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

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 es

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)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...

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 understand

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 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?

Kyo 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 invariant

Kyo 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 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)

Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 6 / 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)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

Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 7 / 24

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

Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 7 / 24

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

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)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?

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 s

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 ts

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 X

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,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 ...

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 Theorem

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 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

Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 11 / 24

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

Kyo Nishiyama (Kyoto Univ.) Group Representations and Invariant Theory 2007/09/27 11 / 24

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

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 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)

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 � #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

Kyo 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)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))

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-orbit

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)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

Kyo 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 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)

Kyo 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;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 ...

Kyo 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:

Kyo 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 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

Kyo 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 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

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)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

Kyo 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) � 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...

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