Invariant theoryFrom Wikipedia, the free encyclopedia

Contents

1 Bracket algebra 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Bracket ring 22.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 Canonizant 33.1 Canonizants of a binary form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4 Capellis identity 44.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Relations with representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.2.1 Case m = 1 and representation Sk Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2.2 The universal enveloping algebra U(gln) and its center . . . . . . . . . . . . . . . . . . . 64.2.3 General m and dual pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.1 Turnbulls identity for symmetric matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3.2 The HoweUmedaKostantSahi identity for antisymmetric matrices . . . . . . . . . . . . 104.3.3 The CaraccioloSportielloSokal identity for Manin matrices . . . . . . . . . . . . . . . . 104.3.4 The MukhinTarasovVarchenko identity and the Gaudin model . . . . . . . . . . . . . . 104.3.5 Permanents, immanants, traces higher Capelli identities . . . . . . . . . . . . . . . . . 11

4.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Catalecticant 145.1 Binary forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Catalecticants of quartic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Cayleys process 16

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6.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 ChevalleyIwahoriNagata theorem 187.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 ChevalleyShephardTodd theorem 198.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9 Dierential invariant 219.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

10 Evectant 2310.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

11 Geometric invariant theory 2411.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2411.2 Mumfords book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2511.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2711.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

12 Glossary of invariant theory 2812.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.2 !$@ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2912.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.5 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3012.6 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.7 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.8 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.9 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.10H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.11I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.12J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

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12.13K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3412.14L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.15M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.16N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.17O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3512.18P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.19Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.20R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.21S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3812.22T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.23U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.24V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.25W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.26XYZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.27See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4012.29External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

13 Grams theorem 4213.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

14 Grbner basis 4314.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

14.1.1 Polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4314.1.2 Monomial ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4414.1.3 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

14.2 Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4514.3 Example and counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.4 Properties and applications of Grbner bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

14.4.1 Equality of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.4.2 Membership and inclusion of ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4714.4.3 Solutions of a system of algebraic equations . . . . . . . . . . . . . . . . . . . . . . . . . 4714.4.4 Dimension, degree and Hilbert series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4.5 Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.4.6 Intersecting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4.7 Implicitization of a rational curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4.8 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.4.9 Eective Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5014.4.10 Implicitization in higher dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

14.5 Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5114.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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14.8 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.9 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

15 Haboushs theorem 5415.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5415.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

16 Hall algebra 5616.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5616.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

17 Hermite reciprocity 5817.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

18 Hilberts basis theorem 5918.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

18.2.1 First Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5918.2.2 Second Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

18.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.4 Mizar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6118.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

19 Hilberts fourteenth problem 6219.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.2 Zariskis formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6219.3 Nagatas counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6319.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

20 Hilberts syzygy theorem 6420.1 Formal statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6420.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

21 HilbertMumford criterion 6521.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

22 Hodge bundle 6622.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6622.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

23 Invariant estimator 67

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23.1 General setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.1.2 Some classes of invariant estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6723.1.3 Optimal invariant estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.1.4 In classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

23.2 Mathematical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.2.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6823.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.2.3 Example: Location parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6923.2.4 Pitman estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

23.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

24 Invariant of a binary form 7124.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7124.3 The ring of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

24.3.1 Covariants of a binary linear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.2 Covariants of a binary quadric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.3 Covariants of a binary cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.4 Covariants of a binary quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7224.3.5 Covariants of a binary quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.6 Covariants of a binary sextic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.7 Covariants of a binary septic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.8 Covariants of a binary octavic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.9 Covariants of a binary nonic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.10 Covariants of a binary decimic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.11 Covariants of a binary undecimic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7324.3.12 Covariants of a binary duodecimic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

24.4 Invariants of several binary forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.1 Covariants of two linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.2 Covariants of a linear form and a quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.3 Covariants of a linear form and a cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.4 Covariants of a linear form and a quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.5 Covariants of a linear form and a quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.6 Covariants of a linear form and a quantic . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.7 Covariants of several linear forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.8 Covariants of two quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.9 Covariants of two quadratics and a linear form . . . . . . . . . . . . . . . . . . . . . . . . 7424.4.10 Covariants of several linear and quadratic forms . . . . . . . . . . . . . . . . . . . . . . . 7424.4.11 Covariants of a quadratic and a cubic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.12 Covariants of a quadratic and a quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.13 Covariants of a quadratic and a quintic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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24.4.14 Covariants of a cubic and a quartic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.15 Covariants of two quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.4.16 Covariants of many cubics or quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

24.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7524.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

25 Invariant polynomial 7725.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

26 Invariant theory 7826.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7826.2 The nineteenth-century origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.3 Hilberts theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7926.4 Geometric invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8026.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8126.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

27 Invariants of tensors 8227.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8227.2 Calculation of the invariants of symmetric 33 tensors . . . . . . . . . . . . . . . . . . . . . . . . 8227.3 Engineering application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8327.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

28 KempfNess theorem 8428.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

29 Kostant polynomial 8529.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8529.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8529.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8629.4 Steinberg basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8829.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

30 LittlewoodRichardson rule 9030.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

30.1.1 LittlewoodRichardson tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9130.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9130.1.3 A more geometrical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

30.2 An algorithmic form of the rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9230.3 LittlewoodRichardson coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9230.4 Generalizations and special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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30.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9430.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9530.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

31 Modular invariant theory 9731.1 Dickson invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9731.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9731.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

32 Moduli space 9932.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9932.2 Basic examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

32.2.1 Projective space and Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.2.2 Chow variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.2.3 Hilbert scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

32.3 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.3.1 Fine moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10032.3.2 Coarse moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.3.3 Moduli stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

32.4 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.4.1 Moduli of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10132.4.2 Moduli of varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.4.3 Moduli of vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

32.5 Methods for constructing moduli spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10232.6 In physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10332.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

33 Molien series 10533.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.2 Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10533.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

34 Newtons identities 10734.1 Mathematical statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

34.1.1 Formulation in terms of symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . 10734.1.2 Application to the roots of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 10834.1.3 Application to the characteristic polynomial of a matrix . . . . . . . . . . . . . . . . . . . 10934.1.4 Relation with Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

34.2 Related identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11034.2.1 A variant using complete homogeneous symmetric polynomials . . . . . . . . . . . . . . . 11034.2.2 Expressing elementary symmetric polynomials in terms of power sums . . . . . . . . . . . 110

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34.2.3 Expressing complete homogeneous symmetric polynomials in terms of power sums . . . . 11034.2.4 Expressing power sums in terms of elementary symmetric polynomials . . . . . . . . . . . 11134.2.5 Expressing power sums in terms of complete homogeneous symmetric polynomials . . . . 11134.2.6 Expressions as determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

34.3 Derivation of the identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11334.3.1 From the special case n = k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11334.3.2 Comparing coecients in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11434.3.3 As a telescopic sum of symmetric function identities . . . . . . . . . . . . . . . . . . . . 115

34.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11534.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11534.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

35 Nullform 11735.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

36 Osculant 11836.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

37 Perpetuant 11937.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

38 Polynomial ring 12138.1 The polynomial ring K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

38.1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12138.1.2 Degree of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12238.1.3 Properties of K[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12238.1.4 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

38.2 Polynomial evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12438.3 The polynomial ring in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

38.3.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12438.3.2 The polynomial ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12538.3.3 Hilberts Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

38.4 Properties of the ring extension R R[X] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12638.4.1 Summary of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

38.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12638.5.1 Innitely many variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12638.5.2 Generalized exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12738.5.3 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12738.5.4 Noncommutative polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12838.5.5 Dierential and skew-polynomial rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

38.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12838.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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39 Quantum invariant 13039.1 List of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13039.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13139.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13139.4 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13139.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

40 Quaternary cubic 13340.1 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13340.2 Sylvester pentahedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13340.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13340.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

41 Quippian 13541.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13541.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

42 Radical polynomial 13642.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

43 Reynolds operator 13743.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

43.1.1 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13743.1.2 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13743.1.3 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

43.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13843.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

44 Ring of symmetric functions 14044.1 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14044.2 The ring of symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

44.2.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14144.2.2 Dening individual symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 14244.2.3 A principle relating symmetric polynomials and symmetric functions . . . . . . . . . . . . 143

44.3 Properties of the ring of symmetric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14344.3.1 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14344.3.2 Structural properties of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14444.3.3 Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

44.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14544.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

45 Schur polynomial 14645.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14645.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

x CONTENTS

45.2.1 JacobiTrudy identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14745.2.2 The Giambelli identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14745.2.3 The MurnaghanNakayama rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14845.2.4 The Littlewood-Richardson rule and Pieris formula . . . . . . . . . . . . . . . . . . . . . 14845.2.5 Specializations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

45.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14845.4 Relation to representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.5 Skew Schur functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14945.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

45.6.1 Double Schur polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15045.6.2 Factorial Schur polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

45.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15145.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

46 Semi-invariant of a quiver 15246.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

46.1.1 Polynomial invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15246.1.2 Semi-invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

46.2 Characterization of representation type through semi-invariant theory . . . . . . . . . . . . . . . . 15446.2.1 SatoKimura theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15446.2.2 SkowronskiWeyman theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

46.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

47 Standard monomial theory 15647.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15647.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15647.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15747.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

48 Symbolic method 15948.1 Symbolic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

48.1.1 Example: the discriminant of a binary quadratic form . . . . . . . . . . . . . . . . . . . . 15948.1.2 Higher degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16048.1.3 More variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

48.2 Symmetric products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16048.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16148.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

49 Ternary cubic 16249.1 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

49.1.1 The ring of invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16249.1.2 The ring of covariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16249.1.3 The ring of contravariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

CONTENTS xi

49.1.4 The ring of concomitants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16349.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16349.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

50 Ternary quartic 16450.1 Hilberts theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16450.2 Invariant theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16450.3 Catalecticant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16450.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16450.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16450.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

51 The Classical Groups 16651.1 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16651.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

52 Trace identity 16852.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16852.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16852.3 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16852.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16852.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

53 Transvectant 16953.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16953.2 Partial transvectants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16953.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16953.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

54 YoungDeruyts development 17054.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17054.2 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 171

54.2.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17154.2.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17354.2.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Chapter 1

Bracket algebra

In mathematics, a bracket algebra is an algebraic system that connects the notion of a supersymmetry algebra witha symbolic representation of projective invariants.Given that L is a proper signed alphabet and Super[L] is the supersymmetric algebra, the bracket algebra Bracket[L]of dimension n over the eld K is the quotient of the algebra Brace{L} obtained by imposing the congruence relationsbelow, where w, w', ..., w" are any monomials in Super[L]:

1. {w} = 0 if length(w) n2. {w}{w'}...{w"} = 0whenever any positive letter a ofL occursmore than n times in themonomial {w}{w'}...{w"}.

3. Let {w}{w'}...{w"} be a monomial in Brace{L} in which some positive letter a occurs more than n times, andlet b, c, d, e, ..., f, g be any letters in L.

1.1 See also Bracket ring

1.2 References Anick, David; Rota, Gian-Carlo (September 15, 1991), Higher-Order Syzygies for the Bracket Algebra andfor the Ring of Coordinates of the Grassmanian, Proceedings of the National Academy of Sciences 88 (18):80878090, doi:10.1073/pnas.88.18.8087, ISSN 0027-8424, JSTOR 2357546.

Huang, Rosa Q.; Rota, Gian-Carlo; Stein, Joel A. (1990), Supersymmetric Bracket Algebra and Invariant The-ory,Acta ApplicandaeMathematicae (Kluwer Academic Publishers) 21: 193246, doi:10.1007/BF00053298.

1

Chapter 2

Bracket ring

In mathematics, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d by dminors of a generic d by n matrix (xij).The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plckerembedding.[1]

For given d n we dene as formal variables the brackets [1 2 ... d] with the taken from {1,...,n}, subject to [12 ... d] = [2 1 ... d] and similarly for other transpositions. The set (n,d) of size

nd

generates a polynomial

ring K[(n,d)] over a eld K. There is a homomorphism (n,d) from K[(n,d)] to the polynomial ring K[xi,j] innd indeterminates given by mapping [1 2 ... d] to the determinant of the d by d matrix consisting of the columnsof the xi,j indexed by the . The bracket ring B(n,d) is the image of . The kernel I(n,d) of encodes the relationsor syzygies that exist between the minors of a generic n by d matrix. The projective variety dened by the ideal I isthe (nd)d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensionalspace.[2]

To compute with brackets it is necessary to determine when an expression lies in the ideal I(n,d). This is achievedby a straightening law due to Young (1928).[3]

2.1 See also Bracket algebra

2.2 References[1] Bjrner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Gnter (1999), Oriented matroids, Ency-

clopedia of Mathematics and Its Applications 46 (2nd ed.), Cambridge University Press, p. 79, ISBN 0-521-77750-X, Zbl0944.52006

[2] Sturmfels (2008) pp.7879

[3] Sturmfels (2008) p.80

Dieudonn, Jean A.; Carrell, James B. (1970), Invariant theory, old and new, Advances in Mathematics 4:180, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525, Zbl 0196.05802

Dieudonn, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102, Zbl 0258.14011

Sturmfels, Bernd (2008), Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation(2nd ed.), Springer-Verlag, ISBN 3211774165, Zbl 1154.13003

Sturmfels, Bernd; White, Neil (1990), Stanley decompositions of the bracket ring,Mathematica Scandinavica67 (2): 183189, ISSN 0025-5521, MR 1096453, Zbl 0727.13005

2

Chapter 3

Canonizant

In mathematical invariant theory, the canonizant or canonisant is a covariant of forms related to a canonical formfor them.

3.1 Canonizants of a binary formThe canonizant of a binary form of degree 2n 1 is a covariant of degree n and order n, given by the catalecticant ofthe penultimate emanant, which is the determinant of the n by n Hadamard matrix with entries aijx + aijy for 0 i,j < n.

3.2 References Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. ClarendonPress, JFM 26.0135.01

3

Chapter 4

Capellis identity

In mathematics, Capellis identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) =det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebragln . It can be used to relate an invariant to the invariant , where is Cayleys process.

4.1 StatementSuppose that xij for i,j = 1,...,n are commuting variables. Write E for the polarization operator

Eij =

nXa=1

xia@

@xja:

The Capelli identity states that the following dierential operators, expressed as determinants, are equal:

E11 + n 1 E1;n1 E1n

... . . . ... ...En1;1 En1;n1 + 1 En1;nEn1 En;n1 Enn + 0

=x11 x1n... . . . ...

xn1 xnn

@@x11

@@x1n... . . . ...@

@xn1 @@xnn

:Both sides are dierential operators. The determinant on the left has non-commuting entries, and is expanded withall terms preserving their left to right order. Such a determinant is often called a column-determinant, since it canbe obtained by the column expansion of the determinant starting from the rst column. It can be formally written as

det(A) =X2Sn

sgn()A(1);1A(2);2 A(n);n;

where in the product rst come the elements from the rst column, then from the second and so on. The determinanton the far right is Cayleys omega process, and the one on the left is the Capelli determinant.The operators E can be written in a matrix form:

E = XDt;

where E;X;D are matrices with elements E , x, @@xij respectively. If all elements in these matrices would becommutative then clearly det(E) = det(X) det(Dt) . The Capelli identity shows that despite noncommutativitythere exists a quantization of the formula above. The only price for the noncommutivity is a small correction:(n i)ij on the left hand side. For generic noncommutative matrices formulas like

4

4.2. RELATIONS WITH REPRESENTATION THEORY 5

det(AB) = det(A) det(B)

do not exist, and the notion of the 'determinant' itself does not make sense for generic noncommutative matrices.That is why the Capelli identity still holds some mystery, despite many proofs oered for it. A very short proof doesnot seem to exist. Direct verication of the statement can be given as an exercise for n' = 2, but is already long for n= 3.

4.2 Relations with representation theoryConsider the following slightly more general context. Suppose that n and m are two integers and xij for i =1; : : : ; n; j = 1; : : : ;m , be commuting variables. Redene Eij by almost the same formula:

Eij =

mXa=1

xia@

@xja:

with the only dierence that summation index a ranges from 1 tom . One can easily see that such operators satisfythe commutation relations:

[Eij ; Ekl] = jkEil ilEkj :Here [a; b] denotes the commutator ab ba . These are the same commutation relations which are satised by thematrices eij which have zeros everywhere except the position (i; j) , where 1 stands. ( eij are sometimes calledmatrix units). Hence we conclude that the correspondence : eij 7! Eij denes a representation of the Lie algebragln in the vector space of polynomials of xij .

4.2.1 Case m = 1 and representation Sk Cn

It is especially instructive to consider the special case m = 1; in this case we have xi1, which is abbreviated as xi:

Eij = xi@

@xj:

In particular, for the polynomials of the rst degree it is seen that:

Eijxk = jkxi:

Hence the action of Eij restricted to the space of rst-order polynomials is exactly the same as the action of matrixunits eij on vectors in Cn . So, from the representation theory point of view, the subspace of polynomials of rstdegree is a subrepresentation of the Lie algebra gln , which we identied with the standard representation in Cn .Going further, it is seen that the dierential operators Eij preserve the degree of the polynomials, and hence thepolynomials of each xed degree form a subrepresentation of the Lie algebra gln . One can see further that the spaceof homogeneous polynomials of degree k can be identied with the symmetric tensor power SkCn of the standardrepresentation Cn .One can also easily identify the highest weight structure of these representations. The monomial xk1 is a highest weightvector, indeed: Eijxk1 = 0 for i < j. Its highest weight equals to (k, 0, ... ,0), indeed: Eiixk1 = ki1xk1 .Such representation is sometimes called bosonic representation of gln . Similar formulas Eij = i @@ j dene theso-called fermionic representation, here i are anti-commuting variables. Again polynomials of k-th degree form anirreducible subrepresentation which is isomorphic to kCn i.e. anti-symmetric tensor power of Cn . Highest weightof such representation is (0, ..., 0, 1, 0, ..., 0). These representations for k = 1, ..., n are fundamental representationsof gln .

6 CHAPTER 4. CAPELLIS IDENTITY

Capelli identity for m = 1

Let us return to the Capelli identity. One can prove the following:

det(E + (n i)ij) = 0; n > 1the motivation for this equality is the following: consider Ecij = xipj for some commuting variables xi; pj . Thematrix Ec is of rank one and hence its determinant is equal to zero. Elements of matrix E are dened by thesimilar formulas, however, its elements do not commute. The Capelli identity shows that the commutative identity:det(Ec) = 0 can be preserved for the small price of correcting matrix E by (n i)ij .Let us also mention that similar identity can be given for the characteristic polynomial:

det(t+ E + (n i)ij) = t[n] + Tr(E)t[n1];where t[k] = t(t+1) (t+k1) . The commutative counterpart of this is a simple fact that for rank = 1 matricesthe characteristic polynomial contains only the rst and the second coecients.Let us consider an example for n = 2.

t+ E11 + 1 E12E21 t+ E22 = t+ x1@1 + 1 x1@2x2@1 t+ x2@2

= (t+ x1@1 + 1)(t+ x2@2) x2@1x1@2= t(t+ 1) + t(x1@1 + x2@2) + x1@1x2@2 + x2@2 x2@1x1@2Using

@1x1 = x1@1 + 1; @1x2 = x2@1; x1x2 = x2x1

we see that this is equal to:

t(t+ 1) + t(x1@1 + x2@2) + x2x1@1@2 + x2@2 x2x1@1@2 x2@2

= t(t+ 1) + t(x1@1 + x2@2) = t[2] + tTr(E):

4.2.2 The universal enveloping algebra U(gln) and its centerAn interesting property of the Capelli determinant is that it commutes with all operators Eij, that is the commutator[Eij ; det(E + (n i)ij)] = 0 is equal to zero. It can be generalized:Consider any elements Eij in any ring, such that they satisfy the commutation relation [Eij ; Ekl] = jkEil ilEkj, (so they can be dierential operators above, matrix units eij or any other elements) dene elements Ck as follows:

det(t+ E + (n i)ij) = t[n] +X

k=n1;:::;0t[k]Ck;

where t[k] = t(t+ 1) (t+ k 1);then:

elements Ck commute with all elements Eij

elements Ck can be given by the formulas similar to the commutative case:

4.2. RELATIONS WITH REPRESENTATION THEORY 7

Ck =X

I=(i1

8 CHAPTER 4. CAPELLIS IDENTITY

CapelliCauchyBinet identitiesFor general m matrix E is given as product of the two rectangular matrices: X and transpose to D. If all elements ofthese matrices would commute then one knows that the determinant of E can be expressed by the so-called CauchyBinet formula via minors of X and D. An analogue of this formula also exists for matrix E again for the same mildprice of the correction E ! (E + (n i)ij) :

det(E + (n i)ij) =X

I=(1i1

4.3. GENERALIZATIONS 9

The only dierential operators which commute with Eij are polynomials in Edualij , and vice versa.

Decomposition of the vector space of polynomials into a direct sum of tensor products of irreducible repre-sentations of GLn and GLm can be given as follows:

C[xij ] = S(Cn Cm) =XD

Dn D0

m :

The summands are indexed by the Young diagrams D, and representations D are mutually non-isomorphic. Anddiagram D determine D0 and vice versa.

In particular the representation of the big group GLn GLm is multiplicity free, that is each irreduciblerepresentation occurs only one time.

One easily observe the strong similarity to SchurWeyl duality.

4.3 GeneralizationsMuch work have been done on the identity and its generalizations. Approximately two dozens of mathematicians andphysicists contributed to the subject, to name a few: R. Howe, B. Kostant[1][2] Fields medalist A. Okounkov[3][4] A.Sokal,[5] D. Zeilberger.[6]

It seems historically the rst generalizations were obtained by Herbert Westren Turnbull in 1948,[7] who found thegeneralization for the case of symmetric matrices (see[5][6] for modern treatments).The other generalizations can be divided into several patterns. Most of them are based on the Lie algebra point of view.Such generalizations consist of changing Lie algebra gln to simple Lie algebras [8] and their super[9][10] (q),[11][12] andcurrent versions.[13] As well as identity can be generalized for dierent reductive dual pairs.[14][15] And nally one canconsider not only the determinant of the matrix E, but its permanent,[16] trace of its powers and immanants.[3][4][17][18]Let us mention few more papers;[19][20][21] [22] [23] [24] [25] still the list of references is incomplete. It has been believedfor quite a long time that the identity is intimately related with semi-simple Lie algebras. Surprisingly a new purelyalgebraic generalization of the identity have been found in 2008[5] by S. Caracciolo, A. Sportiello, A. D. Sokal whichhas nothing to do with any Lie algebras.

4.3.1 Turnbulls identity for symmetric matrices

Consider symmetric matrices

X =

x11 x12 x13 x1nx12 x22 x23 x2nx13 x23 x33 x3n... ... ... . . . ...

x1n x2n x3n xnn

; D =

2 @@x11@

@x12@

@x13 @@x1n

@@x12

2 @@x22@

@x23 @@x2n

@@x13

@@x23

2 @@x33 @@x3n... ... ... . . . ...@

@x1n@

@x2n@

@x3n 2 @@xnn

Herbert Westren Turnbull[7] in 1948 discovered the following identity:

det(XD + (n i)ij) = det(X) det(D)

Combinatorial proof can be found in the paper,[6] another proof and amusing generalizations in the paper,[5] see alsodiscussion below.

10 CHAPTER 4. CAPELLIS IDENTITY

4.3.2 The HoweUmedaKostantSahi identity for antisymmetric matricesConsider antisymmetric matrices

X =

0 x12 x13 x1nx12 0 x23 x2nx13 x23 0 x3n... ... ... . . . ...

x1n x2n x3n 0

; D =

0 @@x12@

@x13 @@x1n

@@x12 0 @@x23 @@x2n @@x13 @@x23 0 @@x3n

... ... ... . . . ... @@x1n @@x2n @@x3n 0

:

Then

det(XD + (n i)ij) = det(X) det(D):

4.3.3 The CaraccioloSportielloSokal identity for Manin matricesConsider two matrices M and Y over some associative ring which satisfy the following condition

[Mij ; Ykl] = jkQilfor some elements Qil. Or in words: elements in j-th column ofM commute with elements in k-th row of Y unlessj = k, and in this case commutator of the elements Mik and Ykl depends only on i, l, but does not depend on k.Assume that M is a Manin matrix (the simplest example is the matrix with commuting elements).Then for the square matrix case

det(MY +Q diag(n 1; n 2; : : : ; 1; 0)) = det(M) det(Y ):Here Q is a matrix with elements Qil, and diag(n 1, n 2, ..., 1, 0) means the diagonal matrix with the elements n 1, n 2, ..., 1, 0 on the diagonal.See [5] proposition 1.2' formula (1.15) page 4, our Y is transpose to their B.Obviously the original Cappelis identity the particular case of this identity. Moreover from this identity one can seethat in the original Capellis identity one can consider elements

@

@xij+ fij(x11; : : : ; xkl; : : : )

for arbitrary functions j and the identity still will be true.

4.3.4 The MukhinTarasovVarchenko identity and the Gaudin modelStatement

Consider matrices X and D as in Capellis identity, i.e. with elements xij and @ij at position (ij).Let z be another formal variable (commuting with x). Let A and B be some matrices which elements are complexnumbers.

det@

@zAX 1

z BDt

4.3. GENERALIZATIONS 11

= detcommute all if as calculateall Putx and zright the on derivations all while left, the on@

@zAX 1

z BDt

Here the rst determinant is understood (as always) as column-determinant of a matrix with non-commutative entries.The determinant on the right is calculated as if all the elements commute, and putting all x and z on the left, whilederivations on the right. (Such recipe is called a Wick ordering in the quantum mechanics).

The Gaudin quantum integrable system and Talalaevs theorem

The matrix

L(z) = A+X1

z BDt

is a Lax matrix for the Gaudin quantum integrable spin chain system. D. Talalaev solved the long-standing problemof the explicit solution for the full set of the quantum commuting conservation laws for the Gaudin model, discoveringthe following theorem.Consider

det@

@z L(z)

=

nXi=0

Hi(z)

@

@z

i:

Then for all i,j,z,w

[Hi(z);Hj(w)] = 0;

i.e. Hi(z) are generating functions in z for the dierential operators in x which all commute. So they provide quantumcommuting conservation laws for the Gaudin model.

4.3.5 Permanents, immanants, traces higher Capelli identities

The original Capelli identity is a statement about determinants. Later, analogous identities were found for permanents,immanants and traces. Based on the combinatorial approach paper by S.G. Williamson [26] was one of the rst resultsin this direction.

Turnbulls identity for permanents of antisymmetric matrices

Consider the antisymmetric matrices X and D with elements xij and corresponding derivations, as in the case of theHUKS identity above.Then

perm(XtD (n i)ij) = permcommute all if as Calculateall Putxright the on derivations all with left, the on (XtD):

Let us cite:[6] "...is stated without proof at the end of Turnbulls paper. The authors themselves follow Turnbull atthe very end of their paper they write:Since the proof of this last identity is very similar to the proof of Turnbulls symmetric analog (with a slight twist),we leave it as an instructive and pleasant exercise for the reader..The identity is deeply analyzed in paper .[27]

12 CHAPTER 4. CAPELLIS IDENTITY

4.4 References[1] Kostant, B.; Sahi, S. (1991), The Capelli Identity, tube domains, and the generalized Laplace transform, Advances in

Math. 87: 7192, doi:10.1016/0001-8708(91)90062-C

[2] Kostant, B.; Sahi, S. (1993), Jordan algebras andCapelli identities, InventionesMathematicae 112 (1): 7192, doi:10.1007/BF01232451

[3] Okounkov, A. (1996), Quantum Immanants and Higher Capelli Identities, arXiv:q-alg/9602028

[4] Okounkov, A. (1996), Young Basis, Wick Formula, and Higher Capelli Identities, arXiv:q-alg/9602027

[5] Caracciolo, S.; Sportiello, A.; Sokal, A. (2008), Noncommutative determinants, CauchyBinet formulae, and Capelli-typeidentities. I. Generalizations of the Capelli and Turnbull identities, arXiv:0809.3516

[6] Foata, D.; Zeilberger, D. (1993), Combinatorial Proofs of Capellis and Turnbulls Identities from Classical Invariant Theory,arXiv:math/9309212

[7] Turnbull, HerbertWestren (1948), Symmetric determinants and the Cayley and Capelli operators, Proc. EdinburghMath.Soc. 8 (2): 7686, doi:10.1017/S0013091500024822

[8] Molev, A.; Nazarov, M. (1997), Capelli Identities for Classical Lie Algebras, arXiv:q-alg/9712021

[9] Molev, A. (1996), Factorial supersymmetric Schur functions and super Capelli identities, arXiv:q-alg/9606008

[10] Nazarov, M. (1996), Capelli identities for Lie superalgebras, arXiv:q-alg/9610032

[11] Noumi, M.; Umeda, T.; Wakayma, M. (1994), A quantum analogue of the Capelli identity and an elementary dierentialcalculus on GLq(n)", Duke Mathematical Journal 76 (2): 567594, doi:10.1215/S0012-7094-94-07620-5

[12] Noumi, M.; Umeda, T.; Wakayma, M. (1996), Dual pairs, spherical harmonics and a Capelli identity in quantum grouptheory, Compositio Mathematica 104 (2): 227277

[13] Mukhin, E.; Tarasov, V.; Varchenko, A. (2006), A generalization of the Capelli identity, arXiv:math.QA/0610799

[14] Itoh,M. (2004), Capelli identities for reductive dual pairs,Advances inMathematics 194 (2): 345397, doi:10.1016/j.aim.2004.06.010

[15] Itoh, M. (2005), Capelli Identities for the dual pair ( O M, Sp N)", Mathematische Zeitschrift 246 (12): 125154,doi:10.1007/s00209-003-0591-2

[16] Nazarov, M. (1991), Quantum Berezinian and the classical Capelli identity, Letters in Mathematical Physics 21 (2): 123131, doi:10.1007/BF00401646

[17] Nazarov, M. (1996), Yangians and Capelli identities, arXiv:q-alg/9601027

[18] Molev, A. (1996), A Remark on the Higher Capelli Identities, arXiv:q-alg/9603007

[19] Kinoshita, K.; Wakayama, M. (2002), Explicit Capelli identities for skew symmetric matrices, Proceedings of the Edin-burgh Mathematical Society 45 (2): 449465, doi:10.1017/S0013091500001176

[20] Hashimoto, T. (2008),Generating function for GLn-invariant dierential operators in the skewCapelli identity, arXiv:0803.1339

[21] Nishiyama, K.; Wachi, A. (2008), A note on the Capelli identities for symmetric pairs of Hermitian type, arXiv:0808.0607

[22] Umeda, Toru (2008), On the proof of the Capelli identities, Funkcialaj Ekvacioj 51 (1): 115, doi:10.1619/fesi.51.1

[23] Brini, A; Teolis, A (1993), Capellis theory, Koszulmaps, and superalgebras, PNAS 90 (21): 1024510249, doi:10.1073/pnas.90.21.10245

[24] Koszul, J (1981), Les algebres de Lie gradues de type sl (n, 1) et l'oprateur de A. Capelli, C.R. Acad. Sci. Paris (292):139141

[25] Orsted, B; Zhang, G (2001), Capelli identity and relative discrete series of line bundles over tube domains

[26] Williamson, S. (1981), Symmetry operators, polarizations, and a generalized Capelli identity, Linear & Multilinear Al-gebra 10 (2): 93102, doi:10.1080/03081088108817399

[27] Umeda, Toru (2000), On Turnbull identity for skew-symmetric matrices, Proc. Edinburgh Math. Soc. 43 (2): 379393,doi:10.1017/S0013091500020988

4.5. FURTHER READING 13

4.5 Further reading Capelli, Alfredo (1887), Ueber die Zurckfhrung der Cayleyschen Operation auf gewhnliche Polar-Operationen,Mathematische Annalen (Berlin / Heidelberg: Springer) 29 (3): 331338, doi:10.1007/BF01447728,ISSN 1432-1807

Howe, Roger (1989), Remarks on classical invariant theory, Transactions of the American MathematicalSociety (American Mathematical Society) 313 (2): 539570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR2001418, MR 0986027

Howe, Roger; Umeda, Toru (1991), The Capelli identity, the double commutant theorem, and multiplicity-free actions, Mathematische Annalen 290 (1): 565619, doi:10.1007/BF01459261

Umeda, Tru (1998), The Capelli identities, a century after, Selected papers on harmonic analysis, groups,and invariants, Amer. Math. Soc. Transl. Ser. 2 183, Providence, R.I.: Amer. Math. Soc., pp. 5178, ISBN978-0-8218-0840-5, MR 1615137

Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255, retrieved 03/2007/26 Check date values in: |accessdate=(help)

Chapter 5

Catalecticant

But the catalecticant of the biquadratic function of x, y was rst brought into notice as an invariant by Mr Boole;and the discriminant of the quadratic function of x, y is identical with its catalecticant, as also with its Hessian.Meicatalecticizant would more completely express the meaning of that which, for the sake of brevity, I denominatethe catalecticant.Sylvester (1852), quoted by (Miller 2010)

In mathematical invariant theory, the catalecticant of a form of even degree is a polynomial in its coecients thatvanishes when the form is a sum of an unusually small number of powers of linear forms. It was introduced bySylvester (1852); see (Miller 2010). The word catalectic refers to an incomplete line of verse, lacking a syllable atthe end or ending with an incomplete foot.

5.1 Binary formsThe catalecticant of a binary form of degree 2n is a polynomial in its coecients that vanishes when the binary formis a sum of at most n powers of linear forms Sturmfels (1993).The catalecticant of a binary form can be given as the determinant of a catalecticant matrix (Eisenbud 1988), alsocalled a Hankel matrix, that is a square matrix with constant (positive sloping) skew-diagonals, such as

266664a b c d eb c d e fc d e f gd e f g he f g h i

377775:

5.2 Catalecticants of quartic formsThe catalecticant of a quartic form is the resultant of its second partial derivatives. For binary quartics the catalecticantvanishes when the form is a sum of 2 4th powers. For a ternary quartic the catalecticant vanishes when the form is asum of 5 4th powers. For quaternary quartics the catalecticant vanishes when the form is a sum of 9 4th powers. Forquinary quartics the catalecticant vanishes when the form is a sum of 14 4th powers. (Elliot 1915, p.295)

5.3 References Eisenbud, David (1988), Linear sections of determinantal varieties, American Journal of Mathematics 110(3): 541575, doi:10.2307/2374622, ISSN 0002-9327, MR 944327

14

5.4. EXTERNAL LINKS 15

Elliott, Edwin Bailey (1913) [1895], An introduction to the algebra of quantics. (2nd ed.), Oxford. ClarendonPress, JFM 26.0135.01

Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, NewYork: Springer-Verlag, doi:10.1007/978-3-211-77417-5, ISBN978-3-211-82445-0,MR1255980

Sylvester, J. J. (1852), On the principles of the calculus of forms, Cambridge and Dublin MathematicalJournal: 5297

5.4 External links Miller, Je (2010), Earliest Known Uses of Some of the Words of Mathematics (C)

Chapter 6

Cayleys process

This article is about the mathematical process. For the industrial OMEGA process, see OMEGA process.

Inmathematics,Cayleys process, introduced byArthur Cayley (1846), is a relatively invariant dierential operatoron the general linear group, that is used to construct invariants of a group action.As a partial dierential operator acting on functions of n2 variables xij, the omega operator is given by the determinant

=

@

@x11 @@x1n... . . . ...

@@xn1

@@xnn

:For binary forms f in x1, y1 and g in x2, y2 the operator is @

2fg@x1@y2

@2fg@x2@y1 . The r-fold process r(f, g) ontwo forms f and g in the variables x and y is then

1. Convert f to a form in x1, y1 and g to a form in x2, y2

2. Apply the operator r times to the function fg, that is, f times g in these four variables

3. Substitute x for x1 and x2, y for y1 and y2 in the result

The result of the r-fold process r(f, g) on the two forms f and g is also called the r-th transvectant and is commonlywritten (f, g)r.

6.1 ApplicationsCayleys process appears in Capellis identity, which Weyl (1946) used to nd generators for the invariants ofvarious classical groups acting on natural polynomial algebras.Hilbert (1890) used Cayleys process in his proof of nite generation of rings of invariants of the general lineargroup. His use of the process gives an explicit formula for the Reynolds operator of the special linear group.Cayleys process is used to dene transvectants.

6.2 References Cayley, Arthur (1846), On linear transformations, Cambridge and Dublin mathematical journal 1: 104122Reprinted in Cayley (1889), The collected mathematical papers 1, Cambridge: Cambridge University press,pp. 95112

16

6.2. REFERENCES 17

Hilbert, David (1890), Ueber die Theorie der algebraischen Formen, Mathematische Annalen 36 (4): 473534, doi:10.1007/BF01208503, ISSN 0025-5831

Howe, Roger (1989), Remarks on classical invariant theory., Transactions of the American Mathematical So-ciety (American Mathematical Society) 313 (2): 539570, doi:10.1090/S0002-9947-1989-0986027-X, ISSN0002-9947, JSTOR 2001418, MR 0986027

Olver, Peter J. (1999), Classical invariant theory, Cambridge University Press, ISBN 978-0-521-55821-1 Sturmfels, Bernd (1993), Algorithms in invariant theory, Texts and Monographs in Symbolic Computation,Berlin, New York: Springer-Verlag, ISBN 978-3-211-82445-0, MR 1255980

Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations, Princeton UniversityPress, ISBN 978-0-691-05756-9, MR 0000255, retrieved 03/2007/26 Check date values in: |accessdate=(help)

Chapter 7

ChevalleyIwahoriNagata theorem

In mathematics, the ChevalleyIwahoriNagata theorem states that if a linear algebraic group G is acting linearlyon a nite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomialsis an isomorphism if this ring is nitely generated and all orbits of G on V are closed (Dieudonn & Carrell 1970,1971, p.55). It is named after Claude Chevalley, Nagayoshi Iwahori, and Masayoshi Nagata.

7.1 References Dieudonn, Jean A.; Carrell, James B. (1970), Invariant theory, old and new, Advances in Mathematics 4:180, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525

Dieudonn, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press,doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102

18

Chapter 8

ChevalleyShephardTodd theorem

In mathematics, the ChevalleyShephardTodd theorem in invariant theory of nite groups states that the ring ofinvariants of a nite group acting on a complex vector space is a polynomial ring if and only if the group is generatedby pseudoreections. In the case of subgroups of the complex general linear group the theorem was rst proved byG. C. Shephard and J. A. Todd (1954) who gave a case-by-case proof. Claude Chevalley (1955) soon afterwardsgave a uniform proof. It has been extended to nite linear groups over an arbitrary eld in the non-modular case byJean-Pierre Serre.

8.1 Statement of the theoremLet V be a nite-dimensional vector space over a eld K and let G be a nite subgroup of the general linear groupGL(V). An element s of GL(V) is called a pseudoreection if it xes a codimension 1 subspace of V and is not theidentity transformation I, or equivalently, if the kernel Ker (s I) has codimension one in V. Assume that the orderof G is relatively prime to the characteristic of K (the so-called non-modular case). Then the following properties areequivalent:[1]

(A) The group G is generated by pseudoreections. (B) The algebra of invariants K[V]G is a (free) polynomial algebra. (B) The algebra of invariants K[V]G is a regular ring. (C) The algebra K[V] is a free module over K[V]G. (C) The algebra K[V] is a projective module over K[V]G.

In the case when the eld K is the eld C of complex numbers, the rst condition is usually stated as "G is a complexreection group". Shephard and Todd derived a full classication of such groups.

8.2 Examples Let V be one-dimensional. Then any nite group faithfully acting on V is a subgroup of the multiplicativegroup of the eld K, and hence a cyclic group. It follows that G consists of roots of unity of order dividing n,where n is its order, so G is generated by pseudoreections. In this case, K[V] = K[x] is the polynomial ringin one variable and the algebra of invariants of G is the subalgebra generated by xn, hence it is a polynomialalgebra.

LetV =Kn be the standard n-dimensional vector space andG be the symmetric group Sn acting by permutationsof the elements of the standard basis. The symmetric group is generated by transpositions (ij), which act byreections on V. On the other hand, by the main theorem of symmetric functions, the algebra of invariants isthe polynomial algebra generated by the elementary symmetric functions e1, en.

19

20 CHAPTER 8. CHEVALLEYSHEPHARDTODD THEOREM

Let V = K2 and G be the cyclic group of order 2 acting by I. In this case, G is not generated by pseudoreec-tions, since the nonidentity element s of G acts without xed points, so that dim Ker (s I) = 0. On the otherhand, the algebra of invariants is the subalgebra of K[V] = K[x, y] generated by the homogeneous elementsx2, xy, and y2 of degree 2. This subalgebra is not a polynomial algebra because of the relation x2y2 = (xy)2.

8.3 GeneralizationsBroer (2007) gave an extension of the ChevalleyShephardTodd theorem to positive characteristic.There has been much work on the question of when a reductive algebraic group acting on a vector space has apolynomial ring of invariants. In the case when the algebraic group is simple all cases when the invariant ring ispolynomial have been classied by Schwarz (1978)In general, the ring of invariants of a nite group acting linearly on a complex vector space is Cohen-Macaulay, so itis a nite rank free module over a polynomial subring.

8.4 Notes[1] See, e.g.: Bourbaki, Lie, chap. V, 5, n5, theorem 4 for equivalence of (A), (B) and (C); page 26 of for equivalence of

(A) and (B); pages 618 of for equivalence of (C) and (C) for a proof of (B)(A).

8.5 References Bourbaki, Nicolas, lments de mathmatiques : Groupes et algbres de Lie (English translation: Bourbaki,Nicolas, Elements of Mathematics: Lie Groups and Lie Algebras)

Broer, Abraham (2007), On Chevalley-Shephard-Todds theorem in positive characteristic, [], arXiv:0709.0715 Chevalley, Claude (1955), Invariants of nite groups generated by reections, Amer. J. Of Math. 77 (4):778782, doi:10.2307/2372597, JSTOR 2372597

Neusel, Mara D.; Smith, Larry (2002), Invariant Theory of Finite Groups, American Mathematical Society,ISBN 0-8218-2916-5

Shephard, G. C.; Todd, J. A. (1954), Finite unitary reection groups, Canadian J. Math. 6: 274304,doi:10.4153/CJM-1954-028-3

Schwarz, G. (1978), Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49(2): 167191, doi:10.1007/BF01403085

Smith, Larry (1997), Polynomial invariants of nite groups. A survey of recent developments, Bull. Amer.Math. Soc. 34 (3): 211250, doi:10.1090/S0273-0979-97-00724-6, MR 1433171

Springer, T. A. (1977), Invariant Theory, Springer, ISBN 0-387-08242-5

Chapter 9

Dierential invariant

In mathematics, a dierential invariant is an invariant for the action of a Lie group on a space that involves thederivatives of graphs of functions in the space. Dierential invariants are fundamental in projective dierential ge-ometry, and the curvature is often studied from this point of view.[1] Dierential invariants were introduced in specialcases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Lie (1884) was therst general work on dierential invariants, and established the relationship between dierential invariants, invariantdierential equations, and invariant dierential operators.Dierential invariants are contrasted with geometric invariants. Whereas dierential invariants can involve a distin-guished choice of independent variables (or a parameterization), geometric invariants do not. lie Cartan's methodof moving frames is a renement that, while less general than Lies methods of dierential invariants, always yieldsinvariants of the geometrical kind.

9.1 DenitionThe simplest case is for dierential invariants for one independent variable x and one dependent variable y. Let Gbe a Lie group acting on R2. Then G also acts, locally, on the space of all graphs of the form y = (x). Roughlyspeaking, a k-th order dierential invariant is a function

I

x; y;

dy

dx; : : : ;

dky

dxk

depending on y and its rst k derivatives with respect to x, that is invariant under the action of the group.The group can act on the higher-order derivatives in a nontrivial manner that requires computing the prolongation ofthe group action. The action of G on the rst derivative, for instance, is such that the chain rule continues to hold: if

(x; y) = g (x; y);

then

g x; y;

dy

dx

def=

x; y;

dy

dx

:

Similar considerations apply for the computation of higher prolongations. This method of computing the prolongationis impractical, however, and it is much simpler to work innitesimally at the level of Lie algebras and the Lie derivativealong the G action.More generally, dierential invariants can be considered for mappings from any smooth manifold X into anothersmooth manifold Y for a Lie group acting on the Cartesian product XY. The graph of a mapping X Y is asubmanifold of XY that is everywhere transverse to the bers over X. The group G acts, locally, on the space of

21

22 CHAPTER 9. DIFFERENTIAL INVARIANT

such graphs, and induces an action on the k-th prolongation Y (k) consisting of graphs passing through each pointmodulo the relation of k-th order contact. A dierential invariant is a function on Y (k) that is invariant under theprolongation of the group action.

9.2 Applications Dierential invariants can be applied to the study of systems of partial dierential equations: seeking similaritysolutions that are invariant under the action of a particular group can reduce the dimension of the problem (i.e.yield a reduced system).[2]

Noethers theorem implies the existence of dierential invariants corresponding to every dierentiable sym-metry of a variational problem.

Flow characteristics using computer vision[3]

Geometric integration

9.3 See also Cartans equivalence method

9.4 Notes[1] Guggenheimer 1977

[2] Olver 1994, Chapter 3

[3] http://dspace.mit.edu/bitstream/handle/1721.1/3348/P-2219-29812804.pdf?sequence=1

9.5 References Guggenheimer, Heinrich (1977), Dierential Geometry, New York: Dover Publications, ISBN 978-0-486-63433-3.

Lie, Sophus (1884), "ber Dierentialinvarianten, Gesammelte Adhandlungen 6, Leipzig: B.G. Teubner,pp. 95138; English translation: Ackerman, M; Hermann, R (1975), Sophus Lies 1884 Dierential InvariantPaper, Brookline, Mass.: Math Sci Press.

Olver, Peter J. (1993),Applications of Lie groups to dierential equations (2nd ed.), Berlin, NewYork: Springer-Verlag, ISBN 978-0-387-94007-6.

Olver, Peter J. (1995), Equivalents, Invariants, and Symmetry, Cambridge University Press, ISBN 978-0-521-47811-3.

Manseld, Elizabeth Louise (2009), A Practical Guide to the Invariant Calculus (PDF); to be published byCambridge 2010, ISBN 978-0-521-85701-7.

9.6 External links Invariant Variation Problems

Chapter 10

Evectant

In mathematical invariant theory, an evectant is a contravariant constructed from an invariant by acting on it with adierential operator called an evector. Evectants and evectors were introduced by Sylvester (1854, p.95).

10.1 References Sylvester, James Joseph (1853), On the calculus of forms, otherwise the theory of invariants, The Cambridgeand Dublin Mathematical Journal 8: 257269

Sylvester, James Joseph (1854), On the calculus of forms, otherwise the theory of invariants, The Cambridgeand Dublin Mathematical Journal 9: 85103

23

Chapter 11

Geometric invariant theory

In mathematics Geometric invariant theory (or GIT) is a method for constructing quotients by group actions inalgebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas fromthe paper (Hilbert 1893) in classical invariant theory.Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides tech-niques for forming the 'quotient' of X by G as a scheme with reasonable properties. One motivation was to constructmoduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980sthe theory developed interactions with symplectic geometry and equivariant topology, and was used to constructmoduli spaces of objects in dierential geometry, such as instantons and monopoles.

11.1 BackgroundMain article: Invariant theory

Invariant theory is concerned with a group action of a group G on an algebraic variety (or a scheme) X. Classicalinvariant theory addresses the situation when X = V is a vector space and G is either a nite group, or one of theclassical Lie groups that acts linearly on V. This action induces a linear action of G on the space of polynomialfunctions R(V) on V by the formula

g f(v) = f(g1v); g 2 G; v 2 V:The polynomial invariants of the G-action on V are those polynomial functions f on V which are xed under the'change of variables due to the action of the group, so that gf = f for all g in G. They form a commutative algebraA = R(V)G, and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' V //G. In thelanguage of modern algebraic geometry,

V //G = SpecA = SpecR(V )G:

Several diculties emerge from this description. The rst one, successfully tackled by Hilbert in the case of a generallinear group, is to prove that the algebra A is nitely generated. This is necessary if one wanted the quotient to be anane algebraic variety. Whether a similar fact holds for arbitrary groups G was the subject of Hilberts fourteenthproblem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course ofdevelopment of representation theory in the rst half of the twentieth century, a large class of groups for which theanswer is positive was identied; these are called reductive groups and include all nite groups and all classical groups.The nite generation of the algebra A is but the rst step towards the complete description of A, and the progressin resolving this more delicate question was rather modest. The invariants had classically been described only in arestricted range of situations, and the complexity of this description beyond the rst few cases held out little hope forfull understanding of the algebras of invariants in general. Furthermore, it may happen that all polynomial invariantsf take the same value on a given pair of points u and v in V, yet these points are in dierent orbits of the G-action. A

24

11.2. MUMFORDS BOOK 25

simple example is provided by themultiplicative groupC* of non-zero complex numbers that acts on an n-dimensionalcomplex vector space Cn by scalar multiplication. In this case, every polynomial invariant is a constant, but there aremany dierent orbits of the action. The zero vector forms an orbit by itself, and the non-zeromultiples of any non-zerovector form an orbit, so that non-zero orbits are paramatrized by the points of the complex projective spaceCPn1. Ifthis happens, one says that invariants do not separate the orbits, and the algebra A reects the topological quotientspace X /G rather imperfectly. Indeed, the latter space is frequently non-separated. In 1893 Hilbert formulated andproved a criterion for determining those orbits which are not separated from the zero orbit by invariant polynomials.Rather remarkably, unlike his earlier work in invariant theory, which led to the rapid development of abstract algebra,this result of Hilbert remained little known and little used for the next 70 years. Much of the development of invarianttheory in the rst half of the twentieth century concerned explicit computations with invariants, and at any rate,followed the logic of algebra rather than geometry.

11.2 Mumfords bookGeometric invariant theory was founded and developed by Mumford in a monograph, rst published in 1965, thatapplied ideas of nineteenth century invariant theory, including some results of Hilbert, to modern algebraic geometryquestions. (The book was greatly expanded in two later editions, with extra appendices by Fogarty and Mumford,and a chapter on symplectic quotients by Kirwan.) The book uses both scheme theory and computational techniquesavailable in examples. The abstract setting used is that of a group action on a scheme X. The simple-minded idea ofan orbit space

G\X,

i.e. the quotient space of X by the group action, runs into diculties in algebraic geometry, for reasons that areexplicable in abstract terms. There is in fact no general reason why equivalence relations should interact well with the(rather rigid) regular functions (polynomial functions), which are at the heart of algebraic geometry. The functionson the orbit space G\X that should be considered are those on X that are invariant under the action of G. The directapproach can be made, by means of the function eld of a variety (i.e. rational functions): take the G-invariantrational functions on it, as the function eld of the quotient variety. Unfortunately this the point of view ofbirational geometry can only give a rst approximation to the answer. As Mumford put it in the Preface to thebook:

The problem is, within the set of all models of the resulting birational class, there is one model whosegeometric points classify the set of orbits in some action, or the set of algebraic objects in some moduliproblem.

In Chapter 5 he isolates further the specic technical problem addressed, in a moduli problem of quite classicaltype classify the big 'set' of all algebraic varieties subject only to being non-singular (and a requisite condition onpolarization). The moduli are supposed to describe the parameter space. For example for algebraic curves it has beenknown from the time of Riemann that there should be connected components of dimensions

0, 1, 3, 6, 9,

according to the genus g =0, 1, 2, 3, 4, , and the moduli are functions on each component. In the coarse moduliproblem Mumford considers the obstructions to be:

non-separated topology on the moduli space (i.e. not enough parameters in good standing) innitely many irreducible components (which isn't avoidable, but local niteness may hold) failure of components to be representable as schemes, although respectable topologically.

It is the third point that motivated the whole theory. As Mumford puts it, if the rst two diculties are resolved

[the third question] becomes essentially equivalent to the question of whether an orbit space of some locallyclosed subset of the Hilbert or Chow schemes by the projective group exists.

26 CHAPTER 11. GEOMETRIC INVARIANT THEORY

To deal with this he introduced a notion (in fact three) of stability. This enabled him to open up the previouslytreacherous area much had been written, in particular by Francesco Severi, but the methods of the literaturehad limitations. The birational point of view can aord to be careless about subsets of codimension 1. To havea moduli space as a scheme is on one side a question about characterising schemes as representable functors (asthe Grothendieck school would see it); but geometrically it is more like a compactication question, as the stabilitycriteria revealed. The restriction to non-singular varieties will not lead to a compact space in any sense as modulispace: varieties can degenerate to having singularities. On the other hand the points that would correspond to highlysingular varieties are denitely too 'bad' to include in the answer. The correct middle ground, of points stable enoughto be admitted, was isolated by Mumfords work. The concept was not entirely new, since certain aspects of it wereto be found in David Hilbert's nal ideas on invariant theory, before he moved on to other elds.The books Preface also enunciated the Mumford conjecture, later proved by William Haboush.

11.3 StabilityStable point redirects here. It is not to be confused with Stable xed point.

If a reductive group G acts linearly on a vector space V, then a non-zero point of V is called

unstable if 0 is in the closure of its orbit, semi-stable if 0 is not in the closure of its orbit, stable if its orbit is closed, and its stabilizer is nite.

There are equivalent ways to state these (this criterion is known as the HilbertMumford criterion):

A non-zero point x is unstable if and only if there is a 1-parameter subgroup of G all of whose weights withrespect to x are positive.

A non-zero point x is unstable if and only if every invariant polynomial has the same value on 0 and x.