Multilinear AlgebraThe prototypical multilinear operation is multiplication. Indeed, every multilinear mapping can be factored through a tensor product. Apart from its intrinsic interest, the tensor product is of fundamental importance in a variety of disciplines, ranging from matrix inequalities and group representation theory, to the combinatorics of symmetric functions, and all these subjects appear in this book. Another attraction of multilinear algebra lies in its power to unify such seemingly diverse topics. This is done in the final chapter by means of the rational representations of the full linear group. Arising as characters of these representations, the classical Schur polynomials are one of the keys to unification. Prerequisites for the book are minimized by self-contained introductions in the early chapters. Throughout the text, some of the easier proofs are left to the exercises, and some of the more difficult ones to the references. |
Contents
Inner Product Spaces | 27 |
Permutation Groups | 53 |
Group Representation Theory | 75 |
Tensor Spaces | 121 |
Symmetry Classes of Tensors | 151 |
Generalized Matrix Functions | 213 |
The Rational Representations of GLn C | 265 |
| 305 | |
| 325 | |
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character of G Cm,m Cn,n compute Corollary corresponding coset decomposable tensors defined DEFINITION Denote det(A Dias da Silva eigenvalues entries Equation equivalent EXAMPLE Exercise Ferrers diagram Figure finite group fixed but arbitrary follows G of Sm GL(n group G hermitian Hint homogeneous polynomial independent indeterminates Inequality inequivalent inner product space invertible irreducible character irreducible constituent irreducible representation isomorphic j)-entry Lemma Let G linear character linear operator linearly independent m-linear matrix representation modulo Multilinear Algebra multiplicity n-by-n matrix nonisomorphic graphs nonzero orthogonal orthonormal basis partition permutation group permutation matrix polynomial invariants positive integer positive semidefinite principal character Proof Let Prove representation of G rotational Schur polynomials Show subgroup G subgroup of Sm subspace Suppose symmetric functions symmetry classes Theorem transitive unitary V₁ vector space vertex vertices Vx G zero σεσ