| Introduction | p. 1 |
| The three historical aspects of the Coxeter transformation | p. 1 |
| A brief review of this work | p. 3 |
| The spectrum and the Jordan form | p. 6 |
| The Jordan form and the golden pair of matrices | p. 6 |
| An explicit construction of eigenvectors | p. 7 |
| Study of the Coxeter transformation and the Cartan matrix | p. 8 |
| Monotonicity of the dominant eigenvalue | p. 8 |
| Splitting formulas and the diagrams T[subscript p,q,r] | p. 9 |
| Splitting formulas for the characteristic polynomial | p. 9 |
| An explicit calculation of characteristic polynomials | p. 10 |
| Formulas for the diagrams T[subscript 2,3,r], T[subscript 3,3,r], T[subscript 2,4,r] | p. 12 |
| Coxeter transformations and the McKay correspondence | p. 13 |
| The generalized R. Steinberg theorem | p. 13 |
| The Kostant generating functions and W. Ebeling's theorem | p. 14 |
| The affine Coxeter transformation | p. 16 |
| The R. Steinberg trick | p. 16 |
| The defect and the Dlab-Ringel formula | p. 18 |
| The regular representations of quivers | p. 19 |
| The regular and non-regular representations of quivers | p. 19 |
| The necessary and sufficient regularity conditions | p. 20 |
| Preliminaries | p. 23 |
| The Cartan matrix and the Tits form | p. 23 |
| The generalized and symmetrizable Cartan matrix | p. 23 |
| The Tits form and diagrams T[subscript p,q,r] | p. 25 |
| The simply-laced Dynkin diagrams | p. 27 |
| The multiply-laced Dynkin diagrams. Possible weighted edges | p. 28 |
| The multiply-laced Dynkin diagrams. A branch point | p. 30 |
| The extended Dynkin diagrams. Two different notation | p. 36 |
| Three sets of Tits forms | p. 37 |
| The hyperbolic Dynkin diagrams and hyperbolic Cartan matrices | p. 38 |
| Representations of quivers | p. 38 |
| The real and imaginary roots | p. 38 |
| A category of representations of quivers and the P. Gabriel theorem | p. 41 |
| Finite-type, tame and wild quivers | p. 42 |
| The V. Kac theorem on the possible dimension vectors | p. 43 |
| The quadratic Tits form and vector-dimensions of representations | p. 44 |
| Orientations and the associated Coxeter transformations | p. 45 |
| The Poincare series | p. 46 |
| The graded algebras, symmetric algebras, algebras of invariants | p. 46 |
| The invariants of finite groups generated by reflections | p. 49 |
| The Jordan normal form of the Coxeter transformation | p. 51 |
| The Cartan matrix and the Coxeter transformation | p. 51 |
| A bicolored partition and a bipartite graph | p. 51 |
| Conjugacy of Coxeter transformations | p. 52 |
| The Cartan matrix and the bicolored Coxeter transformation | p. 52 |
| The dual graphs and dual forms | p. 54 |
| The eigenvalues of the Cartan matrix and the Coxeter transformation | p. 54 |
| An application of the Perron-Frobenius theorem | p. 56 |
| The pair of matrices D D[superscript t] and D[superscript t] D (resp. D F and F D) | p. 56 |
| The Perron-Frobenius theorem applied to D D[superscript t] and D[superscript t] D (resp. D F and F D) | p. 59 |
| The basis of eigenvectors and a theorem on the Jordan form | p. 61 |
| An explicit construction of the eigenvectors | p. 61 |
| Monotonicity of the dominant eigenvalue | p. 63 |
| A theorem on the Jordan form | p. 65 |
| Eigenvalues, splitting formulas and diagrams T[subscript p,q,r] | p. 67 |
| The eigenvalues of the affine Coxeter transformation | p. 67 |
| Bibliographical notes on the spectrum of the Coxeter transformation | p. 71 |
| Splitting and gluing formulas for the characteristic polynomial | p. 74 |
| Formulas of the characteristic polynomials for the diagrams T[subscript p,q,r] | p. 80 |
| The diagrams T[subscript 2,3,r] | p. 81 |
| The diagrams T[subscript 3,3,r] | p. 84 |
| The diagrams T[subscript 2,4,r] | p. 86 |
| Convergence of the sequence of eigenvalues | p. 90 |
| R. Steinberg's theorem, B. Kostant's construction | p. 95 |
| R. Steinberg's theorem and a (p,q,r) mystery | p. 95 |
| The characteristic polynomials for the Dynkin diagrams | p. 99 |
| A generalization of R. Steinberg's theorem | p. 102 |
| The folded Dynkin diagrams and branch points | p. 102 |
| R. Steinberg's theorem for the multiply-laced case | p. 103 |
| The Kostant generating function and Poincare series | p. 105 |
| The generating function | p. 105 |
| The characters and the McKay operator | p. 109 |
| The Poincare series and W. Ebeling's theorem | p. 113 |
| The orbit structure of the Coxeter transformation | p. 116 |
| The Kostant generating functions and polynomials z(t)[subscript i] | p. 116 |
| One more observation of McKay | p. 121 |
| The affine Coxeter transformation | p. 129 |
| The Weyl Group and the affine Weyl group | p. 129 |
| The semidirect product | p. 129 |
| Two representations of the affine Weyl group | p. 130 |
| The translation subgroup | p. 133 |
| The affine Coxeter transformation | p. 136 |
| R. Steinberg's theorem again | p. 137 |
| The element of the maximal length in the Weyl group | p. 138 |
| The highest root and the branch point | p. 140 |
| The orbit of the highest root. Examples | p. 143 |
| The linear part of the affine Coxeter transformation | p. 145 |
| Two generalizations of the branch point | p. 147 |
| The defect | p. 148 |
| The affine Coxeter transformation and defect | p. 148 |
| The necessary regularity conditions | p. 150 |
| The Dlab-Ringel formula | p. 152 |
| The Dlab-Ringel defect and the [Omega]-defect coincide | p. 153 |
| The McKay correspondence and the Slodowy correspondence | p. 155 |
| Finite subgroups of SU(2) and SO(3, R) | p. 155 |
| The generators and relations in polyhedral groups | p. 156 |
| The Kleinian singularities and the Du Val resolution | p. 158 |
| The McKay correspondence | p. 160 |
| The Slodowy generalization of the McKay correspondence | p. 161 |
| The Slodowy correspondence | p. 162 |
| The binary tetrahedral group and the binary octahedral group | p. 164 |
| Representations of the binary octahedral and tetrahedral groups | p. 167 |
| The induced and restricted representations | p. 174 |
| The characters of the binary polyhedral groups | p. 179 |
| The cyclic groups | p. 179 |
| The binary dihedral groups | p. 179 |
| The binary icosahedral group | p. 181 |
| Regularity conditions for representations of quivers | p. 183 |
| The Coxeter functors and regularity conditions | p. 183 |
| The reflection functor F[subscript a superscript +] | p. 184 |
| The reflection functor F[subscript a superscript -] | p. 185 |
| The Coxeter functors [Phi superscript +], [Phi superscript -] | p. 186 |
| The preprojective and preinjective representations | p. 187 |
| The regularity condition | p. 187 |
| The necessary regularity conditions | p. 188 |
| Transforming elements and sufficient regularity conditions | p. 191 |
| The sufficient regularity conditions for the bicolored orientation | p. 191 |
| A theorem on transforming elements | p. 192 |
| The sufficient regularity conditions for an arbitrary orientation | p. 194 |
| The invariance of the defect | p. 195 |
| Examples of regularity conditions | p. 197 |
| The three equivalence classes of orientations of D[subscript 4] | p. 197 |
| The bicolored and central orientations of E[subscript 6] | p. 198 |
| The multiply-laced case. The two orientations of G[subscript 21] and G[subscript 22] = G[subscript 21 superscript V] | p. 199 |
| The case of indefinite B. The oriented star *[subscript n+1] | p. 200 |
| Miscellanea | p. 203 |
| The triangle groups and Hurwitz groups | p. 203 |
| The algebraic integers | p. 204 |
| The Perron-Frobenius Theorem | p. 206 |
| The Schwartz inequality | p. 207 |
| The complex projective line and stereographic projection | p. 208 |
| The prime spectrum, the coordinate ring, the orbit space | p. 210 |
| Hilbert's Nullstellensatz (Theorem of zeros) | p. 210 |
| The prime spectrum | p. 212 |
| The coordinate ring | p. 213 |
| The orbit space | p. 214 |
| Fixed and anti-fixed points of the Coxeter transformation | p. 215 |
| The Chebyshev polynomials and the McKay-Slodowy matrix | p. 215 |
| A theorem on fixed and anti-fixed points | p. 217 |
| References | p. 221 |
| Index | p. 233 |
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