! C n by �(v) = (e1(v); e2(v); : : : ; en(v))Lemma� : V ! C n is surje tive & Sn-invariantEvery �ber ��1(y) (y 2 C n ) is a single Sn-orbit


First Fundamental Theorem (FFT)Example: symmetri group a tion ( ontinued)De�ne a qutient map � : V ! C n by �(v) = (e1(v); e2(v); : : : ; en(v))Lemma� : V ! C n is surje tive & Sn-invariantEvery �ber ��1(y) (y 2 C n ) is a single Sn-orbitProof.� surje tive ( the fundamental theorem of algebra (Gauss's theorem)i.e., giving 8 oe� of the degree n equation,9 n-solutions ounting with multipli ity (( �ber)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 17 / 24

First Fundamental Theorem (FFT)Example: symmetri group a tion ( ontinued)De�ne a qutient map � : V ! C n by �(v) = (e1(v); e2(v); : : : ; en(v))Lemma� : V ! C n is surje tive & Sn-invariantEvery �ber ��1(y) (y 2 C n ) is a single Sn-orbitProof.� surje tive ( the fundamental theorem of algebra (Gauss's theorem)i.e., giving 8 oe� of the degree n equation,9 n-solutions ounting with multipli ity (( �ber)Thus we on lude V =Sn ' C n via the quotient map �� : V ! C n=Sn = C n : generi ally [Sn : 1℄ map (Galois overing)Generi �ber ' Sn, inherits regular representation of SnKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 17 / 24

First Fundamental Theorem (FFT)Orthogonal invariantsG = On y V = C n : ve tor representation (mult of matrix against ve tor)ProblemDes ribe the invariants for G y V � � � � � V = V�mU := C m =) V�m ' V U ' Mn;m oordinates xij on Mn;m


First Fundamental Theorem (FFT)Orthogonal invariantsG = On y V = C n : ve tor representation (mult of matrix against ve tor)ProblemDes ribe the invariants for G y V � � � � � V = V�mU := C m =) V�m ' V U ' Mn;m oordinates xij on Mn;mFFT for orthogonal invariants:Theorem (H. Weyl 1939)C [V�m ℄On = C [zij j 1 � i � j � m℄ where zij =Pnk=1 xkixkjX = (xij) 2 Mn;m =) Z = (zij) = tXX 2 SymmKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 18 / 24

First Fundamental Theorem (FFT)Orthogonal invariantsG = On y V = C n : ve tor representation (mult of matrix against ve tor)ProblemDes ribe the invariants for G y V � � � � � V = V�mU := C m =) V�m ' V U ' Mn;m oordinates xij on Mn;mFFT for orthogonal invariants:Theorem (H. Weyl 1939)C [V�m ℄On = C [zij j 1 � i � j � m℄ where zij =Pnk=1 xkixkjX = (xij) 2 Mn;m =) Z = (zij) = tXX 2 SymmExamplem = 1 : C [V ℄On = C [�℄ � = x21 + � � �+ x2nKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 18 / 24

First Fundamental Theorem (FFT)Contra tion invariantsG = GLn y V = C n =) C [V�m ℄G = C : trivial (NO invariants)


First Fundamental Theorem (FFT)Contra tion invariantsG = GLn y V = C n =) C [V�m ℄G = C : trivial (NO invariants)Right setting using V � : ontragredient (= dual) of VProblemDes ribe invariants for G y V�p � V ��qU := C p ;U 0 := C q =) V�p �V ��q ' V U �V � U 0 ' Mn;p �Mn;q oordinates xij on Mn;p , yij on Mn;qKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 19 / 24

First Fundamental Theorem (FFT)Contra tion invariantsG = GLn y V = C n =) C [V�m ℄G = C : trivial (NO invariants)Right setting using V � : ontragredient (= dual) of VProblemDes ribe invariants for G y V�p � V ��qU := C p ;U 0 := C q =) V�p �V ��q ' V U �V � U 0 ' Mn;p �Mn;q oordinates xij on Mn;p , yij on Mn;qFFT for ontra tion invariants:Theorem (H. Weyl 1939)C [V�p � V ��q℄GLn = C [zij j 1 � i � p; 1 � j � q℄ wherezij =Pnk=1 xkiykj : ontra tion of X and YX = (xij) 2 Mn;p;Y = (yij ) 2 Mn;q =) Z = (zij) = tXY 2 Mp;qKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 19 / 24

Se ond Fundamental Theorem (SFT)Se ond Fundamental Theorem = SFTdes ribing relations among generators ...


Se ond Fundamental Theorem (SFT)Se ond Fundamental Theorem = SFTdes ribing relations among generators ... easiest ase is:


Se ond Fundamental Theorem (SFT)Se ond Fundamental Theorem = SFTdes ribing relations among generators ... easiest ase is:Theorem (Shephard-Todd 1954)Assume #G <1 , G y V : linear representationC [V ℄G is a polynomial ring (no relations)() G is a pseudo-re e tion groupRemarks : pseudo-re e tion = 9U � V : (n � 1)-dim s.t. s��U = idUpseudo-re e tion group = �nite group generated by pseudo-re e tionsKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 20 / 24

Se ond Fundamental Theorem (SFT)Se ond Fundamental Theorem = SFTdes ribing relations among generators ... easiest ase is:Theorem (Shephard-Todd 1954)Assume #G <1 , G y V : linear representationC [V ℄G is a polynomial ring (no relations)() G is a pseudo-re e tion groupRemarks : pseudo-re e tion = 9U � V : (n � 1)-dim s.t. s��U = idUpseudo-re e tion group = �nite group generated by pseudo-re e tionsIf 9 relations, what we an do? NamelyProblemHow to des ribe relations among generators?Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 20 / 24

Se ond Fundamental Theorem (SFT)Return to the general situation G y X (X � C N )G : redu tive ; X : aÆne variety (solutions of polynomial equations)f�1; : : : ;�mg � C [X ℄G : generators of invariants


Se ond Fundamental Theorem (SFT)Return to the general situation G y X (X � C N )G : redu tive ; X : aÆne variety (solutions of polynomial equations)f�1; : : : ;�mg � C [X ℄G : generators of invariants =)Algebra morphism:�� : C [y1 ; : : : ; ym℄ 3 F (y) 7! F (�1; : : : ;�m) 2 C [X ℄G9 relation F (�1; : : : ;�m) � 0 () F (y) 2 Ker �� =: Ii.e., I = Ker �� C [y ℄ des ribes relations ompletelyKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 21 / 24

Se ond Fundamental Theorem (SFT)Return to the general situation G y X (X � C N )G : redu tive ; X : aÆne variety (solutions of polynomial equations)f�1; : : : ;�mg � C [X ℄G : generators of invariants =)Algebra morphism:�� : C [y1 ; : : : ; ym℄ 3 F (y) 7! F (�1; : : : ;�m) 2 C [X ℄G9 relation F (�1; : : : ;�m) � 0 () F (y) 2 Ker �� =: Ii.e., I = Ker �� C [y ℄ des ribes relations ompletelyTheorem (Hilbert's basis theorem)8 ideal I � C [y ℄ admits �nite # of generators fF1; : : : ;F`gNotationI = (F1; : : : ;F`) =Pj̀=1 C [y ℄Fj : ideal generated by fF1; : : : ;F`gKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 21 / 24

Se ond Fundamental Theorem (SFT)SFT des ribes the generators of relations fF1; : : : ;F`g ompletely,whi h are satis�ed by invariants f�1; : : : ;�mg


Se ond Fundamental Theorem (SFT)SFT des ribes the generators of relations fF1; : : : ;F`g ompletely,whi h are satis�ed by invariants f�1; : : : ;�mgExample (orthogonal invariants)Re all G = On y V�m ' Mn;m (V = C n ) oordinates xij on Mn;m =) orthog invariants : zij =Pnk=1 xkixkjX = (xij) 2 Mn;m =) Z = (zij) = tXX 2 Symm1 m � n =) fzijg : alg independent (no relation)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 22 / 24

Se ond Fundamental Theorem (SFT)SFT des ribes the generators of relations fF1; : : : ;F`g ompletely,whi h are satis�ed by invariants f�1; : : : ;�mgExample (orthogonal invariants)Re all G = On y V�m ' Mn;m (V = C n ) oordinates xij on Mn;m =) orthog invariants : zij =Pnk=1 xkixkjX = (xij) 2 Mn;m =) Z = (zij) = tXX 2 Symm1 m � n =) fzijg : alg independent (no relation)2 m > n =) Z = (zij) is of rank n(i.e., relations are (n + 1)-th minors in Symm)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 22 / 24

Geometri invariant theory (a �rst step)Geometri point of viewI � C [y ℄ : ideal of relations (prime) ! Y = fy 2 Cm j F (y) = 0 (8F 2 I )g � C m : variety


Geometri invariant theory (a �rst step)Geometri point of viewI � C [y ℄ : ideal of relations (prime) ! Y = fy 2 Cm j F (y) = 0 (8F 2 I )g � C m : varietyKey theorem:Theorem (Hilbert's Nullstellensatz)(redu ed ideals in C [y ℄) 3 I bije t !Y 2 (zero pt sets of polynomial equations) = (Z losed sets) � CmKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 23 / 24

Geometri invariant theory (a �rst step)Geometri point of viewI � C [y ℄ : ideal of relations (prime) ! Y = fy 2 Cm j F (y) = 0 (8F 2 I )g � C m : varietyKey theorem:Theorem (Hilbert's Nullstellensatz)(redu ed ideals in C [y ℄) 3 I bije t !Y 2 (zero pt sets of polynomial equations) = (Z losed sets) � CmI = I(Y) = fF 2 C [y ℄ j F ��Y � 0g � C [y ℄Y = V(I ) = fy 2 Cm j F (y) = 0 (8F 2 I )g � CmKyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 23 / 24

Geometri invariant theory (a �rst step)Geometri point of viewI � C [y ℄ : ideal of relations (prime) ! Y = fy 2 Cm j F (y) = 0 (8F 2 I )g � C m : varietyKey theorem:Theorem (Hilbert's Nullstellensatz)(redu ed ideals in C [y ℄) 3 I bije t !Y 2 (zero pt sets of polynomial equations) = (Z losed sets) � CmI = I(Y) = fF 2 C [y ℄ j F ��Y � 0g � C [y ℄Y = V(I ) = fy 2 Cm j F (y) = 0 (8F 2 I )g � CmI : prime ideal ! Y : irredu ible(i.e., Y = Y1 [ Y2 (Yi : Z losed) =) Y = Y1 or Y = Y2)Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 23 / 24

Geometri invariant theory (a �rst step)Con lusion:SFT = des ribe algebrai variety de�ned by relationsamong invariantsTo be ontinued...


rep - 青山学院大学kyo/preprint/ohp1_karnataka2007.pdf1 intro duction 2 first f undamental theo...

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