Multilinear algebra.

*(English)*Zbl 0892.15020
Algebra, Logic and Applications. 8. Langhorne, PA: Gordon & Breach. viii, 332 p. (1997).

This book bears the succinct title “Multilinear algebra”. But it is much more than that. Multilinear algebra is a subject with roots in linear algebra, group representation theory, and combinatorics, in particular enumeration and graph theory. And thus this book touches all these fields and in fact contains excellent introductions in the first four chapters:

(1) Partitions, (2) Inner product spaces, (3) Permutation groups, (4) Group representation theory. Nearly all of these and the later chapters contain sections on applications to enumeration and graph theory. Chapter 5 comprises a complete development of the tensor product, while Chapter 6 and 7, forming the heart of the book, deal with (6) Symmetry classes of tensors, including the exterior algebra, and (7) Generalized matrix functions. Many of the topics developed throughout the book are unified in the final chapter (8) The rational representations of \(GL(n,C)\), in which the classical Schur polynomials emerge in the role of characters associated with these representations and form one of the keys to the overall unification.

Despite the book’s broad scope, very little prior experience apart from a certain mathematical maturity is expected from the reader. It is ideally suited for the final undergraduate or first graduate year as well as being attractive as a research reference. Some of the easier proofs are relegated to the extensive exercise sections in each chapter, and some of the more difficult ones to the references. It is an eminently readable book by a specialist in the field, and can be thoroughly recommended to student and researcher alike.

(1) Partitions, (2) Inner product spaces, (3) Permutation groups, (4) Group representation theory. Nearly all of these and the later chapters contain sections on applications to enumeration and graph theory. Chapter 5 comprises a complete development of the tensor product, while Chapter 6 and 7, forming the heart of the book, deal with (6) Symmetry classes of tensors, including the exterior algebra, and (7) Generalized matrix functions. Many of the topics developed throughout the book are unified in the final chapter (8) The rational representations of \(GL(n,C)\), in which the classical Schur polynomials emerge in the role of characters associated with these representations and form one of the keys to the overall unification.

Despite the book’s broad scope, very little prior experience apart from a certain mathematical maturity is expected from the reader. It is ideally suited for the final undergraduate or first graduate year as well as being attractive as a research reference. Some of the easier proofs are relegated to the extensive exercise sections in each chapter, and some of the more difficult ones to the references. It is an eminently readable book by a specialist in the field, and can be thoroughly recommended to student and researcher alike.

Reviewer: R.von Randow (Bonn)

##### MSC:

15A69 | Multilinear algebra, tensor calculus |

15-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra |

20Gxx | Linear algebraic groups and related topics |

05A15 | Exact enumeration problems, generating functions |

05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |

15A75 | Exterior algebra, Grassmann algebras |