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Notes on Coxeter transformations and the McKay correspondence

Author: R Stekolshchik
Publisher: Berlin : Springer, ©2008.
Series: Springer monographs in mathematics.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"One of the beautiful results in the representation theory of the finite groups is McKay's theorem on a correspondence between representations of the binary polyhedral group of SU(2) and vertices of an extended simply-laced Dynkin diagram." "The Coxeter transformation is the main tool in the proof of the McKay correspondence, and is closely interrelated with the Cartan matrix and Poincare series. The Coxeter  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: R Stekolshchik
ISBN: 3540773983 9783540773986 9783540773993 3540773991
OCLC Number: 209333269
Description: xx, 239 pages : illustrations ; 25 cm.
Contents: 1.1 The three historical aspects of the Coxeter transformation 1 --
1.2 A brief review of this work 3 --
1.3 The spectrum and the Jordan form 6 --
1.3.1 The Jordan form and the golden pair of matrices 6 --
1.3.2 An explicit construction of eigenvectors 7 --
1.3.3 Study of the Coxeter transformation and the Cartan matrix 8 --
1.3.4 Monotonicity of the dominant eigenvalue 8 --
1.4 Splitting formulas and the diagrams T[subscript p, q, r] 9 --
1.4.1 Splitting formulas for the characteristic polynomial 9 --
1.4.2 An explicit calculation of characteristic polynomials 10 --
1.4.3 Formulas for the diagrams T[subscript 2,3,r], T[subscript 3,3,r], T[subscript 2,4,r] 12 --
1.5 Coxeter transformations and the McKay correspondence 13 --
1.5.1 The generalized R. Steinberg theorem 13 --
1.5.2 The Kostant generating functions and W. Ebeling's theorem 14 --
1.6 The affine Coxeter transformation 16 --
1.6.1 The R. Steinberg trick 16 --
1.6.2 The defect and the Dlab-Ringel formula 18 --
1.7 The regular representations of quivers 19 --
1.7.1 The regular and non-regular representations of quivers 19 --
1.7.2 The necessary and sufficient regularity conditions 20 --
2.1 The Cartan matrix and the Tits form 23 --
2.1.1 The generalized and symmetrizable Cartan matrix 23 --
2.1.2 The Tits form and diagrams T[subscript p, q, r] 25 --
2.1.3 The simply-laced Dynkin diagrams 27 --
2.1.4 The multiply-laced Dynkin diagrams. Possible weighted edges 28 --
2.1.5 The multiply-laced Dynkin diagrams. A branch point 30 --
2.1.6 The extended Dynkin diagrams. Two different notation 36 --
2.1.7 Three sets of Tits forms 37 --
2.1.8 The hyperbolic Dynkin diagrams and hyperbolic Cartan matrices 38 --
2.2 Representations of quivers 38 --
2.2.1 The real and imaginary roots 38 --
2.2.2 A category of representations of quivers and the P. Gabriel theorem 41 --
2.2.3 Finite-type, tame and wild quivers 42 --
2.2.4 The V. Kac theorem on the possible dimension vectors 43 --
2.2.5 The quadratic Tits form and vector-dimensions of representations 44 --
2.2.6 Orientations and the associated Coxeter transformations 45 --
2.3 The Poincare series 46 --
2.3.1 The graded algebras, symmetric algebras, algebras of invariants 46 --
2.3.2 The invariants of finite groups generated by reflections 49 --
3 The Jordan normal form of the Coxeter transformation 51 --
3.1 The Cartan matrix and the Coxeter transformation 51 --
3.1.1 A bicolored partition and a bipartite graph 51 --
3.1.2 Conjugacy of Coxeter transformations 52 --
3.1.3 The Cartan matrix and the bicolored Coxeter transformation 52 --
3.1.4 The dual graphs and dual forms 54 --
3.1.5 The eigenvalues of the Cartan matrix and the Coxeter transformation 54 --
3.2 An application of the Perron-Frobenius theorem 56 --
3.2.1 The pair of matrices D D[superscript t] and D[superscript t] D (resp. D F and F D) 56 --
3.2.2 The Perron-Frobenius theorem applied to D D[superscript t] and D[superscript t] D (resp. D F and F D) 59 --
3.3 The basis of eigenvectors and a theorem on the Jordan form 61 --
3.3.1 An explicit construction of the eigenvectors 61 --
3.3.2 Monotonicity of the dominant eigenvalue 63 --
3.3.3 A theorem on the Jordan form 65 --
4 Eigenvalues, splitting formulas and diagrams T[subscript p, q, r] 67 --
4.1 The eigenvalues of the affine Coxeter transformation 67 --
4.2 Bibliographical notes on the spectrum of the Coxeter transformation 71 --
4.3 Splitting and gluing formulas for the characteristic polynomial 74 --
4.4 Formulas of the characteristic polynomials for the diagrams T[subscript p, q, r] 80 --
4.4.1 The diagrams T[subscript 2,3,r] 81 --
4.4.2 The diagrams T[subscript 3,3,r] 84 --
4.4.3 The diagrams T[subscript 2,4,r] 86 --
4.4.4 Convergence of the sequence of eigenvalues 90 --
5 R. Steinberg's theorem, B. Kostant's construction 95 --
5.1 R. Steinberg's theorem and a (p, q, r) mystery 95 --
5.2 The characteristic polynomials for the Dynkin diagrams 99 --
5.3 A generalization of R. Steinberg's theorem 102 --
5.3.1 The folded Dynkin diagrams and branch points 102 --
5.3.2 R. Steinberg's theorem for the multiply-laced case 103 --
5.4 The Kostant generating function and Poincare series 105 --
5.4.1 The generating function 105 --
5.4.2 The characters and the McKay operator 109 --
5.4.3 The Poincare series and W. Ebeling's theorem 113 --
5.5 The orbit structure of the Coxeter transformation 116 --
5.5.1 The Kostant generating functions and polynomials z(t)[subscript i] 116 --
5.5.2 One more observation of McKay 121 --
6 The affine Coxeter transformation 129 --
6.1 The Weyl Group and the affine Weyl group 129 --
6.1.1 The semidirect product 129 --
6.1.2 Two representations of the affine Weyl group 130 --
6.1.3 The translation subgroup 133 --
6.1.4 The affine Coxeter transformation 136 --
6.2 R. Steinberg's theorem again 137 --
6.2.1 The element of the maximal length in the Weyl group 138 --
6.2.2 The highest root and the branch point 140 --
6.2.3 The orbit of the highest root. Examples 143 --
6.2.4 The linear part of the affine Coxeter transformation 145 --
6.2.5 Two generalizations of the branch point 147 --
6.3 The defect 148 --
6.3.1 The affine Coxeter transformation and defect 148 --
6.3.2 The necessary regularity conditions 150 --
6.3.3 The Dlab-Ringel formula 152 --
6.3.4 The Dlab-Ringel defect and the [Omega]-defect coincide 153 --
A The McKay correspondence and the Slodowy correspondence 155 --
A.1 Finite subgroups of SU(2) and SO(3, R) 155 --
A.2 The generators and relations in polyhedral groups 156 --
A.3 The Kleinian singularities and the Du Val resolution 158 --
A.4 The McKay correspondence 160 --
A.5 The Slodowy generalization of the McKay correspondence 161 --
A.5.1 The Slodowy correspondence 162 --
A.5.2 The binary tetrahedral group and the binary octahedral group 164 --
A.5.3 Representations of the binary octahedral and tetrahedral groups 167 --
A.5.4 The induced and restricted representations 174 --
A.6 The characters of the binary polyhedral groups 179 --
A.6.1 The cyclic groups 179 --
A.6.2 The binary dihedral groups 179 --
A.6.3 The binary icosahedral group 181 --
B Regularity conditions for representations of quivers 183 --
B.1 The Coxeter functors and regularity conditions 183 --
B.1.1 The reflection functor F[subscript a superscript +] 184 --
B.1.2 The reflection functor F[subscript a superscript -] 185 --
B.1.3 The Coxeter functors [Phi superscript +], [Phi superscript -] 186 --
B.1.4 The preprojective and preinjective representations 187 --
B.1.5 The regularity condition 187 --
B.2 The necessary regularity conditions 188 --
B.3 Transforming elements and sufficient regularity conditions 191 --
B.3.1 The sufficient regularity conditions for the bicolored orientation 191 --
B.3.2 A theorem on transforming elements 192 --
B.3.3 The sufficient regularity conditions for an arbitrary orientation 194 --
B.3.4 The invariance of the defect 195 --
B.4 Examples of regularity conditions 197 --
B.4.1 The three equivalence classes of orientations of D[subscript 4] 197 --
B.4.2 The bicolored and central orientations of E[subscript 6] 198 --
B.4.3 The multiply-laced case. The two orientations of G[subscript 21] and G[subscript 22] = G[subscript 21 superscript V] 199 --
B.4.4 The case of indefinite B. The oriented star *[subscript n+1] 200 --
C Miscellanea 203 --
C.1 The triangle groups and Hurwitz groups 203 --
C.2 The algebraic integers 204 --
C.3 The Perron-Frobenius Theorem 206 --
C.4 The Schwartz inequality 207 --
C.5 The complex projective line and stereographic projection 208 --
C.6 The prime spectrum, the coordinate ring, the orbit space 210 --
C.6.1 Hilbert's Nullstellensatz (Theorem of zeros) 210 --
C.6.2 The prime spectrum 212 --
C.6.3 The coordinate ring 213 --
C.6.4 The orbit space 214 --
C.7 Fixed and anti-fixed points of the Coxeter transformation 215 --
C.7.1 The Chebyshev polynomials and the McKay-Slodowy matrix 215 --
C.7.2 A theorem on fixed and anti-fixed points 217.
Series Title: Springer monographs in mathematics.
Responsibility: R. Stekolshchik.

Abstract:

Here is a key text on the subject of representation theory in finite groups. The pages of this excellent little book, prepared by Rafael Stekolshchik, contain a number of new proofs relating to  Read more...

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