Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.

Classical Algebraic Geometry: A Modern View

/pdf/handbook-of-magma-functions-5anhrdxxj3.pdf

Handbook of magma functions

On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.

Representations of finite groups of Lie type

1 CONTENTS 1. Introduction 2. Molien's theorem 3. Cohen-Macaulay rings 4. Groups generated by pseudo-reflections 5. Three applications 6. Syzygies 7. The canonical module 8. Gorenstein rings 9. Complete intersections 10. Monomial groups 1. Introduction. The theory of invariants of finite groups forms an interesting and relatively self-contained nook in the imposing edifice of commuta-tive algebra. Moreover, there are close connections between this subject and combinatorics, for two reasons: (a) the highly combinatorial tool of generating functions pervades the study of invariants of finite groups, and (b) several direct applications of invariants of finite groups have recently been given to combinatorics. Here we give an exposition of the theory of invariants of finite groups with emphasis on the connections with combinatorics, which assumes a minimal background in commutative algebra and combinatorics on the part of the reader. It is hoped that such an exposition will appeal to several types of readers, (a) Those who simply wish to see a self-contained treatment of an elegant and fascinating subject. This might include coding theorists, physicists, and others who are beginning to use invariant theory as a tool in their own work, (b) Those who are interested in learning something about the revolutionary developments in present-day combinatorics. Until recently combinatorics has been regarded as a disparate collection of ad hoc tricks, but this picture is slowly changing under a determined effort to unify various branches of combinatorics and to understand their relationship with other branches of mathematics, (c) Finally, those who would like a relatively painless glimpse of certain topics of current interest in commutative algebra, such as the theory of Cohen-Macaulay rings and Gorenstein rings. For a really adequate understanding of these concepts it would be necessary to work in far greater generality and to introduce sophisticated machinery from

https://www.ams.org/bull/1979-01-03/S0273-0979-1979-14597-X/S0273-0979-1979-14597-X.pdf

Invariants of finite groups and their applications to combinatorics

Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors.

J. A. Todd

Papers

Finite Unitary Reflection Groups