Tensor Spaces - Seminar Resources

The material below presents the basic theory of tensor products over vector spaces at the level of beginning graduate students. This material was used by me in seminars in combinatorics (UCSD, Mathematics and CSE) where tensor spaces were used. W1 contains the most basic material and was presented first, followed by M2 (plus a review of M1 which is an alternative to W1). The remaining articles (M3, M4, M5, M6) are more specialized, useful for topics in algebraic combinatorics.*

Tensor spaces - the basics

W1: This material defines tensor products of vector spaces, develops the critical notational conventions for this subject, and contains the discrete mathematics background material. Section topics: (1) Introduction; (2) Basic multilinear algebra; (3)Tensor products of vector spaces; (4)Tensor products of matrices; (5) Inner products on tensor spaces; (6) Direct sums and tensor products; (7) Basic concepts and notation. (Seminar notes by S. Gill Williamson)

Tensor Spaces

M1: (1) Multilinear functions, (2) Free spaces, factor spaces, tensor products; (3) Properties of tensor spaces; (4) Contraction, extension, inner product. (Seminar notes by M. Marcus)

Tensor Transformations M2 Sections 1, 2

M2: (1) Tensor Products of Transformations; (2) Properties of Mappings on Tensor Spaces; (3) Symmetry Classes; (4) Induced Transformations. (Seminar notes by M. Marcus)

Tensor Algebras

M3: (1) The Mixed Graded Tensor Algebra; (2) Derivations. (Seminar notes by M. Marcus)

Exterior/Grassmann Algebras

M4: (1) Decomposibility; (2) Duality in Exterior Algebras; (3) Transformations on Grassmann Algebras. (Seminar notes by M. Marcus)

Clifford Algebras

M5: (1) Compatible Algebras; (2) The Structure of Clifford Algebras; (3) Orthogonal Groups. (Seminar notes by M. Marcus)

Representation Theory

M6: (1) Rational Representations; (2) The Regular Representation; (3) The Symmetric Group.(Seminar notes by M. Marcus)

* The seminar notes by Professor Marcus (my thesis advisor) were provided to me as they were developed. M1, M2, M3 were combined by the author and published by Marcel Dekker in book form as Finite Dimensional Linear Algebra, Part I, 1973 (out of print). M4, M5, M6 were expanded and became part of Finite Dimensional Linear Algebra, Part II, 1975, Marcel Dekker (out of print). The individual chapter indices to these seminar notes were provided by me for the convenience of my students.

Corrigenda (by Dr. Tony Trojanowski):

M3 Corrigenda M2 Corrigenda

Linear Algebra Review: A review of linear algebra concepts was found to be very helpful in getting students "on the same page" for the seminar on tensor spaces. The following handout evolved from this need:

Matrix Canonical Forms